$m=n$ and $x_{r}=x_{r}^{/}$. We will write $\sum^{*}x_{r}$ to denote the
formal sum $\sum_{r=1}^{N}x_{i}$ (recall $N$ is fixed).

C2:

From an element $x_{X},$we can construct the pull back function (or
functional) $x_{(XY)Y}$ of $x_{X}$ by $x\phi\coloneqq\phi x$ for each
$\phi_{XY}$.

C3:

From a bifunction $f_{(X\times Y)Z}$ we can construct the two curried
functions $f_{X(YZ)}^{c1}\coloneqq x\mapsto(y\mapsto f(x,y))$ so that
$\left(fx\right)(y)=f(x,y)$, and $f_{Y(XZ)}^{c2}\coloneqq y\mapsto(x\mapsto
f(x,y))$ so that $\left(fy\right)(x)=f(x,y)$.

Let us equip $Y$ with a binary operator $+_{Y}$, ie an element of the set of
bifunctions $(Y\times Y)Y$, so that $Y=(Y,+_{Y})$ is a semi-group. Recall we
can always add a unit if necessary to make a semi-group into a monoid, so we
assume $Y$ is a monoid.

C4:

From $+_{Y}$ we construct the pull back (binary) operator
$\left(+_{Y}\right)_{XY}$ on $XY$ by
$(\phi_{1}+_{Y}\phi_{2})x\coloneqq\phi_{1}x+_{Y}\phi_{2}x$. We call
$(XY,+_{Y})$ the pull back monoid. We write $\sum^{Y}\phi_{r}$ to denote a sum
in $XY$ using the pull back $+_{Y}$.

Note that we now have two monoids constructed on $XY$, namely the free monoid
$\left((XY)^{*},+_{(XY)^{*}}\right)$ and the pull back monoid $(XY,+_{XY})$.
The general element of $(XY)^{*}$ has the form
$\left(\sum^{*}\phi_{r}\right)_{(XY)^{*}}$ and
$\left(\sum^{Y}\phi_{r}\right)_{XY}$ respectively.

C5:

Given the two monoid constructions $(XY)^{*}$ and $XY$ on the underlying set
$XY$, we construct the natural homomorphism $\theta_{(XY)^{*}(XY)}$ by
$\theta\left(\sum^{*}\phi_{r}\right)\coloneqq\sum^{Y}\phi_{r}$ which is
surjective but not generally injective. This also allows us to define
$\left(\sum^{*}\phi_{r}\right)x\coloneqq\left(\sum^{Y}\phi_{r}\right)x=\sum\left(\phi_{r}x\right)$.
Important case is when $Y=W^{*}$, ie $Y$ is already a free monoid. Then the
natural homomorphism is
$\left(\sum^{*}\phi_{r}\right)_{(XY)^{*}}\mapsto\sum^{*}\phi_{r}$ which is now
an isomorphism, ie $(XW^{*})^{*}\cong XW^{*}$

C6:

Finally, given a set function $\phi_{XY}$ we construct the monoid homomorphism
$\phi_{X^{*}Y}\coloneqq\left(\sum^{*}x_{i}\right)\mapsto\sum(\phi x_{i})$.
Note that this is well defined since $X^{*}$ is free. If this were not the
case $\phi_{X^{*}Y}$ may be multi-valued (ie no longer strictly a function).

######  Theorem 38.

Let $X,Y$ be sets and $Z=(Z,+_{Z})$ a monoid, then for any set of bifunctions
$f_{r}$ in $(X\times Y)Z$ we have
$\left(\sum^{*}f_{r}\right)(x,y)=\left((\sum^{*}f_{r}^{c1})(x)\right)y=x\left((\sum^{*}f_{r}^{c2})(y)\right)$

###### Proof.

$\left(\sum^{*}f_{r}\right)(x,y)=_{C5}\left(\sum^{Z}f_{r}\right)(x,y)=_{C4}\sum
f_{r}(x,y)=_{C3}\sum\left((f_{r}^{c2}y)(x)\right)=_{C2}\sum\left(x(f_{r}^{c2}y))\right)=_{C5}x\left(\sum^{*}(f_{r}^{c2}y)\right)=_{C5}x\left((\sum^{*}f_{r}^{c2})y\right)$
∎

######  Corollary 39.

Given a Dynamical System $(X,T_{XX})$ with a Value System $(V,+_{V})$, we have
$S_{N}(\phi,x)=(S_{N}\phi)x=\phi(S_{N}x)$

###### Proof.

From $T$ we construct $T_{(\Phi\times X)V}(\phi,x)\coloneqq\phi(Tx)$. From the
theorem we get
$\left(\sum^{*}T_{r}\right)(\phi,x)=_{Thm}\phi\left((\sum^{*}T_{r}^{c2})(x)\right)=_{C6}\sum\phi\left(T_{r}^{c_{2}}x\right)$.
Now
$\phi\left(T_{r}^{c_{2}}x\right)=_{C2}\left(T_{r}^{c_{2}}x\right)\phi=_{C3}T_{r}(\phi,x)=_{\textrm{def}}\phi(Tx)$
and so
$\sum\phi\left(T_{r}^{c_{2}}x\right)=\sum\phi(T_{r}x)=_{C6}\phi\left(\sum^{*}(T_{r}x)\right)=_{C5}\phi\left((\sum^{*}T_{r})x)\right)$.

Now put $T_{r}=T^{r}$ and $S_{N}=\sum^{*}T_{XX}^{r}$,
$S_{N}=\sum^{*}(T^{r})_{(\Phi\times X)V}.$

Alternate form gives $S_{N}(\phi,x)=(S_{N}\phi)x$ where $S_{N}$ is
$\Phi\Phi^{*}$ but under $\sim$ this becomes $\Phi\Phi$ , ie $S_{N}\phi$ can
be regarded simply as another observable. ∎

###  4.3. Proof by Ansatz

Define $S_{N}x\coloneqq\sum_{r=1}^{N}T^{r}x$ as a formal sum, and
$\phi(\sum_{r=1}^{N}x_{r})\coloneqq\sum_{r=1}^{N}\phi(x_{r})$ then
$\phi(S_{N}x)=\sum_{r=1}^{N}\phi(T^{r}x)=S_{N}(\phi,x)$

##  5\. Distribution of orbit segments on the circle

Given a Dynamical System $(X,T)$ we now develop the theory of orbit segments
$S_{N}x\coloneqq\sum_{r=1}^{N}T^{r}x$ which we introduced in the previous
section, but specifically for the case where $T$ is a rotation $R_{\alpha}$ of
the circle through $\alpha_{\mathbb{R}}$ revolutions. In this case for a
circle point $x_{\mathbb{T}}$ we have
$S_{N}x=\sum_{r=1}^{N}T^{r}x=\left(T^{r}x\right)_{r=1}^{N}=\left(x+_{\mathbb{T}}r\alpha\right)_{r=1}^{N}$
(where $+_{\mathbb{T}}$ is addition on the circle). Note that we will find it
useful to be able to switch between the formal sum notation
$\sum_{r=1}^{N}T^{r}x$ and the sequence notation
$\left(T^{r}x\right)_{r=1}^{N}=\left(x+_{\mathbb{T}}r\alpha\right)_{r=1}^{N}$.

The distribution of the orbit
$(r\alpha)_{r=1}^{N}=(\alpha,2\alpha,3\alpha\ldots,N\alpha)$ is well
understood classically when considered simply as a set, and is a primary
example of equidistribution. In this section we instead study the distribution
of the orbit as a sequence. This introduces some additional and quite elegant
structure. In later sections we shall exploit this structure in deriving
estimates for our anergodic Birkhoff sums. However the results of this section
seem of interest in their own right.

###  5.1. QuasiPeriod Decomposition

Introduction/Motivation Intuitively if $\alpha$ and $\beta$ lie “close” to
each other, then the orbits of a circle point $x$ under the rotations
$R_{\alpha}$ and $R_{\beta}$ will continue to lie close to one another over
suitably short initial orbit segments. We make this notion precise. If
$\frac{p_{r}}{q_{r}}$ is a convergent of $\alpha$, then we know (see 2.1) that
$\alpha-\frac{p_{r}}{q_{r}}=\frac{(-1)^{r}}{q_{r}q_{r+1}^{/}}$, and so the
$t-$th points of the orbits of $x$ under $R_{\alpha},R_{\frac{p_{r}}{q_{r}}}$
are separated from each other by a distance of
$\left\|t\alpha-t\frac{p_{r}}{q_{r}}\right\|=\left\|\frac{t}{q_{r}q_{r+1}^{/}}\right\|$.
Further for $1\leq t\leq q_{r}$ we have
$0<(-1)^{r}(t\alpha-\frac{tp_{r}}{q_{r}})=\frac{t}{q_{r}q_{r+1}^{/}}\leq\frac{1}{q_{r+1}^{/}}<\frac{1}{q_{r}}$.
But since the sequence $(tp_{r})_{t=1}^{q_{r}}$ is just a permutation $\bmod
q_{r}$ of $(u)_{u=1}^{q_{r}}$, this tells us that the orbital points
$\\{t\alpha\\}_{t=1}^{q_{r}}$ are ’pigeon-holed’ by the circle partition
defined by the points $(\frac{u}{q_{r}})_{u=1}^{q_{r}}$: in other words, for
each $1\leq u\leq q_{r}$ the partition interval
$(\frac{u-1}{q_{r}},\frac{u}{q_{r}})$ contains precisely one orbital point
$t\alpha$. Recall that each $q_{r}$ we call a quasiperiod of $\alpha$. The
preceding discussion now motivates decomposing the initial orbit segment of
length $N$ into sub-segments of quasiperiod length. A sub-segment of length
$q_{r}$ we will then relate (using the observations above) to rational
(periodic) orbits of $q_{r}$. The Ostrowski representation (see 2.5) will be
used to give us precisely this decomposition.

What we have said so far is a classical strategy developed by Koksma(##paper)
and further developed by Herman(##paper). However we now introduce two new
elements:

  1. (1)

Previously the approach was used directly in the decomposition of the Birkhoff
sum under study. However our strategy is to decouple the Birkhoff sum itself
from the underlying dynamics of the irrational rotation. This is an important
conceptual distinction which bears fruit later. It comes at the expense of
slightly more theory and notation, in order to describe the decomposition of
orbits rather than scalar sums.

  2. (2)

Previously the approach required the sum-function in the Birkhoff sum to be of
bounded variation. This is because the periodic orbits could be allowed to
float to begin at the start of a quasiperiod segment. In order to study
unbounded functions, we are forced instead to use periodic orbits anchored to
the same fixed initial point. This requires more analysis, but also results in
a deeper understanding of the distribution of the points of an initial orbit
segment.

Ostrowski Decomposition of Orbit Segments Given $\alpha$, the Ostrowski
Representation (see 2.5) $OR_{\alpha}(N)$ gives us a canonical way of
representing an integer $N\geq 0$ as a sum of quasiperiods of $\alpha$, which
we write $N=\sum_{r=0}^{n}b_{r}q_{r}$. With a little care over notation we can
use this to induce a corresponding canonical decomposition of an orbit segment
of length $N$ into segments of quasiperiod length.

We first introduce a way of representing other integers $M$ in terms of the
Ostrowski representation of $N$:

######  Definition 40 (Ostrowski Triples).

Let $\alpha,N$ be fixed, and $N$ has Ostrowski Representation
$OR_{\alpha}(N)=\sum_{r=0}^{n}b_{r}q_{r}$ wrt $\alpha$. We say
$(r,s,t)_{\mathbb{Z}^{3}}$ is an Ostrowski triple representing the integer
$M_{\mathbb{Z}}$ (with respect to $\alpha,N$) if
$M=\sum_{u=r+1}^{n}b_{u}q_{u}+sq_{r}+t$. For convenience we will use the
homonymous formal notation $rst$ to represent both the triple $(r,s,t)$ and
the represented integer $M$ so that we can write
$M=rst=\sum_{u=r+1}^{n}b_{u}q_{u}+sq_{r}+t$.

Note from the definition that
$rst=rs0+t=r00+sq_{r}+t=\sum_{u=r+1}^{n}b_{u}q_{u}+sq_{r}+t$. In particular
$n00=0$, $n01=1$, $(-1)00=N$, and
$000=\sum_{u=1}^{n}b_{u}q_{u}=N-b_{0}q_{0}=N-b_{0}$. For $0\leq r\leq n-1$,
$r00=b_{n}q_{n}+b_{n-1}q_{n-1}\ldots+b_{r+1}q_{r+1}$ so that that $rst$
_increases_ as $r$ _decreases_ from $n$ to $-1$. Note also that $M$ may in
general be represented by many Ostrowski triples with respect to $N,\alpha$
but we can define a distinguished representation as follows.

######  Definition 41.

For $1\leq M\leq N$ we define _the_ Ostrowski triple of $M$ (wrt $N,\alpha)$
to be the unique Ostrowski triple representing $M$ defined by
$r=\min\\{k:k00<M\\}$, $s=\left\lfloor\frac{M-1-r00}{q_{r}}\right\rfloor$,
$t=M-rs0$. For $M=1..N$ we call the Ostrowski triple of $M$ a canonical triple
of $(\alpha,N)$.

Note that if $rst$ is a canonical triple then from the definition $sq_{r}\leq
M-1-r00$ giving $t\geq 1$. In particular it is easy to see then that for
canonical triples we have $0\leq r\leq n,0\leq s\leq b_{r}-1<a_{r+1},1\leq
t\leq q_{r}$, and and $1\leq rst\leq N$. $N$ itself is represented by the
canonical triple $=0(b_{n}-1)q_{n}$ , and $rs0$ is _never_ a canonical triple
(since $t\geq 1$).

We are now ready to decompose the orbit segment $S_{N}(x_{0})$ into segments
of quasiperiod length. If $M=rst$ we have the equivalent notations
$x_{M}=T^{M}x_{0}=T^{rst}x_{0}=x_{rst}$.

######  Definition 42 (Ostrowski Decomposition).

Given $\alpha$ irrational, $N\geq 0$ and a Dynamical system $(X,T)$, we define
the Ostrowski decomposition of the orbit segment
$S_{N}(x_{0})=\sum_{u=1}^{N}T^{u}x_{0}=\sum_{u=1}^{N}x_{u}$ to be the reverse
formal sum
$S_{N}(x_{0})=\underleftarrow{\sum_{r=0}^{n}}S_{b_{r}q_{r}}(x_{r00})$. We
further decompose each $S_{b_{r}q_{r}}(y)$ into $b_{r}$ segments of length
$q_{r}$ to obtain the Extended Ostrowski Decomposition
$S_{N}(x_{0})=\underleftarrow{\sum_{r=0}^{n}}\left(\sum_{s=0}^{b_{r}-1}S_{q_{r}}(x_{rs0})\right)$.

We can write this more explicitly (recalling $n00=0)$ as

$\displaystyle S_{N}(x_{0})$
$\displaystyle=S_{b_{n}q_{n}}(x_{0})+S_{b_{n-1}q_{n-1}}(x_{b_{n}q_{n}})+S_{b_{n-2}q_{n-2}}(x_{b_{n}q_{n}+b_{n-1}q_{n-1}})+\ldots
S_{b_{0}q_{0}}(x_{N-b_{0}q_{0}})$
$\displaystyle=S_{q_{n}}(x_{n00})+S_{q_{n}}(x_{n10})+S_{q_{n}}(x_{n20})\ldots+S_{q_{n}}(x_{n(b_{n}-1)0})$
$\displaystyle+S_{q_{n-1}}(x_{(n-1)00})+S_{q_{n-1}}(x_{(n-1)10})+S_{q_{n-1}}(x_{(n-1)20})\ldots+S_{q_{n-1}}(x_{(n-1)(b_{n-1}-1)0})$
$\displaystyle\vdots$
$\displaystyle+S_{q_{0}}(x_{000})+S_{q_{0}}(x_{010})+S_{q_{0}}(x_{020})\ldots+S_{q_{0}}(x_{0(b_{n}-1)0})$

Note that if $b_{r}=0$ the corresponding line is empty.

Also note that $0s0=(N-b_{0}q_{0})+s+0$ and $q_{0}=1$ so the last line
simplifies to
$x_{001}+x_{011}+\ldots+x_{0(b_{n}-1)1}=x_{(N-b_{0})+1}+x_{(N-b_{0})+2}+\ldots+x_{N}$.

###  5.2. Application to irrational rotations

When $T$ is an irrational rotation $R_{\alpha}$ of the circle we have
$T^{M}x_{0}=\left\\{x_{0}+M\alpha\right\\}$ so that if $M=rst$ we have
$x_{rst}=\left\\{x_{0}+(rst)\alpha\right\\}$. In particular we will write
$\alpha_{rst}\coloneqq\left\\{(rst)\alpha\right\\}$ and then the Ostrowski
Decomposition is
$S_{N}(x_{0})=\underleftarrow{\sum_{r=0}^{n}}\sum_{s=0}^{b_{r}-1}S_{q_{r}}(x_{0}+\alpha_{rs0})$
and $S_{q_{r}}(x_{0}+\alpha_{rs0})$ is the sequence of points
$\left(\left\\{x_{0}+\alpha_{rst}\right\\}\right)_{t=1}^{q_{r}}$. We are now
in a position to extend our discussion to each orbit segment $S_{q_{r}}$ in
the Ostrowski decomposition. In section (5.1) we introduced the notion of
tracking quasiperiod segments of the orbit under $R_{\alpha}$ by means of
quasiperiod partitions. We introduce a quantity which will help us measure the
precision of this tracking behaviour.

######  Definition 43 (Tracking error).

We define the tracking error $(\epsilon_{rst})_{\mathbb{R}}$of $\alpha_{rst}$
to be the signed real number

$\epsilon_{rst}=(-1)^{r}\alpha_{r00}+\left(\frac{s}{q_{r+1}^{/}}+\frac{t}{q_{r}q_{r+1}^{/}}\right)$

We shall see later that the tracking error $\epsilon_{rst}$ is the signed
smallest distance of the circle point $\alpha_{rst}$ from the circle point
$t\text{$\frac{p_{r}}{q_{r}}$}$. Our goal is to develop bounds to show that
this error is always suitably small, although at this point we cannot even
rule out $\left|\epsilon_{rst}\right|>1$. However we can observe immediately
that for $r=n$ we have $\alpha_{n0t}=t\alpha$ and so our earlier discussion
(5.1) tells us that for $t\in 1..q_{n}$ we have
$0<\epsilon_{n0t}=(-1)^{n}\left(t\alpha-t\frac{p_{n}}{q_{n}}\right)\leq\frac{1}{q_{n+1}^{/}}<\frac{1}{a_{n+1}q_{n}}$.
.

The general case becomes more complex as we have to take into account the
effects of quasiperiod segments which do not start from the origin. We first
show that we can in fact recover $\alpha_{rst}$ from the definition of
$\epsilon_{rst}$, via the following simple result:

######  Proposition 44.

$\alpha_{rst}=\left\\{t\frac{p_{r}}{q_{r}}+(-1)^{r}\epsilon_{rst}\right\\}$

###### Proof.

By definition
$\alpha_{rst}=\left\\{\alpha_{r00}+(sq_{r}+t)\alpha\right\\}=\alpha_{r00}+(sq_{r}+t)\alpha+M$
for some integer $M$. But
$\alpha=\frac{p_{r}}{q_{r}}+\frac{(-1)^{r}}{q_{r}q_{r+1}^{/}}$ so that
$\alpha_{rst}=\alpha_{r00}+(-1)^{r}\left(\frac{s}{q_{r+1}^{/}}+\frac{t}{q_{r}q_{r+1}^{/}}\right)+t\frac{p_{r}}{q_{r}}+M=(-1)^{r}\epsilon_{rst}+t\frac{p_{r}}{q_{r}}+M$
and the result follows since $\alpha_{rst}=\left\\{\alpha_{rst}\right\\}$. ∎

We now investigate the two components of $\epsilon_{rst}$ separately, namely
$(-1)^{r}\alpha_{r00}$ and $\frac{s}{q_{r+1}^{/}}+\frac{t}{q_{r}q_{r+1}^{/}}$.
We start with the latter as it is the simpler.

######  Lemma 45.

For canonical $rst$ we have
$0<\frac{s}{q_{r+1}^{/}}+\frac{t}{q_{r}q_{r+1}^{/}}\leq\frac{s+1}{q_{r+1}^{/}}<\frac{1}{q_{r}^{/}}$

###### Proof.

For canonical $rst$ we have $0\leq s\leq b_{r}-1$and $1\leq t\leq q_{r}$ so
that
$0<\frac{s}{q_{r+1}^{/}}+\frac{t}{q_{r}q_{r+1}^{/}}\leq\frac{s+1}{q_{r+1}^{/}}\leq\frac{b_{r}}{q_{r+1}^{/}}\leq\frac{a_{r+1}}{q_{r+1}^{/}}<\frac{a_{r+1}^{/}}{q_{r+1}^{/}}=\frac{1}{q_{r}^{/}}$
∎

We now turn to investigate the term $\alpha_{r00}$. Recall the definition
$\alpha_{r00}=\left\\{\sum_{u=r+1}^{n}b_{u}q_{u}\alpha\right\\}$. We might
expect this sum to be capable of taking a good range of values within $[0,1)$
but fortunately for our purposes it turns out to be surprisingly constrained,
and this rigidity is the foundational result of this section.

######  Lemma 46.

If $b_{r}\neq 0$ then $\left\|\alpha_{r00}\right\|<\frac{1}{q_{r+1}^{/}}$ and
more precisely
$\frac{-1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}<(-1)^{r}\left\\{\left\\{\alpha_{r00}\right\\}\right\\}<\frac{1}{q_{r+2}^{/}}$

###### Proof.

For $0\leq r\leq n$ we have
$\alpha_{r00}=\sum_{u=r+1}^{n}b_{u}q_{u}\alpha=\sum_{u=r+1}^{n}b_{u}p_{u}+\sum_{u=r+1}^{n}\frac{\left(E_{u}-O_{u}\right)b_{u}}{q_{u+1}^{/}}$.
Hence

(5.1)
$\left\\{\left\\{\alpha_{r00}\right\\}\right\\}=\left\\{\left\\{\sum_{u=r+1}^{n}\frac{E_{u}b_{u}}{q_{u+1}^{/}}-\sum_{u=r+1}^{n}\frac{O_{u}b_{u}}{q_{u+1}^{/}}\right\\}\right\\}$

We now estimate the even and odd sums on the right side. We will primarily use
the fact that $b_{u}\leq a_{u+1}$ but with an important refinement. Recall
(Lemma 18) that if $b_{r}\neq 0$ then $b_{r+1}<a_{r+2}$ and so we have
$0\leq\sum_{u=r+1}^{n}\frac{E_{u}b_{u}}{q_{u+1}^{/}}\leq\left(\sum_{u=r+1}^{n}\frac{E_{u}a_{u+1}}{q_{u+1}^{/}}\right)-\frac{E_{r+1}}{q_{r+2}^{/}}=\left(\sum_{u=r+1+E_{r}}^{n-O_{n}}\frac{E_{u}a_{u+1}}{q_{u+1}^{/}}\right)-\frac{O_{r}}{q_{r+2}^{/}}$,
and
$0\leq\sum_{u=r+1}^{n}\frac{O_{u}b_{u}}{q_{u+1}^{/}}\leq\left(\sum_{u=r+1}^{n}\frac{O_{u}a_{u+1}}{q_{u+1}^{/}}\right)-\frac{O_{r+1}}{q_{r+2}^{/}}=\left(\sum_{u=r+1+O_{r}}^{n-E_{n}}\frac{O_{u}a_{u+1}}{q_{u+1}^{/}}\right)-\frac{E_{r}}{q_{r+2}^{/}}$.
We now estimate the sums over $a_{u}$. Since
$\frac{a_{u+1}}{q_{u+1}^{/}}<\frac{1}{q_{u}^{/}}$, and considering the parity
of $r,n$ we can refine the sums to:

(5.2)
$\sum_{u=r+1}^{n}\frac{E_{u}a_{u+1}}{q_{u+1}^{/}}<\sum_{u=r+1+E_{r}}^{n-O_{n}}\frac{E_{u}}{q_{u}^{/}}<\frac{2}{q_{r+1+E_{r}}^{/}}$

(5.3)
$\sum_{u=r+1}^{n}\frac{O_{u}a_{u+1}}{q_{u+1}^{/}}<\sum_{u=r+1+O_{r}}^{n-E_{n}}\frac{O_{u}}{q_{u}^{/}}<\frac{2}{q_{r+1+O_{r}}^{/}}$

Now $r+1+E_{r}\geq 2$ and $q_{2}^{/}>2$ and so
$\sum_{u=r+1}^{n}\frac{E_{u}a_{u+1}}{q_{u+1}^{/}}=\left\\{\sum_{u=r+1}^{n}\frac{E_{u}a_{u+1}}{q_{u+1}^{/}}\right\\}$.
Similarly $r+1+O_{r}\geq 2$ for $r\geq 1$ and so
$\sum_{u=r+1}^{n}\frac{O_{u}a_{u+1}}{q_{u+1}^{/}}=\left\\{\sum_{u=r+1}^{n}\frac{O_{u}a_{u+1}}{q_{u+1}^{/}}\right\\}$
unless $r=0$ and $q_{1}^{/}<2$. But in the latter case $q_{1}=1$ and hence
$\alpha>\frac{1}{2}$ which means $b_{0}=0$. Hence
$\sum_{u=r+1}^{n}\frac{O_{u}a_{u+1}}{q_{u+1}^{/}}=\left\\{\sum_{u=r+1}^{n}\frac{O_{u}a_{u+1}}{q_{u+1}^{/}}\right\\}$
whenever $b_{r}\neq 0$. We now know both these sums are less than $1$.

Now recall $\alpha=\frac{p_{u}}{q_{u}}+\frac{(-1)^{u}}{q_{u}q_{u+1}^{/}}$ so
that $\frac{1}{q_{u+1}^{/}}=(-1)^{u}\left(q_{u}\alpha-p_{u}\right)$. Hence
$\frac{a_{u+1}}{q_{u+1}^{/}}=(-1)^{u}a_{u+1}\left(q_{u}\alpha-
p_{u}\right)=(-1)^{u}\left(q_{u+1}-q_{u-1}\right)\alpha-(-1)^{u}a_{u+1}p_{u}$.
Summing over $u$ even gives the telescoping sum:

$\sum_{u=r+1}^{n}\frac{E_{u}a_{u+1}}{q_{u+1}^{/}}=\left\\{\sum_{u=r+1}^{n}\frac{E_{u}a_{u+1}}{q_{u+1}^{/}}\right\\}=\left\\{\sum_{u=r+1+O_{r+1}}^{n-O_{n}}E_{u}\left(q_{u+1}-q_{u-1}\right)\alpha\right\\}=\left\\{\left(q_{n+1-O_{n}}-q_{r+O_{r+1}}\right)\alpha\right\\}=\left\\{-\frac{1}{q_{n+2-O_{n}}^{/}}+\frac{1}{q_{r+1+O_{r+1}}^{/}}\right\\}<\frac{1}{q_{r+1+E_{r}}^{/}}$

Again using $r+1+E_{r}\geq 2$ gives
$\frac{1}{q_{r+1+E_{r}}^{/}}<\frac{1}{2}$and so
$0\leq\sum_{u=r+1}^{n}\frac{E_{u}b_{u}}{q_{u+1}^{/}}<\frac{1}{q_{r+1+E_{r}}^{/}}-\frac{E_{r+1}}{q_{r+2}^{/}}<\frac{1}{2}$.
Following the same argument for $u$ odd gives
$0\leq\sum_{u=r+1}^{n}\frac{O_{u}b_{u}}{q_{u+1}^{/}}<\frac{1}{q_{r+1+O_{r}}^{/}}-\frac{O_{r+1+O_{r}}}{q_{r+2}^{/}}<\frac{1}{2}$
but again with the proviso that $b_{r}\neq 0$

We can now use these results in (5.1) to obtain (for $b_{r}\neq 0$):

$-\frac{1}{2}<\left(-\frac{1}{q_{r+1+O_{r}}^{/}}+\frac{E_{r}}{q_{r+2}^{/}}\right)<\left(-\sum_{u=r+1}^{n}\frac{O_{u}b_{u}}{q_{u+1}^{/}}\right)<\,\,\left\\{\left\\{\alpha_{r00}\right\\}\right\\}\,\,<\left(\sum_{u=r+1}^{n}\frac{E_{u}b_{u}}{q_{u+1}^{/}}\right)<\left(\frac{1}{q_{r+1+E_{r}}^{/}}-\frac{O_{r}}{q_{r+2}^{/}}\right)<\frac{1}{2}$

It is easily checked that this is a restatement of the theorem result. ∎

We now combine the previous lemmas of this section to obtain:

######  Lemma 47.

For $rst$ canonical, $\epsilon_{rst}$ has lower and upper bounds
$\epsilon_{rst}^{L}$ and
$\epsilon_{rst}^{U}=\epsilon_{rst}^{L}+\frac{1}{q_{r+1}^{/}}$ respectively,
satisfying:

$-\frac{1}{q_{r+1}^{/}}<\left(\frac{s-1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)<\left(\frac{(s-1)q_{r}+t}{q_{r}q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)=\epsilon_{rst}^{L}<\epsilon_{rst}<\epsilon_{rst}^{U}=\left(\frac{sq_{r}+t}{q_{r}q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)\leq\left(\frac{s+1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)\leq\frac{1}{q_{r}^{/}}$

.

In particular $\left|\epsilon_{rst}\right|<\frac{1}{q_{r}^{/}}$ and if $s\neq
0$ $\epsilon_{rst}>0$. Further for $r=n$
$\epsilon_{rst}=\frac{sq_{r}+t}{q_{r}q_{r+1}^{/}}>0$, and for
$r<n,s=0,t=q_{r}$ $\epsilon_{rst}>\frac{1}{q_{r+2}^{/}}>0$

###### Proof.

Using $0\leq s\leq b_{r}-1,1\leq t\leq q_{r}$ for $rst$ canonical gives
$0<\frac{sq_{r}+t}{q_{r}q_{r+1}^{/}}\leq\frac{s+1}{q_{r+1}^{/}}$. Further
$\frac{1}{q_{r+2}^{/}}+\frac{s+1}{q_{r+1}^{/}}\leq\frac{1}{q_{r+2}^{/}}+\frac{a_{r+1}}{q_{r+1}^{/}}=\frac{1}{q_{r}^{/}}$.
Hence $\frac{-1}{q_{r+1}^{/}}<\epsilon<\frac{1}{q_{r}^{/}}$ and for $q_{r}\geq
2$ this means $\frac{-1}{2}<\epsilon<\frac{1}{2}$ so that
$\epsilon_{rst}=\\{\\{\epsilon_{rst}\\}\\}$. The various inequalities of the
first result then follow and we turn to the remaining results. For $r=n$ we
have $\alpha_{n00}=0$ and so
$\epsilon_{rst}=\frac{sq_{r}+t}{q_{r}q_{r+1}^{/}}>0$. For $s\neq 0$,
$\epsilon_{rst}>\frac{s-1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}>0$. If $s=0$ but
$t=q_{r}$,
$\epsilon_{rst}>\frac{(s-1)q_{r}+t}{q_{r}q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}=\frac{1}{q_{r+2}^{/}}>0$.

∎

The results can be packaged into one simple and quite elegant result which
includes all of the cases above:

######  Corollary 48 (Parity Duality).

_Writing
$l_{rst}=\left\\{(-1)^{r}\frac{tp_{r}}{q_{r}}\right\\}+\epsilon_{rst}^{L}$ and
$u_{rst}=\left\\{(-1)^{r}\frac{tp_{r}}{q_{r}}\right\\}+\epsilon_{rst}^{U}$,
then $0<\frac{1}{q_{r+2}^{/}}<l_{rst}=u_{rst}-\frac{1}{q_{r+1}^{/}}$ and
$u_{rst}\leq 1-\frac{1}{q_{r}}+\frac{1}{q_{r}^{/}}<1$ ($r>0)$ and $u_{rst}\leq
1-\frac{1}{q_{r+1}^{/}}<1$ ($r=0$). In addition: _

__

$\displaystyle l_{rst}$ $\displaystyle<\text{$\alpha_{rst}$$<$$u_{rst}$}\qquad
r\,\mathrm{even}$ $\displaystyle l_{rst}$
$\displaystyle<\text{$\overline{\alpha_{rst}}$$<$$u_{rst}$ }\qquad
r\,\mathrm{odd}$

###### Proof.

Note that $\left\\{(-1)^{r}\frac{tp_{r}}{q_{r}}\right\\}\leq
1-\frac{1}{q_{r}}$ and by (47) $\epsilon_{rst}^{U}\leq\frac{1}{q_{r}^{/}}$
giving $u_{rst}\leq 1-\frac{1}{q_{r}}+\frac{1}{q_{r}^{/}}$. But
$\frac{1}{q_{r}^{/}}<\frac{1}{q_{r}}$ for $r>0$ and then $u_{rst}<1$.

For $r=0$ we have $\left\\{(-1)^{r}\frac{tp_{r}}{q_{r}}\right\\}=0$ but we now
have $q_{0}^{/}=q_{0}=1$ which does not achieve the result. However also from
(47) if $r=0$
$\epsilon_{0st}^{U}\leq\frac{1}{q_{0}^{/}}-\frac{1}{q_{1}^{/}}=1-\frac{1}{q_{1}^{/}}$,
and $q_{1}^{/}>1$ so that $u_{0st}<1$ and so $u_{rst}<1$ for any $r\geq 0$. We
now consider $l_{rst}$.

If $\left\\{\frac{tp_{r}}{q_{r}}\right\\}\neq 0$ then $q_{r}>1$ and we have
$\left\\{(-1)^{r}\frac{tp_{r}}{q_{r}}\right\\}\geq\frac{1}{q_{r}}$. From (47)
$\epsilon_{rst}^{U}>\frac{t}{q_{r}q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}$ and so
$l_{rst}>\frac{1}{q_{r}}+\frac{1}{q_{r+2}^{/}}-\frac{1}{q_{r+1}^{/}}>\frac{1}{q_{r+2}^{/}}>0$.

If $\left\\{\frac{tp_{r}}{q_{r}}\right\\}=0$ then
$l_{rst}=\epsilon_{rst}^{U}-\frac{1}{q_{r+1}^{/}}$ and either $q_{r}=1$ (and
then $t=1=q_{r}$) or $t=q_{r}$ \- in either case $t=q_{r}$. Then
$\epsilon_{rsq_{r}}^{U}>\frac{1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}$ and hence
$l_{rst}>\frac{1}{q_{r+2}^{/}}>0$.

Finally $\epsilon_{rst}^{L}=\epsilon_{rst}^{U}-\frac{1}{q_{r+1}^{/}}$ in (47),
which gives us $l_{rst}=u_{rst}-\frac{1}{q_{r+1}^{/}}$.

The first result follows, and we turn to consideration of the results
involving $\alpha_{rst}$.

Recall from (44)
$\alpha_{rst}=\left\\{\frac{tp_{r}}{q_{r}}+(-1)^{r}\epsilon_{rst}\right\\}$
and $\left|\epsilon_{rst}\right|<\frac{1}{q_{r}^{/}}$ and
$\epsilon_{rst}^{L}<\epsilon_{rst}<\epsilon_{rst}^{U}$

For $\left\\{\frac{tp_{r}}{q_{r}}\right\\}\neq 0$ (so that $q_{r}>1$)
$\left\\{\frac{tp_{r}}{q_{r}}+(-1)^{r}\epsilon_{rst}\right\\}=\left\\{\frac{tp_{r}}{q_{r}}\right\\}+(-1)^{r}\epsilon_{rst}$.

For $r$ even this gives
$\alpha_{rst}=\left\\{\frac{tp_{r}}{q_{r}}\right\\}+\epsilon_{rst}$.

If $r$ is odd then
$\overline{\alpha_{rst}}=1-\alpha_{rst}=1-\left(\left\\{\frac{tp_{r}}{q_{r}}\right\\}-\epsilon_{rst}\right)=\left\\{-\frac{tp_{r}}{q_{r}}\right\\}+\epsilon_{rst}$.

The results follow in both cases from
$\epsilon_{rst}^{L}<\epsilon_{rst}<\epsilon_{rst}^{U}$ and the definitions of
$u_{rst},l_{rst}$.

If $\left\\{\frac{tp_{r}}{q_{r}}\right\\}=0$ we have
$\left\\{\frac{tp_{r}}{q_{r}}+(-1)^{r}\epsilon_{rst}\right\\}=\left\\{(-1)^{r}\epsilon_{rst}\right\\}$.
We also have $u_{rst}=\epsilon_{rst}^{U},l_{rst}=\epsilon_{rst}^{L}$. Hence
$0<l_{rst}=\epsilon_{rst}^{L}<\epsilon_{rst}<\epsilon_{rst}^{U}=u_{rst}<1$
which also gives us $\left\\{\epsilon_{rst}\right\\}=\epsilon_{rst}$.

For $r$ even this gives
$\alpha_{rst}=\left\\{\epsilon_{rst}\right\\}=\epsilon_{rst}$.

If $r$ is odd then
$\overline{\alpha_{rst}}=1-\left\\{-\epsilon_{rst}\right\\}=\left\\{\epsilon_{rst}\right\\}=\epsilon_{rst}$.

The results follow directly. ∎

Note that we can rewrite the second inequality using $\overline{x}=1-x$ to
obtain the dual result $\overline{l_{rst}}>\alpha_{rst}>\overline{u_{rst}}$
for $r$ odd.

We have now developed a lot of understanding of where the point $\alpha_{rst}$
lies in the partition defined by the points $t\frac{p_{r}}{q_{r}}$, but there
are also some important special cases where we can go further (and which are
also important when we come to consider the application to anergodic Birkhoff
sums).

Corollary 48 tells us that $\alpha_{rst}$ always lies in one of the two
intervals of length $1/q_{r}$either side of $tp_{r}/q_{r}$, and in the
majority of cases ($s\neq 0$or $r=n$ or $t=q_{r}$) it lies in the primary
interval. With a little extra work we can identify the interval more
precisely. The partition interval of $\alpha_{rst}$

######  Definition 49.

Fixing $r$, the circle points $\frac{k}{q_{r}}$ ($k=1..q_{r}$) form a
partition defining $q_{r}$ distinct open intervals of length
$\frac{1}{q_{r}}$. For $q_{r}=1$ there is only such one point and interval,
but for $q_{r}>1$ each point is surrounded by two distinct intervals, one on
each side, which we will label $I_{k}$ and $-I_{k}$ (the positive and negative
intervals of the point $\frac{k}{q_{r}}$), which we will write generically as
$(-1)^{u}I_{k}$ for some $u_{\mathbb{Z}}$. We define $(-1)^{u}I_{k}$
specifically as the set of points
$\left\\{\frac{k\,+\,(-1)^{u}\nu}{q_{r}}\right\\}_{Set}$ for $\nu\in(0,1)$.

Note that in fact the latter definition also holds for $q_{r}=1$, but in this
case the two intervals coincide ($I_{k}=-I_{k}$).

Recall that circle points $\frac{k}{q_{r}},\frac{k+mq_{r}}{q_{r}}$ coincide
for any integer $m_{\mathbb{Z}}$, so that
$(-1)^{u}I_{k}=(-1)^{u}I_{k+mq_{r}}$.

The set of positive intervals and the set of negative intervals represent two
different ways of labelling the same underlying set of intervals which make up
the partition. There is therefore a natural bijection between the two sets of
labels. We can capture this explicitly as follows:

######  Lemma 50.

$(-1)^{u}I_{k}=(-1)^{u+1}I_{k+(-1)^{u}}$

###### Proof.

$(-1)^{u}I_{k}=\left\\{\frac{k\,+\,(-1)^{u}\nu}{q_{r}}\right\\}=\left\\{\frac{k+(-1)^{u}\,-\,(-1)^{u}(\nu-1)}{q_{r}}\right\\}=(-1)^{u+1}I_{k+(-1)^{u}}$.
∎

######  Lemma 51.

, $\alpha_{rst}\in(-1)^{r}\operatorname{sgn}(\epsilon_{rst})I_{k}$ where
$k=tp_{r}\bmod q_{r}$.

###### Proof.

This follows immediately from
$\alpha_{rst}=\left\\{t\frac{p_{r}}{q_{r}}+(-1)^{r}\epsilon_{rst}\right\\}$
and $\left|\epsilon_{rst}\right|<\frac{1}{q_{r}}$. ∎

######  Proposition 52.

For $r\geq 0$, $p_{r}=(-1)^{r+1}q_{r-1}^{-1}\bmod q_{r}$

###### Proof.

This follows immediately from the
identity$q_{r}p_{r-1}-p_{r}q_{r-1}=(-1)^{r}$for $r\geq 0$ ∎

######  Lemma 53.

For canonical $rst$, and any integer $u$, the points $\alpha_{rst}$ which lie
in $(-1)^{u}I_{k}$ are precisely those for which either (a)
$t=(-1)^{r+1}kq_{r-1}\bmod q_{r}$ and
$\operatorname{sgn}(\epsilon_{rst})=(-1)^{u+r}$, or (b)
$t=(-1)^{r+1}\left(k+(-1)^{u}\right)q_{r-1}\bmod q_{r}$ and
$\operatorname{sgn}(\epsilon_{rst})=(-1)^{u+r+1}$

###### Proof.

From previous 3 lemmas ∎

Special case - the intervals around the origin

######  Corollary 54.

Given $r$, the canonical triples for which $\alpha_{rst}$ lies in an interval
around the origin (ie $\pm I_{0}$) are given by:

  1. (1)

For $(-1)^{r}I_{0}$: $t=q_{r}$ (for any $s$) or $r<n,s=0,t=q_{r}-q_{r-1}$ and
$\operatorname{sgn}(\epsilon_{r0(q_{r}-q_{r-1})})=-1$). Then we get the lower
bound
$l_{r0(q_{r}-q_{r-1})}>\frac{1}{q_{r}}+\frac{-1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}>\frac{1}{2q_{r}}$.

  2. (2)

For $(-1)^{r+1}I_{0}$: $t=q_{r-1}\bmod q_{r}$ and
$\operatorname{sgn}(\epsilon_{rst})=+1$. Then we get the upper bound
$u_{rsq_{r-1}}<1-\frac{a_{r+1}^{/}-s}{q_{r+1}^{/}}$.

###### Proof.

For canonical $rst$, the points $\alpha_{rst}$ which lie in $(-1)^{u}I_{0}$
are precisely those for which either (a) $t=0\bmod q_{r}$ and
$\operatorname{sgn}(\epsilon_{rst})=(-1)^{u+r}$, or (b)
$t=(-1)^{r+1}\left((-1)^{u}\right)q_{r-1}\bmod q_{r}$ and
$\operatorname{sgn}(\epsilon_{rst})=(-1)^{u+r+1}$

(a) For $t=q_{r}$ we have from (47) that always
$\operatorname{sgn}(\epsilon_{rst})=+1$ and so $u+r=0\bmod 2$ and
$\alpha_{rsq_{r}}$ always lies in $(-1)^{r}I_{0}$.

(b) If $u=r\bmod 2$, the possibilities are
$\operatorname{sgn}(\epsilon_{rst})=-1$ and $t=q_{r}-q_{r-1}$ giving
$\alpha_{rst}$ in $(-1)^{r}I_{0}$. But in this case we can only have
$\operatorname{sgn}(\epsilon_{rst})=-1$ if $r<n,s=0$. Then
$l_{r0(q_{r}-q_{r-1})}>\frac{1}{q_{r}}+\left(\frac{-1}{q_{r+1}^{/}}+\frac{q_{r}-q_{r-1}}{q_{r}q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)>\frac{1}{2q_{r}}$.

If $u=r+1\bmod 2$, the possibilities are
$\operatorname{sgn}(\epsilon_{rst})=+1$ and $t=q_{r-1}$ giving $\alpha_{rst}$
in $(-1)^{r+1}I_{0}$. But in this case we can only have
$\operatorname{sgn}(\epsilon_{rst})\neq+1$ if $s=0$. And
$u_{rsq_{r-1}}=\left(1-\frac{1}{q_{r}}\right)+\left(\frac{sq_{r}+q_{r-1}}{q_{r}q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)<1-\frac{a_{r+1}^{/}-s}{q_{r+1}^{/}}$
by (48). ∎

__

##  6\. Analysis of spaces of unbounded observables

The analysis of spaces of unbounded (and non integrable) functions is not well
developed, and so in this section we lay some foundations suitable for our
purposes. The usual approaches to functional analysis via the theory of Banach
spaces and Schauder bases do not seem immediately helpful here. Instead,
noting that fundamentally we are interested in summing functions over large
orbits, we look for help from the elementary theory of integration. In
particular we find that extending the ideas of Bounded Variation and Jordan
Decomposition leads to a natural set of function spaces suitable for our use.

For precise definitions of circle terminology in this section, see 2.4.

###  6.1. Definitions and Basic Topology

Recall that a real observable on the circle is a (total) function
$\phi_{\mathbb{T}\mathbb{R}}$. Our first inclination may be to define an
unbounded observable as a function which can take values on the extended real
line $\mathbb{R}\bigcup\\{\pm\infty\\}$. However this is not necessary, and
instead it proves technically simpler to restrict the codomain of unbounded
observables to $\mathbb{R}$ for the following reasons:

  1. (1)

The sum $\sum_{i=1}^{n}\phi(x_{i})$ is always well defined (we avoid the
difficulty of defining $\infty-\infty$)

  2. (2)

The function $\sum_{i=1}^{n}\phi_{i}$ is always well defined (again we avoid
the difficulty of defining $\infty-\infty$)

  3. (3)

We can derive observables (total functions) $\phi_{\mathbb{T}\mathbb{R}}$ from
partial functions by using the simple convention that the resulting observable
vanishes outside the domain of the partial function. We will write
$\phi_{X\mathbb{R}}(x)=\psi_{Y\mathbb{R}}(x)$ for $Y\subseteq X$ to mean
$\phi$ coincides with $\psi$ on $Y$ and vanishes on $X$ outside $Y$.

######  Example 55.

The function $\phi_{\mathbb{I}\mathbb{R}}(x)=\frac{1}{x}$ on
$\mathbb{I}=(0,1)$ is a partial function on $\mathbb{T}=[0,1)$ whose domain is
$(0,1).$ It extends to the observable $\phi_{\mathbb{T}\mathbb{R}}$, a total
function on $[0,1)$ by $\phi_{\mathbb{T}\mathbb{R}}(0)\coloneqq 0$. The
observable $\phi$ is unbounded but defined on the whole of $[0,1)$, and has
codomain $\mathbb{\mathbb{R}}$ (there is no need to extend
$\mathbb{\mathbb{R}}$ with points at infinity). We allow ourselves to write
$\phi_{\mathbb{T}\mathbb{R}}(x)=\frac{1}{x}$ with the convention that
$\phi(x)=0$ where $\frac{1}{x}$ is undefined.

The following treatment of locally bounded variation is slightly non-standard
but again allows us to avoid some technical difficulties. We first extend the
definition of bounded variation to non-closed intervals:

######  Definition 56 (Bounded Variation on an Interval).

Let $J$ be any interval (not necessarily closed) of the real line or the
circle, and let $Part(J)$ the set of partitions of $J$ (ie ordered sequences
of the form $(x_{i})_{i=1}^{n}$ for $n\geq 1$). Given a function
$\phi_{J\mathbb{\mathbb{R}}}$, if $J$ is not the full circle, we define the
variation of $\phi$ over $J$ as
$\operatorname{Var}_{J}\phi=\sup_{Part(J)}\sum_{r=1}^{n-1}\left|\phi(x_{r+1})-\phi(x_{r})\right|$.
If $\operatorname{Var}_{J}\phi$ exists we say $\phi$ is of Bounded Variation
$(\phi$ is BV) on $J$, otherwise $\phi$ is of Unbounded Variation ($\phi$ is
UBV) on $J$. If $J$ is the full circle, we define
$\operatorname{Var}_{\mathbb{T}}\phi=\sup_{Part(\mathbb{T})}\sum_{r=1}^{n}\left|\phi(x_{r+1})-\phi(x_{r})\right|$
where $x_{n+1}\coloneqq x_{1}$.

Note that this definition coincides with the classical definitions of Bounded
Variation on closed intervals and on the full circle.

######  Example 57.

Given the observable $\phi x=\frac{1}{x}$ on the circle (using our convention
that $\phi(0)=0$) then $\phi$ is BV on the directed intervals
$[\frac{1}{2},1)$,$[\frac{1}{2},1]$ but UBV on $[0,\frac{1}{2})$ and
$(0,\frac{1}{2})$.

We now give a simple but powerful principle which is not specifically related
to unboundedness, but includes it.

######  Definition 58 (Localisation).

Given a topological space $(X,\mathscr{F})$, and a proposition $P$ defined on
the open sets of $X$. We derive the proposition locally $P_{X}$ (written
$LP_{X}$) which is defined at points of $X$ as follows: $x$ is locally $P$ (ie
$LP_{X}(x)$) iff $x$ has an open neighbourhood $N_{x}\in\mathscr{F}$
satisfying $P_{X}(N_{x})$. In particular if the proposition is defined in
terms of a function $\phi_{XY}$, ie $P\coloneqq P(\phi_{XY})$, then when $x$
is locally $P$ (ie $LP(x)$) we will also say that $\phi$ is locally $P$ at
$x$. We will denote the set of points at which $x$ is locally $P$ as
$LP(\phi)$.

######  Example 59.

Let $(J,\mathscr{F})$ be an interval of the real line or the circle equipped
with the ambient subspace topology, and $\phi_{J\mathbb{R}}$ a fixed function.
Let $P=BV$, the proposition defined on each $K\in\mathscr{F}$ that $\phi$ is
of Bounded Variation on $K$. Then $\phi_{J\mathbb{R}}$ is of Locally Bounded
Variation (ie $LP=LBV$) at $x_{J}$ iff $x$ has a neighbourhood
$K\in\mathscr{F}$ on which $\phi$ is of Bounded Variation. We then write
$LBV_{J}(x)$. Note that if $\phi$ is not $LBV$ at $x$, then there is no
neighbourhood of $J$ containing $x$ on which $\phi$ is $LBV$, which means in
particular that $\phi$ is $UBV$ on _every_ open interval containing $x$ and we
will write $UBV(x)$. Finally $LBV_{J}(\phi)$ is the set of points in $J$ at
which $\phi$ is of locally bounded variation, and $UBV_{J}(\phi)$ is the set
of points at which $\phi$ is of Unbounded Variation.

######  Proposition 60.

Given a topological space $(X,\mathscr{F})$, an observable
$\phi_{X\mathbb{R}}$, and a proposition $P_{X}$ defined on open sets of $X$,
then the set of points $LP_{X}(\phi)$ (the points at which $\phi$ is locally
$P$) is open, and its boundary belongs to the complement of $LP(\phi)$, the
closed set of points at which $\phi$ is not locally $P$

###### Proof.

Let $x\in LP(\phi)$ so there is a neighbourhood $N_{x}$ with $x\in N_{x}$ and
$P(N_{x})$. By definition there is an open set $O_{x}\in\mathscr{F}$ with
$x\in O_{x}\subseteq N_{x}$. Now for any $y\in O_{x}$, $N_{x}$ is also a
neighbourhood of $y$, and since $P(N_{x})$ we have $y\in LP(\phi)$. Hence
$O_{x}\subseteq LP(\phi)$ and so $x$ is an interior point of $LP(\phi)$ and
$LP(\phi)$ is open.

Now $LP(x)$ is defined for every $x_{X}$ and so either $LP(x)$ or $!LP(x)$,
and so the points at which $!LP(x)$ form the complement of $LP(\phi)$. Since
$LP(\phi)$ is open, its complement is closed and the boundary of $LP(\phi)$
lies completely in that complement. ∎

######  Corollary 61 (LBV topology).

In an interval $J$ with the subspace topology, the points $LBV_{J}(\phi)$ at
which a function $\phi_{J\mathbb{\mathbb{R}}}$ is of locally bounded
variation, form an at most countable set of disjoint open intervals in $J$,
and the boundaries of these intervals in $J$ are of unbounded variation.

###### Proof.

Applying the Proposition to example 59 gives us that $LBV_{J}(\phi)$ (the set
of points of $J$ at which $\phi$ is of locally bounded variation) is an open
set in $J$, and hence (by Proposition 8) is an at most countable union of
disjoint open intervals. The boundary of the set $LBV_{J}(\phi)$ in $J$ is
therefore the set of boundaries of these intervals and the proposition tells
us these lie in $!LBV(\phi)$, ie they are in $UBV(\phi)$.

∎

Note that the endpoints of an interval $J$ itself are _not_ boundary points in
the subspace topology, so that the endpoints of $J$ may still be
$LBV_{J}(\phi).$ For example let $\phi x=1/x$ with $J=[\frac{3}{4},1],$then
$\phi_{J\mathbb{\mathbb{R}}}$ is $LBV$ on the whole of $J$ including both
endpoints, and so $LBV(\phi)$ is the single interval $J$ which is open without
boundary in the subspace topology of $J$. Contrast this with the case of
$J=[0,\frac{1}{4}]$, when $LBV(\phi)$ is now $(0,\frac{1}{4}]$ and this
interval has the boundary point $0$ in $J$ and is $UBV$, but $\frac{1}{4}$ is
$LBV$ and is not a boundary point of $LBV(\phi)$ in $J$.

Also from Corollary 61 we have $!LBV(\phi)=UBV(\phi)$, ie $\phi$ is of
Unbounded Variation at all points (including endpoints) outside the open
intervals of $LBV(\phi)$. Note that $UBV(\phi)$ itself may have positive
measure. For example the Dirichlet observable (the indicator function on the
rationals in $[0,1)$) is $UBV$ everywhere and $LBV(\phi)$ is empty.

We summarise our findings in the following Theorem:

######  Theorem 62.

For any interval $J$ of the circle or real line (with the subspace topology),
and observable $\phi_{J\mathbb{\mathbb{R}}}$, there is a canonical
decomposition of $J$ into the disjoint union of the closed set $UBV_{J}(\phi)$
with an at most countable set of open subintervals $\\{I_{k}\\}$ on which
$\phi$ is $LBV$. Further, writing $\phi_{U}=\left\llbracket x\in
UBV(\phi)\right\rrbracket\phi,\phi_{k}=\left\llbracket x\in
I_{k}\right\rrbracket\phi$ we have

(6.1) $\phi=\phi_{U}+\sum_{k}\phi_{k}$

where each $\phi_{k}$ is $LBV$ with disjoint open support $I_{k}$, and the
boundary (endpoints) of each $I_{k}$ are in $UBV(\phi)$

######  Definition 63 (Primitive functions).

Given intervals $I\subseteq J$, a function $\phi_{J\mathbb{R}}$ is a primitive
on $I$ if it vanishes888this means $I$ contains the _set theoretic_ support of
$\phi$ in $J$, but not necessarily the _topological_ support as this is the
closure of the set theoretic support) outside $I$, is monotone on $I$, and has
a bound on $I$ (ie $\inf_{I}\phi$ or $\sup_{I}\phi$ exists, or both). If it
has both bounds it is a bounded primitive, otherwise it is an upper or lower
bounded primitive.

Notes:

  1. (1)

A monotone function on an interval is $LBV$ on that interval, so that any
primitive on $I$ is $LBV$ on $I$.

  2. (2)

The monotone constraint rules out the existence of primitives which are
bounded functions of unbounded variation (eg $\sin 1/x$ on $(0,1)$).

  3. (3)

We need to take the same care over primitives and interval endpoints as we do
for any $LBV$functions. So a primitive may be unbounded on non-included
endpoints, eg $\phi_{\mathbb{TR}}x=1/x$ is a lower bounded primitive on
$I=(0,1)$ but unbounded at $0$ (where its value is $0$). If $I$ includes an
endpoint $a$, then $\phi$ must be bounded on $I$ at $a$ though it may be
unbounded outside $I$ at $a$, eg $\phi_{\mathbb{TR}}x=1/x$ is also a bounded
primitive on $I=[\frac{1}{2},1]$, where it is bounded at $1$ in $I$, but
unbounded at $1$ in $\mathbb{T}$.

###  6.2. Some initial function classes

Normalised functions

######  Definition 64.

We say an observable $\phi$ is normalised if $\phi(x)=0$ on $UBV(\phi)$. (Note
this requires $UBV(\phi)$ to have empty interior). The normalisation of $\phi$
is the normalised function $\phi^{Norm}(x)=\left\llbracket x\not\in
UBV(\phi)\right\rrbracket\phi(x)$. We say $\phi$ is equivalent to $\psi$ under
normalisation if the equivalence relation $\phi^{Norm}=\psi^{Norm}$ holds.

######  Example 65.

The observable $\phi x=1/x$ is normalised (since $UBV(\phi)=\\{0\\}$ and $\phi
0=0$). However the Dirichlet function is not normalised, since it is UBV
everywhere but not $0$ everywhere. Its normalisation is the $0$ function.

When considering Birkhoff sums $S_{N}\phi=\sum_{r=1}^{N}\phi(x_{0}+r\alpha)$
we are primarily interested only in orbits which avoid the unbounded points of
$\phi$. In these cases the value of $\phi$ at each unbounded point has no
effect on the Birkhoff sum, ie $S_{N}\phi=S_{N}\phi^{Norm}$. We will
henceforth regard observables as normalised unless otherwise indicated.

Note that if $\phi$ is normalised then $\phi_{U}=0$ in (6.1), and so this
equation simplifies to $\phi=\sum_{k}\phi_{k}$. In the next section we will
investigate the decomposition of the functions $\phi_{k}$.

Banach congruence classes Given an additive group $G$ of real valued
functions, let $G_{B}=\\{\phi:\sup|\phi|<\infty\\}$ be the Banach space (and
subgroup) of bounded functions. Then each congruence class of $G/G_{B}$
consists of unbounded functions whose differences lie in $G_{B}$.

######  Definition 66.

We say that two observables $\phi,\psi$ are of finite difference if
$\phi-\psi$ is bounded, and we write ($\phi\stackrel{{\scriptstyle
B}}{{\sim}}\psi$) noting that this is an equivalence relation.

######  Example 67.

On $[0,1)$ it is easily shown that $\pi\cot\pi x\stackrel{{\scriptstyle
B}}{{\sim}}\frac{1-2x}{x(1-x)}\stackrel{{\scriptstyle
B}}{{\sim}}\frac{1}{\\{\\{x\\}\\}}$ but $\csc\pi x\not\sim\cot\pi x$ as their
difference is unbounded as $x\rightarrow 1$.

If $\phi,\psi$ are of bounded difference, this means $S_{N}\phi-
S_{N}\psi=O(N)$ and further if $\oint(\phi-\psi)=0$ then $S_{N}\phi-
S_{N}\psi=O(\log N)^{2+\epsilon}$ (**) for ae $\alpha$. In particular this
means if $S_{N}\phi$ has a super-linear growth rate (ie greater than $O(N)$),
then $S_{N}\psi$ has the same growth rate and the same set of unbounded
points, and that this super-linear growth rate is determined purely by the
behaviour of $\phi,\psi$ at unbounded points.

We now refine some existing understanding and terminology related to
boundedness to enable us to study unboundedness. Monotone functions We will
use the word “increasing” always to mean monotone increasing unless we specify
otherwise. The word “decreasing” is dual to increasing, and can be substituted
in the following paragraph.

###  6.3. Classification of primitive functions on an interval

Figure 6.1. Quadrants of Primitives

Let $I\subseteq J$ be an interval with endpoint coordinates $a,b$, such that
$J$ contains the endpoints $a,b$. (If $J$ is the circle we allow $a,b$ to
represent the same circle point, but require $b=a+1$). Recall
$\phi_{J\mathbb{\mathbb{R}}}$ is a primitive on $I$ if $\phi$ vanishes outside
$I$, is monotone on $I$, and has a bound (ie at least one of $\inf\phi$ or
$\sup\phi$ exists). Note that a primitive function which is constant is both
ascending and descending on $I$, but any other primitive is either ascending
or descending.

A primitive is called upper bounded if $\sup\phi$ exists, and lower bounded if
$\inf\phi$ exists. It is called a bounded primitive if both bounds exist, and
an unbounded primitive otherwise. We can therefore classify an unbounded
primitive by whether it is ascending or descending (A or D) and whether it is
upper or lower bounded (U or L).

Let us write the set of Descending, Lower bounded primitives on $I$ as DL, we
similarly define the sets AL, AU, and DU. An unbounded primitive belongs to
precisely one of these sets, but a bounded primitive is both upper and lower
bounded and so will belong to both AU and AL, or to both DU and DL, except for
the case of a constant function which belongs to all 4 sets. Note that DL is
closed under addition and contains the constant primitive 0. It therefore
forms a commutative monoid. Further it forms a module over the semi-ring
$\mathbb{R}^{+}=\\{x\in\mathbb{R}:x\geq 0\\}$. The same applies to each of the
other 3 sets.

Note that if $\phi$ is a primitive on $I$, then so is $-\phi$. Further, if $I$
is either open or closed (ie not semi-open) then
$\overline{\phi}(x)\coloneqq\phi(a+b-x)$ is also a primitive on $I$, and hence
so is $-\overline{\phi}$. Finally
$\overline{-\phi}(x)=(-\phi)(a+b-x)=-\overline{\phi}(x)$.

Let $\mathcal{F}$ be a family of primitives of $I$. Then
$-\mathcal{F}\coloneqq\\{\phi:-\phi\in\mathcal{F}\\}$ is clearly also a family
of primitives of $I$. If $I$ is either open or closed we can also define
another family of primitives
$\overline{\mathcal{F}}\coloneqq\\{\phi:\overline{\phi}(a+b-x)\in\mathcal{F}\\}$,
and then
$\overline{-\mathcal{F}}=-\overline{\mathcal{F}}=\\{\phi:-\overline{\phi}\in\mathcal{F}\\}$
is a 4th family. We call the set
$\\{\mathcal{F},-\mathcal{F}.\overline{\mathcal{F}},-\overline{\mathcal{F}}\\}$
the dual quadrants of $\mathcal{F}$ (under the dualities $-,\bar{.}$). Note
that the set $\\{DL,AU,DU,AL\\}$ comprises the dual quadrants of each of its 4
members when $I$ is open or closed (but not semi-open), and fixing $DL$ this
gives $AU=-DL,AL=\overline{DL},DU=-\overline{DL}$.

If $\phi$ is a primitive of $I$ which has a lower bound of 0 (ie $\phi\geq 0$
on $I$) then we say $\phi$ is a positive primitive on $I$, and if $\phi\leq 0$
we say $\phi$ is negative. We denote the set of descending positive primitives
on $I$ as $\Phi(I)$, and note this is a sub-monoid of $DL$ ($\Phi(I)\subset
DL(I)$). Then the dual quadrant $-\Phi(I)\subset AU(I)$ is the monoid of
ascending negative primitives. If $I$ is open or closed, we can also identify
the other dual quadrants as the monoids $\overline{\Phi}(I)\subset
AL(I),-\overline{\Phi}(I)\subset DU(I)$.

Positive and negative primitives have the important properties that they can
be extended and translated in $J$.

If $\phi\in\Phi(I)$ and $K$ is a right extension of $I$ (ie $x\in K\backslash
I\,\Rightarrow\,x\geq b$ where $b$ is the right endpoint of $I$), then
$\phi\in\Phi(K)$ so that $\Phi(I)\subset\Phi(K)$. Similarly
$-\Phi(I)\subset-\Phi(K)$, and if $K^{\prime}$ is a left extension of $I$ (and
open or closed with $I$) then also
$\overline{\Phi}(I)\subset\overline{\Phi}(K^{\prime}),-\overline{\Phi}(I)\subset-\overline{\Phi}(K^{\prime})$.

We now assume $I=(a,b)$ is open. Let $K=|c,d|$ (where we set $c=0,d=1$ when
$K$ is the circle $[0,1)$). If $\phi\in\Phi(I)$ we can define the translation
$T\phi(x)=\phi(x+(a-c))$ and then $T\phi\in\Phi(c,c+(b-a))$. By the remarks on
extension above we can write $T\phi\in\Phi(c,d)$ and $-T\phi\in-\Phi(c,d)$. In
particular when $K$ is the circle, $T\phi\in\Phi(0,1),-T\phi\in-\Phi(0,1)$.
Similarly if $\phi\in\overline{\Phi}(I)$ we define the translation
$\overline{T}\phi(x)=\phi((d-b)+x))$ and then
$\overline{T}\phi\in\overline{\Phi}(d-(b-a),d)$ which we can extend to
$\overline{T}\phi\in\overline{\Phi}(c,d)$ and
$-\overline{T}\phi\in-\overline{\Phi}(c,d)$. In particular when $J$ is the
circle,
$\overline{T}\phi\in\overline{\Phi}(0,1),-\overline{T}\phi\in-\overline{\Phi}(0,1)$.
In summary, positive and negative primitives on an open interval $I$ can be
translated to primitives on the interior of $K$.

Note that a monotone function on a closed interval is BV. Since a primitive
$\phi$ on $I$ vanishes outside $I$, if $I$ is closed $\phi$ is BV (and hence
LBV) on the whole of $J$. If $I$ is open then $\phi$ is LBV on $I$ (by 61) and
$LBV$ outside the closure of $I$, and also $LBV$ at the bounded endpoint of
$I$.

###  6.4. Representation of functions by primitives

Recall we denote the set of points of locally bounded variation $LBV(\phi)$,
and its complement $UBV(\phi)$. We have shown $LBV(\phi)$ is a union of
disjoint open intervals, although it is important to note that this includes
the possibilities $LBV(\phi)=\mathbb{T}$ (when $\phi$ is BV on the whole
circle) and $LBV=\emptyset$ (for example $\phi(x)=\left\llbracket
x=\frac{p}{q},(p,q)=1\right\rrbracket q$ is unbounded everywhere and so is not
locally bounded anywhere). Otherwise there is an at most countable set of
disjoint open intervals $I_{k}=(x_{k}^{L},x_{k}^{R})$ with
$LBV(\phi)=\bigcup_{k}I_{k}$ and $x_{k}^{L},x_{k}^{R}\in UBV(\phi)$. We will
assume $LBV(\phi),UBV(\phi)\neq\emptyset$ going forward. We can now define the
observables $\phi_{U}x=\left\llbracket x\in UBV(\phi)\right\rrbracket\phi x$
and $\phi_{i}=\left\llbracket x\in(x_{i}^{L},x_{i}^{R})\right\rrbracket\phi$,
which enables us to write $\phi=\phi_{U}+\sum_{i}\phi_{i}$ with
$UBV(\phi_{i})=\\{x_{i}^{L},x_{i}^{R}\\}$. In the special case of $\phi$
having a single UBV point $x$, we get $x_{1}^{L}=x_{1}^{R}=x$, otherwise the
endpoints are all distinct.

Note that for a normalised function, by definition $\phi_{U}=0$.

We can now extend the Jordan Decomposition theorem from BV functions to LBV
functions:

######  Lemma 68 (Jordan Decomposition of LBV).

If $\phi$ is LBV on an (non-empty) open interval $I=(a,b)$then for each point
$m\in(a,b)$ there is a canonical decomposition into primitives on $I$, namely
$\phi=\phi(m)+\sum_{i=1}^{4}\phi_{i}$ where
$\phi_{1}\in\Phi(a,m),\,\phi_{2}\in-\Phi(a,m),\,\phi_{3}\in\overline{\Phi}(m,b),\,\phi_{4}\in-\overline{\Phi}(m,b)$.

###### Proof.

We partition $\phi$ into left and right functions $\phi^{L}=\left\llbracket
x\in(a,m)\right\rrbracket(\phi(x)-\phi(m)),\,\phi^{R}=\left\llbracket
x\in(m,b)\right\rrbracket(\phi(x)-\phi(m))$.We now define the left and right
variations of $\phi$ for $a<x<b$ by $V^{L}(x)=\left\llbracket
x\in(a,m)\right\rrbracket\operatorname{Var}_{[x,m]}\phi,\,V^{R}(x)=\left\llbracket
x\in(m,b)\right\rrbracket\operatorname{Var}_{[m,x]}\phi$ (the total variation
of $\phi$ over $(x,m),(m,x)$ respectively). These functions are well defined
since $\phi$ is BV on each interval $(x,m)$ or $(m,x)$. We now define the
difference functions
$D^{L}(x)=V^{L}(x)-\phi^{L}(x),\,D^{R}(x)=V^{R}(x)-\phi^{R}(x)$. Note that
these 4 defined functions all vanish outside $(a,m)$ or $(m,b)$ and are
monotone and positive and hence positive primitives on their domains.
$V^{L},D^{L}$ are both decreasing and so $V^{L},D^{L}\in\Phi(a,m)$, whereas
$V^{R},D^{R}$ are increasing and so $V^{R},D^{R}\in\overline{\Phi}(m,b)$. But
now $\left\llbracket
x\in(a,b)\right\rrbracket\phi(x)=\phi(m)+V^{L}(x)-D^{L}(x)+V^{R}(x)-D^{R}(x)$
and the result follows. ∎

We can combine (6.1) and Lemma (68) to obtain:

######  Theorem 69 (Decomposition Theorem).

Given an observable $\phi_{J\mathbb{R}}$ then $LBV(\phi)$ is an at most
countable union of disjoint open intervals $I_{k}=(a_{k},b_{k})$, and
$\phi=\phi_{U}+\sum_{k}\phi_{k}$ where $\phi_{U}=\left\llbracket x\in
UBV(\phi)\right\rrbracket\phi$ and $\phi_{k}$ is an LBV function on $I_{k}$.
Further, given $m_{k}\in I_{k}$ we have the representation of $\phi_{k}$ in
terms of primitives given by $\phi_{k}=\phi(m_{k})+\sum_{j=1}^{4}\phi_{kj}$
where
$\phi_{k1}\in\Phi(a_{k},m_{k}),\,\phi_{k2}\in-\Phi(a_{k},m_{k}),\,\phi_{k3}\in\overline{\Phi}(a_{k},m_{k}),\,\phi_{k4}\in-\overline{\Phi}(a_{k},m_{k})$.

##  7\. Homogeneous Birkhoff Sums of unbounded observables

###  7.1. Introduction

In the last section we established that if $\phi_{\mathbb{TR}}$ is an
observable on the circle we have the decomposition $\phi=\phi_{U}+\sum_{k\in
K}\phi_{k}$ into (at most countable) functions of disjoint support, where each
$\phi_{k}$ is LBV with its support in an open interval $I_{k}$. We also showed
we can decompose each $\phi_{k}$ into a sum of 4 primitive functions
$\phi_{ki}$ from each quadrant of $\Phi(I)$. Since $S_{N}$ is linear this
means we have reduced the study of $S_{N}(\phi,x)$ to the study of
$S_{N}(\phi_{U},x)$ and the study of $S_{N}(\phi_{ki},x)$ with $\phi_{ki}$
primitive. We will generally be concerned with sums for which $x+r\alpha$
never coincides with an unbounded point of $\phi$ so that
$S_{N}(\phi_{U},x)=0$. This means we can replace $\phi$ by its normalised form
$\phi^{Norm}$ (which vanishes on the set of unbounded points of $\phi$)
without changing the overall sum. In this section we concentrate our attention
on the study of $S_{N}(\phi,x)$ for primitive $\phi$.

Now suppose $\phi$ is a primitive on the directed interval $(a,b)$. If $\phi$
is unbounded at $a$ then $\phi_{a}\coloneqq
x_{\mathbb{T}}\rightarrow\phi\\{x+a\\}$ is a primitive on $(0,1)$, unbounded
at $0^{+}$. Similarly if $\phi$ is unbounded at $b$ then $\phi_{b}\coloneqq
x_{\mathbb{T}}\rightarrow\phi\\{x-b\\}$ is a primitive on $(0,1)$, unbounded
at $1^{-}$. Then $S_{N}(\phi,x_{0})=S_{N}(\phi_{a},x_{0}-a)$ and
$S_{N}(\phi,x_{0})=S_{N}(\phi_{b},x_{0}+b)$ which means we can reduce the
study of Birkhoff sums of general primitives to the study of Birkhoff sums of
primitives on $(0,1)$. We will now assume $\phi$ is a primitive on $(0,1)$.

Recall that we call $S_{N}(\phi,x)$ a homogeneous sum when $x=0$ and
inhomogeneous otherwise.

The general case of an inhomogeneous sum $S_{N}(\phi,x)$ is challenging. This
may seem surprising because we can easily convert an inhomogeneous sum to a
homogeneous sum by defining $\phi_{c}(x)=\phi(x+c)$, and then
$S_{N}(\phi,x_{0})=S_{N}(\phi_{x_{0}},0)$. The problem is that in general
$\phi_{x_{0}}$ will not then be a primitive on $(0,1)$, and the underlying
problem has not been simplified.

Fortunately homogeneous sums are sufficient to address a high proportion of
problems of interest (including the cases described in the abstract of this
paper).. Given that this paper is already somewhat lengthy, we therefore
content ourselves with postponing consideration of inhomogeneous sums to
another paper, and restrict this paper to the homogeneous case.

We will now use the results on the sequential distribution of $(r\alpha)$ from
section 5 to develop estimates of the homogeneous sum $S_{N}\phi\coloneqq
S_{N}(\phi,0)=\sum_{r=1}^{N}\phi(r\alpha)$ when $\phi$ is a monotonic function
on $(0,1)$. We will use another variant of notation $S_{N}(x)\coloneqq
S_{N}(\phi,x)=\sum_{r=1}^{N}\phi(x+r\alpha)$, noting carefully that this is
semantically distinct from the homonymous notation
$S_{N}(x)\coloneqq\sum_{r=1}^{N}x_{r}$ of section 4.

The restriction to monotone functions might at first seem restrictive.
Classical studies on sums of bounded functions tend to focus on functions of
bounded variation as the primary objects of interest. However since any
function of bounded variation can be represented as the sum of two monotonic
functions (one increasing, one decreasing), a focus on monotone functions is
not a limitation, but rather a recognition of the fact that monotonic
functions in fact lie slightly deeper.

We will first establish some duality results which reduce the number of cases
we will need to consider. We will use the Source-Target Magma theory developed
in Section 3, where we put the source $S=(0,1)$ with involution
$\sigma_{S}\coloneqq x\mapsto(1-x)$ and target $T=\mathbb{R}$ with involution
$\tau_{S}\coloneqq x\mapsto-x$.

###  7.2. Dual Relations for Birkhoff Functionals

In this section we will explore the use of dualities to reduce the number of
cases we will need to consider when we look at bounds for Birkhoff sums. We
will exploit both the quite general Source and Target dualities investigated
in Section 3, and also a more specialised quasiperiodic duality which arises
in the context of Continued Fraction theory.

These dualities arise under the operation of various involutions, and we will
call fixed points of an involution self-conjugate under that involution.

We will also define a duality which combines the effects of the Source and
Quasiperiodic involutions. As this will be an important duality for us, we
will call it simply the Double Duality, and its fixed points are the double
self-conjugate points.

Source-Target Dualities in Birkhoff Sums on the circle Recall that a Birkhoff
sum $S_{N}\phi$ can be regarded as a functional operator
$\left(S_{N}\right)_{(\mathbb{TR})\mathbb{R}}$ acting on the function
$\phi_{\mathbb{TR}}$. Note that since the orbit $(r\alpha)$ avoids the origin,
the sum $S_{N}\phi_{\mathbb{TR}}=\sum_{1}^{N}\phi_{\mathbb{TR}}(r\alpha)$ is
identical in value with the sum
$S_{N}\phi_{S\mathbb{R}}\left\\{r\alpha\right\\}$ where $\phi_{\mathbb{TR}}$
has been replaced by its restriction to the interval $S=(0,1)$, and the
Birkhoff Sum functional $\left(S_{N}\right)_{(\mathbb{TR})\mathbb{R}}$ becomes
$\left(S_{N}\right)_{(S\mathbb{R})\mathbb{R}}$.

Using this observation, we can now apply the theory developed in Section 3. In
particular we will build on the example application given in Subsection 3.4.
We take as the real intervals $S\coloneqq(0,1)$ and $T\coloneqq\mathbb{R}$
(again equipped with the endorelations $R_{S}=R_{T}=\leq_{\mathbb{R}}$). We
now set the source involution $\sigma_{S}$ to be the circle involution
$\overline{.}:x\mapsto\\{1-x\\}$ but restricted to $(0,1)$ (where we note it
is still an involution and can be written $x\mapsto 1-x$), and we set the
target involution to be $\tau_{T}\coloneqq x_{\mathbb{R}}\mapsto-x$.

Note that now the source pull up of $\sigma_{S}$ is
$\sigma_{ST}=(\overline{.})_{ST}\coloneqq\phi_{ST}\mapsto\phi\circ\text{$\sigma_{S}$}$,
ie $\overline{\phi}\alpha\coloneqq\phi\overline{\alpha}$. Similarly the target
pull up of $\tau_{T}$ is $\tau_{ST}=-_{ST}\coloneqq\phi\mapsto-_{T}\circ\phi$,
ie
$\left(-_{ST}\phi\right)\alpha\coloneqq-_{\mathbb{R}}\left(\phi\alpha\right)$.
Again we can pull up to Level 2, to get $\sigma_{(ST)T}\coloneqq
A_{(ST)T}\mapsto A\circ\sigma_{ST}$, ie $\overline{A}\phi\coloneqq
A\overline{\phi}$ and similarly
$\left(-_{(ST)T}A\right)\phi\coloneqq-_{\mathbb{R}}\left(A\phi\right)$.

If we further assume that $A$ is a $\tau-$morphism this means
$A(-_{ST}\phi)=-_{\mathbb{R}}\left(A\phi\right)$.

In this source-target environment we also have further structure than in 3.4
since $T=\left(\mathbb{R},+_{\mathbb{R}},\times_{\mathbb{R}}\right)$ is a
field. The pull up the binary operations of $T$ to $ST$ induces an algebra of
functions over $T$ (where $\lambda_{T}\phi_{ST}\,+_{ST}\,\mu_{T}\psi_{ST}$ and
$\phi\,\times_{ST}\,\psi$ have their natural meanings). It is easily verified
that the involutions $\sigma_{ST},\tau_{ST}$ are are then linear in both
$+_{ST},\times_{ST}$. The same argument applies analogously to the algebra
$(ST)T$ of functionals over $T$.

If $\Psi\subset ST$ is self-conjugate under $\sigma\tau$, the Example
application of Subsection 3.4 now applies, giving us:

(7.1) $\displaystyle A_{1}\leq_{\Psi}A_{2}$
$\displaystyle\Leftrightarrow\sigma A_{1}\leq_{\overline{\Psi}}\sigma
A_{2}\Leftrightarrow\sigma^{/}A_{1}\leq_{\overline{\Psi}}\sigma^{/}A_{2}\Leftrightarrow\sigma\sigma^{/}A_{1}\leq_{\Psi}\sigma\sigma^{/}A_{2}$
$\Leftrightarrow\tau A_{1}\leq_{\Psi}^{Op}\tau A_{2}\Leftrightarrow\sigma\tau
A_{1}\leq_{\overline{\Psi}}^{Op}\sigma\tau A_{2}\Leftrightarrow\sigma^{/}\tau
A_{1}\leq_{\overline{\Psi}}^{Op}\sigma^{/}\tau
A_{2}\Leftrightarrow\sigma\sigma^{/}\tau
A_{1}\leq_{\Psi}^{Op}\sigma\sigma^{/}\tau A_{2}$

We can go further in the case that $A_{1},A_{2}$ are $\tau-$morphisms. We use
the results of (3.10) to obtain from $A_{1}\leq_{\Psi}A_{2}$ :

(7.2) $\displaystyle A_{1}\leq_{\Psi}A_{2}\Leftrightarrow\sigma
A_{1}\leq_{\overline{\Psi}}\sigma A_{2}\Leftrightarrow\tau
A_{1}\leq_{\overline{\Psi}}\tau A_{2}\Leftrightarrow\sigma\tau
A_{1}\leq_{\Psi}\sigma\tau A_{2}$ $\Leftrightarrow\tau
A_{1}\leq_{\Psi}^{Op}\tau A_{2}\Leftrightarrow\sigma\tau
A_{1}\leq_{\overline{\Psi}}^{Op}\sigma\tau A_{2}\Leftrightarrow
A_{1}\leq_{\overline{\Psi}}^{Op}A_{2}\Leftrightarrow\sigma
A_{1}\leq_{\Psi}^{Op}\sigma A_{2}$

We pick out the first and last dual relations as being of particular use to us
later in this Section:

(7.3) $A_{1}\leq_{\Psi}A_{2}\Leftrightarrow\sigma A_{2}\leq_{\Psi}\sigma
A_{1}$

######  Remark 70.

Let $\Theta\subset ST$ be a set of functions which are descending (ascending)
on $S$. Now $\sigma_{ST}\tau_{ST}\Theta=-\overline{\Theta}$ and this is also a
set of descending (ascending) functions. Recall
$\Psi(\Theta)\coloneqq\Theta\bigcup\sigma\tau\Theta=\Theta\bigcup-\overline{\Theta}$
is self-conjugate under $\sigma\tau$, as is also
$\sigma\Psi=\overline{\Psi}=\overline{\Theta}\bigcup-\Theta$. So in these
cases of $\Phi$, $\Psi$ is a self-conjugate set of descending (ascending)
functions, and $\overline{\Psi}$ is a self-conjugate set of ascending
(descending) functions for which the dual relations in (7.1) apply, and
further (7.2) apply when $A_{1},A_{2}$ are $\tau-$morphisms. There are
important two special cases:

  1. (1)

When $\Theta=\Psi$ is the set of all monotonic descending (or monotonic
ascending, or simply all monotonic) functions

  2. (2)

When $\Psi=\Phi\bigcup\sigma\tau\Phi$ where $\Phi$ is the monoid of positive
descending primitives from (**).

Quasi-Period Duality

In this section we explore a specialised duality which arises in the context
of continued fraction developments. We will use some of the Source/Target
Duality theory from the previous section, but now in our specific environment
of Birkhoff sums over rotations. Indexed Families Recall that it is sometimes
convenient to call a function $\iota_{IX}:I\rightarrow X$ an index function.
We then call $I$ the index set, and $x_{i}\coloneqq\iota(i)$ an indexed
element. We call the collection $(x_{i})_{i\in I}$ an indexed family (indexed
by $I$). If $I=\\{i\\}_{i=1}^{n}$ we also write the indexed family as
$(x_{i})_{i=1}^{n}$.

######  Remark 71.

It is particularly important to note that $(x_{i})$ is not a set if $\iota$ is
not injective (it is a collection containing identical elements). This means
we must be particularly careful if we wish to define functions on the set
$X=\\{x_{i}\\}$ making use of the index $i$. For example, given an
endomorphism $f_{II}$ of $I$ we can define the pull back function
$g_{IX}=\iota\circ f$ which is a well-defined index function from $I$ to $X$.
It is then tempting to think that we can then also define a derived
endomorphism $h_{XX}:x_{i}\mapsto x_{f(i)}$ but this is not the case in
general: if $h$ exists and $\iota(i)=\iota(j)$ then
$x_{f(i)}=h(x_{i})=h(x_{j})=x_{f(j)}$ and this is not the case for general
$f$, even if $f$ is a bijection. In general then, $h_{XX}$ is a relation (or
multi-valued function).

Quasi-Period Indexed Families Recall (10) that we denote the sequence of
quasiperiods of $\alpha_{\mathbb{R}\backslash\mathbb{Q}}$ as
$(q_{r}^{\alpha}){}_{r=0}^{\infty}$, and that for
$\alpha\in(0,\frac{1}{2})\backslash\mathbb{Q}$ we have
$q_{r}^{\alpha}=q_{\overline{r}}^{\overline{\alpha}}$ where
$\overline{\alpha}=\\{1-\alpha\\}$ and $\overline{r}=r+1$. This result reveals
a non-trivial duality which we will develop in this section.

Note first that we have here two maps with the homonymous notation
${}^{\prime}\overline{.}^{\prime}$ . We can extend both to involutions as
follows. The first is defined on $\mathbb{I}=[0,1)$ by
$\left(\overline{.}\right)_{\mathbb{I}}\coloneqq\alpha\mapsto\\{1-\alpha\\}$
(note that $\overline{0}=0$ is fixed point). The second needs slightly more
care in definition: recalling $\mathbb{N}=\mathbb{N}^{+}\bigcup\\{0\\}$ we
take the disjoint union $\mathbb{N}_{2}=\mathbb{N}\bigsqcup\mathbb{N}^{+}$ and
define the involution $\left(\overline{.}\right)_{\mathbb{N}_{2}}$ by:
$\overline{r_{\mathbb{N}}}\coloneqq(r+1)_{\mathbb{N}^{+}}$and
$\overline{r_{\mathbb{N}^{+}}}\coloneqq(r-1)_{\mathbb{N}}$. We can now use
these two involutions to define a product involution
$\left(\overline{.}\right)_{\mathbb{I}}\times\left(\overline{.}\right)_{\mathbb{N}_{2}}$
on a suitable set as follows:

######  Definition 72.

The QuasiPeriod index set $QPI=\left(QPI_{Set},\overline{.}\right)$ has as its
underlying set the strict subset $QPI_{Set}\subset\mathbb{I}\times\mathbb{N}$
defined by

$QPI\coloneqq\left\\{(\alpha,r):\alpha\in\mathbb{I}\backslash\mathbb{Q}\>\&\>r\geq\left\llbracket\alpha\geq\frac{1}{2}\right\rrbracket\right\\}$

It is equipped with the Quasiperiod involution $Q_{QPI}$ defined by
$Q(\alpha,r)\coloneqq\overline{(\alpha,r)}=(\overline{\alpha},\overline{r})$
where $\overline{\alpha}\coloneqq\\{1-\alpha\\}$ and $\overline{r}\coloneqq
r+(-1)^{\left\llbracket\alpha>\frac{1}{2}\right\rrbracket}$.

A QP indexed family (QPI family)
$X=\left\\{(x_{r}^{\alpha}),\iota_{(QPI)X}\right\\}$ is a family
$X=(x_{r}^{\alpha})$ indexed by the QP index set $QPI$ using the index map
$\iota_{(QPI)X}:(\alpha,r)_{QPI}\mapsto x_{r}^{\alpha}$). There is usually no
need to specify the index map, in which case we simply write
$X=(x_{r}^{\alpha})$. Points satisfying
$x_{\overline{r}}^{\overline{\alpha}}=x_{r}^{\alpha}$ we call QP self-
conjugate. . If the index map is independent of $\alpha$, ie for each $r$,
$\iota(\alpha,r)=x_{r}$ for every $\alpha$, we will also allow ourselves to
write the family as $(x_{r}^{\alpha})=(x_{r})$. Similarly if $\iota$ is
independent of $r$, we will write the family as $(x^{\alpha})$, and as $(x)$
if $\iota$ is a constant map.

######  Remark 73.

Note that $(\alpha,r)\in QPI$ means $r\geq 0$ for $\alpha<\frac{1}{2}$ and
$r\geq 1$ for $\alpha>\frac{1}{2}$. In particular this means that a QPI family
$(x_{r}^{\alpha})$ contains no element $x_{0}^{\alpha}$ for
$\alpha>\frac{1}{2}$.

Note that a QP indexed family may be further indexed by other index sets, eg
the family $(x_{rz}^{\alpha})$ is indexed by $QPI\times Z$, but the QP
involution only affects the QP index within the family, ie
$x_{rz}^{\alpha}=\iota\left((\alpha,r),z\right)$ and
$\iota\left((\overline{\alpha},\overline{r}),z\right)=x_{\overline{r}z}^{\overline{\alpha}}$.

In line with Remark 71 we cannot assume that there is an induced qp involution
$Q_{X}\coloneqq x_{r}^{\alpha}\mapsto x_{\overline{r}}^{\overline{\alpha}}$ on
a QPI family $X=(x_{r}^{\alpha})$ if the index map is not injective. However
in many (but not all ) important cases, the function $x_{r}^{\alpha}\mapsto
x_{\overline{r}}^{\overline{\alpha}}$ is well-defined (and then the induced
function is also an involution since $Q_{QPI}$ is itself an involution). When
this is the case some individual proofs below could be simplified. However our
overall presentation is simplified by having a single set of proofs which work
whether an induced involution is defined or not. Hence we will NOT generally
assume $Q\coloneqq x_{r}^{\alpha}\mapsto x_{\overline{r}}^{\overline{\alpha}}$
is well-defined in the sequel.

We now define a duality which will prove important in later sections. This is
a hybrid using both the QP involution and the source involution
$\left(\overline{.}\right)_{\mathbb{II}}\coloneqq\alpha\mapsto{1-\alpha}$.

######  Definition 74 (Double Dual).

Given a QPI family of functions $(\phi_{r}^{\alpha})$, we call the derived
family $\left(\overline{\phi_{\overline{r}}^{\overline{\alpha}}}\right)$ the
family of the double dual functions of $(\phi_{r}^{\alpha})$. For convenience
we will use the simplified notation
$\overline{\phi}_{r}^{\alpha}\coloneqq\overline{\phi_{\overline{r}}^{\overline{\alpha}}}$.
When $\overline{\phi}_{r}^{\alpha}=\phi_{r}^{\alpha}$ we say
$\phi_{r}^{\alpha}$ is double self-conjugate.

######  Remark 75.

It is important note that following Remark (**) we cannot regard the Double
Dual as a function map because
$\overline{\phi}_{r}^{\alpha}\mapsto\phi_{r}^{\alpha}$ may be multi-valued.

######  Example 76.

Important examples of QPI families

  1. (1)

The quasiperiod index set $QPI$ is itself a QP indexed family using the index
map $Id_{QPI}$. Since $Id$ is injective, in this case the involution
$Q:(\alpha,r)\mapsto\overline{(\alpha,r)}$ exists trivially.

  2. (2)

A scalar qp family $\left(\lambda_{r}^{\alpha}\right)$ with each
$\lambda_{r}^{\alpha}\in\mathbb{R}$. Important examples are
$q_{r}^{\alpha},a_{r+1}^{\alpha},b_{r}^{\alpha}$ (where these symbols have
their standard meanings). Note that these particular examples are also qp
self-conjugate (eg $q_{r}^{\alpha}=q_{\overline{r}}^{\overline{\alpha}}$) with
the single exception of $a_{r+1}^{\alpha}$ for $\alpha<\frac{1}{2},r=0$ when
$a_{2}^{\overline{\alpha}}=a_{1}^{\alpha}-1$. Scalar families will not
generally have injective index maps
($Q_{{}_{\left(\lambda_{r}^{\alpha}\right)}}$is not well defined), although
the identity function serves as a partial qp involution when restricted to the
set of self-conjugate elements.

  3. (3)

The particular scalar qp family $\left(E_{r}^{\alpha}\right)$ where
$E_{r}^{\alpha}\coloneqq E_{r}=\left\llbracket
r\,\mathrm{even}\right\rrbracket$ (so the family $\left(E_{r}^{\alpha}\right)$
is independent of $\alpha$). Note
$E_{\overline{r}}^{\overline{\alpha}}=O_{r}^{\alpha}$ and
$O_{\overline{r}}^{\overline{\alpha}}=E_{r}^{\alpha}$ and so
$\left(E_{r}^{\alpha}\right)=\left(O_{r}^{\alpha}\right)$.The index map
$\iota:(\alpha,r)\mapsto E_{r}^{\alpha}=E_{r}$ is clearly not injective, but
in fact the induced involution $Q_{\left(E_{r}^{\alpha}\right)}$ does exist.

  4. (4)

Given a scalar qp family $\\{\lambda_{r}^{\alpha}\\}$ we can form new scalar
families such as
$\left\\{f_{\mathbb{\mathbb{R}R}}(\lambda_{r}^{\alpha})\right\\}$ or
$\left\\{\phi_{\mathbb{IR}}(\lambda_{r}^{\alpha})\right\\}$ (when
$\lambda_{r}^{\alpha}\in\mathbb{I}$), and $\left\llbracket
P(\lambda_{r}^{\alpha})\right\rrbracket$ (where $P$ is a proposition). If
$\lambda_{r}^{\alpha}$ is qp self conjugate, so are $f(\lambda_{r}^{\alpha})$,
$\phi(\lambda_{r}^{\alpha})$ and $\left\llbracket
P(\lambda_{r}^{\alpha})\right\rrbracket$. In particular
$\phi\left(\frac{1}{q_{r}^{\alpha}}\right)$ is qp self conjugate for $r>0$.

  5. (5)

The qp family of qp index functions
$n^{\alpha}:\mathbb{N}^{+}\rightarrow\mathbb{N}$ defined by
$n^{\alpha}(N)\coloneqq\max\\{t:q_{t}^{\alpha}\leq N\\}$. Note we have dropped
$r$ from the notation to indicate that these functions are independent of $r$,
ie for any $r$, $n_{r}^{\alpha}=n^{\alpha}$. Note also that we can now write
the Ostrowski representation of $N$ as
$N=\sum_{r=\left\llbracket\alpha>\frac{1}{2}\right\rrbracket}^{n^{\alpha}(N)}b_{r}q_{r}$,
and also that we can now write the relationship between the indexes
$n^{\alpha},n^{\overline{\alpha}}$ as
$n^{\overline{\alpha}}(N)=n^{\alpha}(N)+(-1)^{\left\llbracket\alpha>\frac{1}{2}\right\rrbracket}$.
In particular this gives
$q_{n^{\alpha}}^{\alpha}=q_{n^{\overline{\alpha}}}^{\overline{\alpha}}.$

  6. (6)

The qp scalar family $c_{rst}^{\alpha}$ for some symbol $c$:

    1. (a)

When $c$ is the null symbol we will simply write $\alpha_{rst}$. The QP
conjugate is $\overline{\alpha}_{\overline{r}st}$, and since
$\overline{\alpha}_{\overline{r}st}=\\{1-\alpha_{rst}\\}=\overline{\alpha_{rst}}$
qp involution and source involution coincide on
$\mathbb{I}\backslash\mathbb{Q}$

    2. (b)

When $c$ is an observable symbol, eg $\phi$, we write
$\phi_{rst}^{\alpha}\coloneqq\phi(\alpha_{rst})$ with QP conjugate
$\phi_{\overline{r}st}^{\overline{\alpha}}=\phi(\overline{\alpha}_{\overline{r}st})=\phi(\overline{\alpha_{rst}})=\overline{\phi}_{rst}^{\alpha}$,
and again qp involution and source involution coincide

. QPI Families of Operators

Let $\mathscr{X}$ be the set of QPI families of functionals (families whose
elements lie in $(ST)T$), and let
$\left(X_{r}^{\alpha}\right),\left(Y_{r}^{\alpha}\right)$ 2 members of
$\mathscr{X}$. Define $Z_{r}^{\alpha}\coloneqq\lambda X_{r}^{\alpha}+\mu
Y_{r}^{\alpha}$. By this definition
$Z_{\overline{r}}^{\overline{\alpha}}\coloneqq\lambda
X_{\overline{r}}^{\overline{\alpha}}+\mu Y_{\overline{r}}^{\overline{\alpha}}$
and hence $\left(Z_{r}^{\alpha}\right)$ is also a QPI family, and we can write
$\left(Z_{r}^{\alpha}\right)=\lambda_{T}\left(X_{r}^{\alpha}\right)\,+_{\mathscr{X}}\,\mu_{T}\left(Y_{r}^{\alpha}\right)$
to mean for each $(\alpha,r)$ that $Z_{r}^{\alpha}=\lambda X_{r}^{\alpha}+\mu
Y_{r}^{\alpha}$. Note then $\mathscr{X}$ is a vector space over $T$ , and
$+_{\mathscr{X}}$ is the pull up of $+_{T}$ to $\mathscr{X}$. Similarly we can
also pull up $\times_{T}$ to $\mathscr{X}$ so that
$\left(X_{r}^{\alpha}\right)\times\left(Y_{r}^{\alpha}\right)$ is again a QPI
family, and so $\mathscr{X}$ is also an algebra over $T$.

Recall also that that when $ST$ is equipped with an involution
$\upsilon{}_{ST}$ then this induces a pullup involution $\upsilon_{(ST)T}$
defined by $\upsilon_{(ST)T}(X)\coloneqq X\circ\upsilon_{ST}$ which is also
linear in $+_{ST},\times_{ST}$. Again given a QPI family
$\left(X_{r}^{\alpha}\right)$ we can define the pullup involution
$\upsilon_{\mathscr{X}}$ by
$\upsilon_{\mathscr{X}}\left(\,\left(X_{r}^{\alpha}\right)\,\right)=\left(\upsilon_{(ST)T}X_{r}^{\alpha}\right)$
and again this is linear in $+_{\mathscr{X}},\times_{\mathscr{X}}$, so that
$\upsilon_{\mathscr{X}}$ is an algebra automorphism.

In particular we have proved:

######  Proposition 77 (Linearity of Double Dual).

If $Z_{r}^{\alpha}=\sum_{i=1}^{k}X_{ir}^{\alpha}\times Y_{ir}^{\alpha}$ then
$\left(Z_{r}^{\alpha}\right)$ is a QPI family and
$\overline{Z_{\overline{r}}^{\overline{\alpha}}}=\sum_{i=1}^{k}\overline{X_{\overline{r}}^{\overline{\alpha}}}\times\overline{Y_{\overline{r}}^{\overline{\alpha}}}$,
ie the double dual of an algebraic combination of $QPI$ families is the same
algebraic combination of the double duals

###### Proof.

We showed above that $\left(Z_{r}^{\alpha}\right)$ is a QPI family. The double
dual of $Z_{r}^{\alpha}$ is
$\sigma_{(ST)T}Z_{\overline{r}}^{\overline{\alpha}}$ which by definition of
$Z_{r}^{\alpha}$ is
$\sigma_{(ST)T}\left(\sum_{i=1}^{k}X_{\overline{r}}^{\overline{\alpha}}\times
Y_{\overline{r}}^{\overline{\alpha}}\right)$ and the result follows since
$\sigma$ is an algebra homomorphism. ∎

######  Example 78.

Important examples of qp operator families

  1. (1)

Given an operator $X$ we can define the constant qp family $\mathscr{X}_{X}$
in which $X_{r}^{\alpha}\phi=X\phi$ for all $(\alpha,r)\in QPI$. This family
is automatically qp self conjugate
($X_{\overline{r}}^{\overline{\alpha}}\phi=X\phi=X_{r}^{\alpha}\phi$).

  2. (2)

The partition sum operator family $P_{q_{r}^{\alpha}}$ (see Subsection 7.3 for
details) is a linear operator family which is Source self conjugate
($P_{q_{r}^{\alpha}}=\overline{P_{q_{r}^{\alpha}}}$), QP self conjugate
$(P_{q_{r}^{\alpha}}=P_{q_{\overline{r}}^{\overline{\alpha}}})$, and hence
also double self conjugate
($P_{q_{r}^{\alpha}}=\overline{P}_{q_{r}^{\alpha}}\coloneqq\overline{P_{q_{\overline{r}}^{\overline{\alpha}}}}$)

  3. (3)

$\phi\mapsto\phi_{rst}^{\alpha}\coloneqq\phi\left(\alpha_{rst}\right)$ is an
(anonymous) linear operator family which is neither source nor qp self
conjugate, but _is_ double self conjugate
$($$\phi_{rst}^{\alpha}=\overline{\phi}_{rst}^{\alpha}\coloneqq\overline{\phi_{\overline{r}st}^{\overline{\alpha}}}$).

  4. (4)

The family
$S_{rs}^{\alpha}\phi=S_{q_{r}}^{\alpha}(\phi,\alpha_{rs})\coloneqq\sum_{t=1}^{q_{r}}\phi_{rst}^{\alpha}$
is double self conjugate since $\phi\mapsto\phi_{rst}^{\alpha}$ is double self
conjugate, and hence also the sum $\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$ is
double self-conjugate.

  5. (5)

$\phi\mapsto\phi(f(q_{r}^{\alpha}))$ is an (anonymous) linear operator family
which is not source self conjugate, but which is qp self conjugate since the
qp family $\left(q_{r}^{\alpha}\right)$ is qp self conjugate
($q_{r}^{\alpha}=q_{\overline{r}}^{\overline{\alpha}}$). In particular
$\phi\mapsto\phi(\frac{1}{q_{r}^{\alpha}})$ is qp self conjugate.

Recall that in this section we are considering the Source-Target Magma where
$S=(0,1),T=\mathbb{R}$.

######  Definition 79.

Given two QPI operator families
$\left(X_{r}^{\alpha}\right),\left(Y_{r}^{\alpha}\right)$ we will say
$\left(X_{r}^{\alpha}\right)$ is dominated by $\left(Y_{r}^{\alpha}\right)$ on
$\Theta\subseteq ST$ if for every ($\alpha,r)\in QPI$ we have
$X_{r}^{\alpha}\leq_{\Theta}Y_{r}^{\alpha}$. We write this as
$\left(X_{r}^{\alpha}\right)\leq_{\Theta}\left(Y_{r}^{\alpha}\right)$

######  Lemma 80 (Domination Duals).

Let $\Psi\subset ST$ be self-conjugate under the involution
$\sigma_{ST}\tau_{ST}$. Given two _linear_ QPI families
$\left(X_{r}^{\alpha}\right),\left(Y_{r}^{\alpha}\right)$ of functionals from
$(ST)T$, then we have the following dual results:

$\left(X_{r}^{\alpha}\right)\leq_{\Psi}\left(Y_{r}^{\alpha}\right)\Leftrightarrow\left(X_{\overline{r}}^{\overline{\alpha}}\right)\leq_{\Psi}\left(Y_{\overline{r}}^{\overline{\alpha}}\right)\Leftrightarrow\left(\overline{Y_{r}^{\alpha}}\right)\leq_{\Psi}\left(\overline{X_{r}^{\alpha}}\right)\Leftrightarrow\left(\overline{Y_{\overline{r}}^{\overline{\alpha}}}\right)\leq_{\Psi}\left(\overline{X_{\overline{r}}^{\overline{\alpha}}}\right)$

###### Proof.

($1\Leftrightarrow 2$): By definition
$\left(X_{r}^{\alpha}\right)=\left(X_{\overline{r}}^{\overline{\alpha}}\right)$,
and so trivially
$\left(X_{r}^{\alpha}\right)\leq_{\Psi}\left(Y_{r}^{\alpha}\right)\Leftrightarrow\left(X_{\overline{r}}^{\overline{\alpha}}\right)\leq_{\Theta}\left(Y_{\overline{r}}^{\overline{\alpha}}\right)$.

($1\Leftrightarrow 3$): Since $\Psi$ is self-conjugate, and
$X_{r}^{\alpha},Y_{r}^{\alpha}$ are linear (and hence $\tau-$morphisms), then
by 7.3 we have
$X_{r}^{\alpha}\leq_{\Psi}Y_{r}^{\alpha}\Leftrightarrow\overline{Y_{r}^{\alpha}}\leq_{\Psi}\overline{X_{r}^{\alpha}}$,
and the result follows.

($2\Leftrightarrow 4$): This follows by taking the duality ($1\Leftrightarrow
3$) and applying the duality ($1\Leftrightarrow 2$). ∎

Analogous results follow for $\overline{\Psi}$.

Recall that
$S_{rs}^{\alpha}\phi\coloneqq\sum_{t=1}^{q_{r}}\phi_{rst}^{\alpha}$

######  Corollary 81.

If $\Psi$ is self-conjugate and $\left(X_{r}^{\alpha}\right)$ is linear,
$\left(\sum_{s=0}^{b_{r}^{\alpha}-1}S_{rs}^{\alpha}\right)\leq_{\Psi}\left(X_{r}^{\alpha}\right)$
on $\Psi$ if and only if
$\left(\overline{X_{\overline{r}}^{\overline{\alpha}}}\right)\leq_{\Psi}\left(\sum_{s=0}^{b_{r}^{\alpha}-1}S_{rs}^{\alpha}\right)$

###### Proof.

This follows directly from the lemma putting
$Y_{r}^{\alpha}=\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$ and using the fact that
$b_{r}^{\alpha}$and $S_{rs}^{\alpha}$ are double self conjugate, and hence by
linearity of the double dual (Proposition 77) so is the sum $Y_{r}^{\alpha}$ .
∎

###  7.3. Partition Sums

We now introduce a set of linear functionals on observables which will form an
important part of our Birkhoff Sum estimates.

######  Definition 82.

Partition Sums. Consider the partition of $[0,1)$ into $k\geq 1$ equal sub-
intervals starting from $0$. We call the $k-1$ points with coordinates
$\left\\{\frac{t}{k}\right\\}_{t=1}^{k-1}$ (ie not including $0$) the interior
points, and define the partition sum of $\phi$ on the partition to be the sum
of values of $\phi$ over the interior points, namely
$P_{k}(\phi)=\sum_{t=1}^{k-1}\phi(\frac{t}{k})$.

######  Remark 83.

Note that $P_{k}$ is a linear functional of $\phi$, and that $P_{1}(\phi)$ is
an empty sum taking the value $0$, and that $P_{2}(\phi)=\phi(\frac{1}{2})$.

Recall from the previous section that we are using the notation
$\left(\overline{.}\right)_{\mathbb{T}^{\circ}}$ for the Source involution of
the circle $x\mapsto\overline{x}=\\{1-x\\}$. This has the pull ups
$\left(\overline{.}\right)_{\left(\mathbb{TR}\right)^{\circ}}:\phi\mapsto\overline{\phi}$,
$\left(\overline{.}\right)_{\left((\mathbb{TR})T\right)^{\circ}}:A\mapsto\overline{A}$
defined by
$\overline{\phi}x=\phi\overline{x},\overline{A}\phi=A\overline{\phi}$

######  Proposition 84.

$P_{k}$ is self-conjugate under the Source involution
$\left(\overline{\text{.}}\right)_{\mathbb{T}}$, and if $\phi_{\mathbb{TR}}$
is anti-symmetric then $P_{k}(\phi)=0$

###### Proof.

By definition, putting $u=k-t$:

$\overline{P_{k}}(\phi)=P_{k}(\overline{\phi})=\sum_{t=1}^{k-1}\phi(1-\frac{t}{k})=\sum_{u=1}^{k-1}\phi(\frac{u}{k})=P_{k}(\phi)$

Now if $\phi$ is anti-symmetric on the circle we have $\phi=-\overline{\phi}$.
Since $P_{k}$ is linear
$P_{k}\left(\phi\right)=P_{k}\left(-\overline{\phi}\right)=-P_{k}\left(\overline{\phi}\right)=-P_{k}\left(\phi\right)$,
hence $P_{k}(\phi)=0$. ∎

We will usually be concerned with partition sums $P_{q_{r}^{\alpha}}$ where
$q_{r}^{\alpha}$ is a quasiperiod of $\alpha_{\mathbb{T}}$.

######  Proposition 85.

$P_{q_{r}^{\alpha}}$ is self-conjugate under the quasiperiod involution
$(\alpha,r)\mapsto(\overline{\alpha},\overline{r})$

###### Proof.

Recall that $q_{r}^{\alpha}$ is itself self-conjugate, ie
$q_{r}^{\alpha}=q_{\overline{r}}^{\overline{\alpha}}$, and the result follows
trivially. ∎

###  7.4. Birkhoff sums of monotonic functions on $(0,1)$

Recall
$S_{N}^{\alpha}\phi=\sum_{r=0}^{n^{\alpha}}\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}\phi$
where
$S_{rs}^{\alpha}\phi=S_{q_{r}^{\alpha}}(\phi,\alpha_{rs})=\sum_{t=1}^{q_{r}^{\alpha}}\phi(\alpha_{rs}+t\alpha)=\sum_{t=1}^{q_{r}^{\alpha}}\phi_{rst}^{\alpha}$.
We will fix $\alpha$ for the moment and drop it from the annotation.

When $q_{r}=1$ we may have $q_{1}=q_{0}=1$. However in this case we also have
$\alpha>\frac{1}{2}$. Also from (18)$b_{0}=0$ so that the $S_{q_{0}}$ term is
not of interest and we can write
$S_{N}^{\alpha}\phi=\sum_{r=\left\llbracket\alpha>\frac{1}{2}\right\rrbracket}^{n}\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}\phi$

Now for $q_{r}=1$, $t\in(1..q_{r})$ gives $t\in(1)$ and so
$S_{q_{r}}(\alpha_{rs})=\sum_{t=1}^{q_{r}}\phi_{rst}=\phi_{rsq_{r}}$ and
similarly for $q_{r}=2$ we have
$S_{q_{r}}(\alpha_{rs})=\sum_{t=1}^{q_{r}}\phi_{rst}=\phi_{rsq_{r}}+\phi_{rsq_{r-1}}$
(since $q_{r-1}=1$). Combining these results we have for $q_{r}\leq 2$ (and
where $b_{r}=0$ gives the empty sum $0$):

(7.4)
$\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}\phi=\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}^{\alpha}}^{\alpha}+\left\llbracket
q_{r}^{\alpha}>1\right\rrbracket\phi_{rsq_{r-1}^{\alpha}}^{\alpha}\right)$

######  Definition 86 (Bounds Operators).

Given $\alpha,N$, the Bounds Functionals $\left(B_{r}^{\alpha}\right)$ form a
QP family of operators in $\left(\mathbb{TR}\right)\mathbb{R}$ defined on
observables $\phi$ by:

(7.5) $B_{r}^{\alpha}\phi=\left\llbracket
b_{r}^{\alpha}>0\right\rrbracket\left(\left\llbracket
q_{r}^{\alpha}>1\right\rrbracket
b_{r}^{\alpha}\left(P_{q_{r}^{\alpha}}(\phi)-\overline{\phi}(\frac{1}{q_{r}^{\alpha}})\right)+\left\llbracket
2<q_{r}^{\alpha}<q_{n^{\alpha}}^{\alpha}\right\rrbracket
E_{r}^{\alpha}\left(\phi_{r0,q_{r}-q_{r-1}}^{\alpha}-\overline{\phi}\left(\frac{2}{q_{r}^{\alpha}}\right)\right)+\sum_{s=0}^{b_{r}^{\alpha}-1}\left(\phi_{rsq_{r}}^{\alpha}+\left\llbracket
q_{r}^{\alpha}>1\right\rrbracket\phi_{rsq_{r-1}}^{\alpha}\right)\right)$

######  Remark 87.

Note this reduces to (7.4) for $q_{r}^{\alpha}\leq 2$.

Further note that each Bounds Functional is an algebraic combinations of other
linear QPI functionals and hence linear .

######  Lemma 88.

The Bounds Functionals $\left(B_{r}^{\alpha}\right)$ have QP duals
$\left(B_{\overline{r}}^{\overline{\alpha}}\right)$ and Double Duals
$\left(\overline{B_{\overline{r}}^{\overline{\alpha}}}\right)$ which can be
written (when $\alpha$ is fixed):

(7.6) $B_{\overline{r}}^{\overline{\alpha}}\phi=\left\llbracket
b_{r}>0\right\rrbracket\left(\left\llbracket q_{r}>1\right\rrbracket
b_{r}\left(P_{q_{r}}(\phi)-\overline{\phi}(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\overline{\phi}_{r0,q_{r}-q_{r-1}}-\overline{\phi}\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\overline{\phi}_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\overline{\phi}_{rsq_{r-1}}\right)\right)$ (7.7)
$\overline{B_{\overline{r}}^{\overline{\alpha}}}\phi=\left\llbracket
b_{r}>0\right\rrbracket\left(\left\llbracket q_{r}>1\right\rrbracket
b_{r}\left(P_{q_{r}}(\phi)-\phi(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\phi\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\phi_{rsq_{r-1}}\right)\right)$

###### Proof.

Applying the QP involution to the definition (7.5) gives us the QP dual:

$B_{\overline{r}}^{\overline{\alpha}}\phi=\left\llbracket
b_{\overline{r}}^{\overline{\alpha}}>0\right\rrbracket\left(\left\llbracket
q_{\overline{r}}^{\overline{\alpha}}>1\right\rrbracket
b_{\overline{r}}^{\overline{\alpha}}\left(P_{q_{\overline{r}}^{\overline{\alpha}}}(\phi)-\overline{\phi}(\frac{1}{q_{\overline{r}}^{\overline{\alpha}}})\right)+\left\llbracket
2<q_{\overline{r}}^{\overline{\alpha}}<q_{n^{\overline{\alpha}}}^{\overline{\alpha}}\right\rrbracket
E_{\overline{r}}^{\overline{\alpha}}\left(\phi_{\overline{r}0,q_{\overline{r}}^{\overline{\alpha}}-q_{\overline{r}-1}^{\overline{\alpha}}}^{\overline{\alpha}}-\overline{\phi}\left(\frac{2}{q_{\overline{r}}^{\overline{\alpha}}}\right)\right)+\sum_{s=0}^{b_{\overline{r}}^{\overline{\alpha}}-1}\left(\phi_{\overline{r}sq_{\overline{r}}^{\overline{\alpha}}}^{\overline{\alpha}}+\left\llbracket
q_{\overline{r}}^{\overline{\alpha}}>1\right\rrbracket\phi_{\overline{r}sq_{\overline{r}-1}^{\overline{\alpha}}}^{\overline{\alpha}}\right)\right)$

But now we have by (**) that
$b_{r}^{\alpha},q_{r}^{\alpha},P_{q_{r}^{\alpha}}$ and
$q_{n^{\alpha}}^{\alpha}$ are all self-conjugate under the QP involution, so
that for example $b_{r}^{\alpha}=b_{\overline{r}}^{\overline{\alpha}}$ and we
can and will denote either of them by $b_{r}$ when $\alpha$ is fixed. Further
$E_{\overline{r}}^{\overline{\alpha}}=O_{r}^{\alpha}=O_{r}$ and
$\phi_{\overline{r}st}^{\overline{\alpha}}=\overline{\phi}_{rst}^{\alpha}$ and
the QP dual result (7.6) follows. For the double dual we have
$\overline{B_{\overline{r}}^{\overline{\alpha}}}\phi\coloneqq
B_{\overline{r}}^{\overline{\alpha}}\overline{\phi}$. From (7.6) we then
obtain:

$B_{\overline{r}}^{\overline{\alpha}}\overline{\phi}=\left\llbracket
b_{r}>0\right\rrbracket\left(\left\llbracket q_{r}>1\right\rrbracket
b_{r}\left(P_{q_{r}}(\overline{\phi})-\phi(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\phi\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\phi_{rsq_{r-1}}\right)\right)$

Now note that $P_{q_{r}}$ is also self-conjugate under
$\phi\mapsto\overline{\phi}$ and (7.7) follows. ∎

######  Theorem 89 (Bounds for monotonic functions).

Given $\alpha,N,r$ and the associated Bounds Functionals
$B_{r}^{\alpha},\overline{B_{\overline{r}}^{\overline{\alpha}}}$, let $\phi$
be a monotonic decreasing observable on $(0,1)$. Then we have:

(7.8) $\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}\phi$
$\displaystyle\leq\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}\phi\leq
B_{r}^{\alpha}\phi$

Further, these are equalities for constant $\phi$ (and also if $b_{r}=0$ or
$q_{r}\leq 2$).

###### Proof.

Given the right hand inequality $\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}\phi\leq
B_{r}^{\alpha}\phi$ we can deduce the left hand inequality via the corollary
to the domination lemma, Corollary 81. It remains to prove the right hand. We
first deal with the equality claims and $q_{r}\leq 2$.

If $b_{r}=0$ then we have the equality
$0=\overline{B_{\overline{r}}^{\overline{\alpha}}}\phi=B_{r}^{\alpha}\phi=\sum_{s=0}^{b_{r}-1}S_{q_{r}}(\alpha_{rs})$.
For $q_{r}\leq 2$, we have
$P_{2}=\phi(\frac{1}{2})=\overline{\phi}(\frac{1}{2})$ and $P_{1}=0$, and so
for $q_{r}\leq 2$ both the first and mid terms of
$\overline{B_{\overline{r}}^{\overline{\alpha}}},B_{r}^{\alpha}$ vanish and we
are left with (7.4) which is an equality. If $q_{r}>2$ and $\phi(x)=c$ then
the mid terms vanish, the first term is $b_{r}\left((q_{r}-1)c-c\right)$ and
the final term becomes $b_{r}(c+c)$ so that $A_{r}\phi=B_{r}\phi=b_{r}q_{r}c$
which is also $\sum_{s=0}^{b_{r}-1}S_{q_{r}}(\alpha_{rs})$. We now deal with
the general case for $q_{r}>2$.

We now use the results of Section 5 to understand where $\alpha_{rst}$ lies in
the regular $q_{r}$ partition of the circle.

Case: $s>0$ or $s=0\&r=n$.

In this case for each $t$, the co-siting condition holds, ie $\alpha_{rst}$
lies in $I(t\alpha)$ so that $\phi_{rst}$ lies between the values of $\phi$ at
the interval endpoints. Summing over increasing values of $\alpha_{rst}$ (ie
in the direction $0$ to $1$) gives us:

For $r$ even we have
$S_{q_{r}}(\alpha_{rs})=\sum_{t=1}^{q_{r}}\phi(\alpha_{rst})\geq\phi_{rsq_{r}}+\left(P_{q_{r}}-\phi(\frac{1}{q_{r}})\right)+\phi_{rsq_{r-1}}$
and
$S_{q_{r}}(\alpha_{rs})\leq\phi_{rsq_{r}}+\left(P_{q_{r}}-\phi(1-\frac{1}{q_{r}})\right)+\phi_{rsq_{r-1}}$.

Similarly for $r$ odd we have
$\phi_{rsq_{r-1}}+\left(P_{q_{r}}-\phi(1-\frac{1}{q_{r}})\right)+\phi_{rsq_{r}}\geq
S_{q_{r}}(\alpha_{rs})\geq\phi_{rsq_{r-1}}+\left(P_{q_{r}}-\phi(\frac{1}{q_{r}})\right)+\phi_{rsq_{r}}$
which is structurally identical to the $r$ even case, even though
$\alpha_{rsq_{r}},\alpha_{rsq_{r-1}}$ are switched in their positions around
$0$.

Case: $s=0\,\&\,r<n,q_{r}>1$.

In this case for each $t\neq q_{r}$, $\alpha_{rst}$ may be shifted by one
partition interval from $I(t\alpha)$. In particular,
$\alpha_{rs(q_{r}-q_{r-1})}$ may be shifted into the same partition interval
as $\alpha_{rsq_{r}}$.

For $r$ even, the possible shift is backward (towards 0), so that the lower
bound is not affected, but the upper bound increases to
$S_{q_{r}}(\alpha_{rs})\leq\left(P_{q_{r}}-\phi(1-\frac{1}{q_{r}})\right)+\phi_{r0q_{r}}+\phi_{r0,q_{r}-q_{r-1}}+\left(\phi_{rsq_{r-1}}-\phi(1-\frac{2}{q_{r}})\right)=\left(P_{q_{r}}+\phi_{rsq_{r}}+\phi_{rsq_{r-1}}-\phi(1-\frac{1}{q_{r}})\right)+\left(\phi_{r0,q_{r}-q_{r-1}}-\phi(1-\frac{2}{q_{r}})\right)$.

For $r$ odd, the possible shift is forward (towards 1), so that the upper
bound is not affected, but the lower bound decreases to
$S_{q_{r}}(\alpha_{rs})\geq\left(P_{q_{r}}-\phi\left(\frac{1}{q_{r}}\right)\right)+\left(\phi_{rsq_{r-1}}-\phi(\frac{2}{q_{r}})\right)+\phi_{rsq_{r}}+\phi_{r0,q_{r}-q_{r-1}}=\left(\phi_{rsq_{r}}+P_{q_{r}}-\phi(\frac{1}{q_{r}})+\phi_{rsq_{r-1}}\right)+\left(\phi_{r0,q_{r}-q_{r-1}}-\phi(\frac{2}{q_{r}})\right)$.

The results follow on reorganising the terms.

∎

######  Corollary 90.

$\sum_{r=0}^{n}\overline{B_{\overline{r}}^{\overline{\alpha}}}\phi\leq
S_{N}\phi\leq\sum_{r=0}^{n}B_{r}^{\alpha}\phi$

###### Proof.

This follows from
$S_{N}\phi=\sum_{r=0}^{n}\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}\phi$ ∎

######  Corollary 91 (Duality results).

Let $\Psi\subset ST$ be a $\sigma\tau$ self-conjugate set of decreasing
functions on $(0,1)$. Then we have:

$\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}$
$\displaystyle\>\leq_{\Psi}$
$\displaystyle\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$
$\displaystyle\>\leq_{\Psi}\;$ $\displaystyle B_{r}^{\alpha}$
$\displaystyle\sigma\overline{B_{\overline{r}}^{\overline{\alpha}}}$
$\displaystyle\>\leq_{\overline{\Psi}}$
$\displaystyle\sigma\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$
$\displaystyle\>\leq_{\overline{\Psi}}\;$ $\displaystyle\sigma B_{r}^{\alpha}$
$\displaystyle\tau\overline{B_{\overline{r}}^{\overline{\alpha}}}$
$\displaystyle\>\leq_{\overline{\Psi}}$
$\displaystyle\tau\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$
$\displaystyle\>\leq_{\overline{\Psi}}\;$ $\displaystyle\tau B_{r}^{\alpha}$
$\displaystyle\sigma\tau\overline{B_{\overline{r}}^{\overline{\alpha}}}$
$\displaystyle\>\leq_{\Psi}$
$\displaystyle\;\sigma\tau\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$
$\displaystyle\>\leq_{\Psi}\;$ $\displaystyle\sigma\tau B_{r}^{\alpha}$

$\displaystyle\tau\overline{B_{\overline{r}}^{\overline{\alpha}}}$
$\displaystyle\>\geq_{\Psi}$
$\displaystyle\tau\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$
$\displaystyle\>\geq_{\Psi}\;$ $\displaystyle\tau B_{r}^{\alpha}$
$\displaystyle\tau\sigma\overline{B_{\overline{r}}^{\overline{\alpha}}}$
$\displaystyle\>\geq_{\overline{\Psi}}$
$\displaystyle\;\tau\sigma\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$
$\displaystyle\>\geq_{\overline{\Psi}}\;$ $\displaystyle\tau\sigma
B_{r}^{\alpha}$ $\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}$
$\displaystyle\>\geq_{\overline{\Psi}}$
$\displaystyle\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$
$\displaystyle\>\geq_{\overline{\Psi}}\;$ $\displaystyle B_{r}^{\alpha}$
$\displaystyle\sigma\overline{B_{\overline{r}}^{\overline{\alpha}}}$
$\displaystyle\>\geq_{\Psi}$
$\displaystyle\sigma\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}$
$\displaystyle\>\geq_{\Psi}\;$ $\displaystyle\sigma B_{r}^{\alpha}$

###### Proof.

This is an application of the dualities in Subsection 7.2. ∎

######  Corollary 92.

If $\phi$ is a decreasing anti-symmetric function then

(7.9) $\displaystyle B_{r}^{\alpha}\phi$ $\displaystyle=\left\llbracket
b_{r}>0\right\rrbracket\left(\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\left(\phi_{rsq_{r-1}}+\phi(\frac{1}{q_{r}})\right)\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
E_{r}\left(\phi_{r0,q_{r}-q_{r-1}}+\phi\left(\frac{2}{q_{r}}\right)\right)\right)$
(7.10) $\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}\phi$
$\displaystyle=\left\llbracket
b_{r}>0\right\rrbracket\left(\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\left(\phi_{rsq_{r-1}}-\phi(\frac{1}{q_{r}})\right)\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\phi\left(\frac{2}{q_{r}}\right)\right)\right)$

Further for $q_{r}>1$ each sum
$\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\phi_{rsq_{r-1}}\right)$ is positive
for $r$ even, and negative for $r$ odd.

.

###### Proof.

Note if $\phi$ is anti-symmetric then $\phi=-\overline{\phi}$ and
$P_{q_{r}}(\phi)=0$. The expressions (7.9,7.10) then follow easily from the
theorem. We now examine
$X_{r}=\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\phi_{rsq_{r-1}}\right)$. For
$r$ even
$\phi_{rsq_{r}}+\phi_{rsq_{r-1}}\leq\phi\left(\frac{s+1/a_{r+2}^{/}}{q_{r+1}^{/}}\right)-\phi\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$.

Reversing the order of second terms gives us
$X_{r}=\sum_{s=0}^{b_{r}-1}\left(\phi\left(\frac{s+1/a_{r+2}^{/}}{q_{r+1}^{/}}\right)-\phi\left(\frac{a_{r+1}-(b_{r}-1)+s}{q_{r+1}^{/}}\right)\right)$.
Since $b_{r}\leq a_{r+1},$ each term in this sum is positive, and hence
$X_{r}\geq 0$ for $r$ even. Similarly $X_{r}\leq 0$ for $r$ odd. ∎

###  7.5. Further bounds for primitive functions

The results of the previous section hold for any monotonic decreasing $\phi$.
In this section we will develop the upper bound results further for those
functions which are lower bounded, and in particular for the positive
primitives in $\Phi$.. Recall from Subsection 6.4 we can reduce any unbounded
observable to a sum of primitives.

The key new ingredient arises from the fact that if $\phi{}_{\Phi}$ is lower
bounded at say $1^{-}$ with bound $c$, then for irrational
$0<\alpha<\frac{1}{2}$ the sequence of extremal points $\alpha_{rsq_{r}}$ tend
to $1^{-}$ for $r$ odd and hence $\phi_{rsq_{r}}\rightarrow c$.

Recall our notation for limits when they exist, for example
$\phi(1^{-})=\lim_{x\uparrow 1}\phi(x)$. Note the limit exists whenever $\phi$
is LBV left of $1$, and in particular for $\phi\in\Phi\bigcup-\Phi$. Also note
that $\phi\mapsto\phi(1^{-})$ is then a linear functional, and in particular
$(-\phi)(1^{-})=-\phi(1^{-})$. Analogous remarks apply to
$\phi(0^{+})=\lim_{x\downarrow 0}\phi(x)$. Note that if $\phi(1^{-})$ exists,
so does $\overline{\phi}(0^{+})$ and the two limits are equal.

Finally for $\phi$ monotonic descending with $\phi(1^{-})=C<0$ then
$\phi-C\in\Phi$ and $S_{N}\phi=S_{N}(\phi-C)+CN$, which means we can restrict
our attention to $\phi\in\Phi$. Similar remarks apply to all quadrants.

######  Lemma 93 (Upper bounds for primitive functions).

Let $\phi\in\Phi$. Then for $q_{r}>2$ we have:

(7.11) $\displaystyle B_{r}^{\alpha}(\phi)$ $\displaystyle\leq\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}P_{q_{r}}+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
E_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\overline{\phi}\left(\frac{1}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(E_{r}\phi_{rsq_{r}}+O_{r}\phi_{rsq_{r-1}}\right)\right)$
(7.12)
$\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}\overline{\phi}$
$\displaystyle\leq\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}P_{q_{r}}+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\overline{\phi}_{r0,q_{r}-q_{r-1}}-\overline{\phi}\left(\frac{1}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(O_{r}\overline{\phi}_{rsq_{r}}+E_{r}\overline{\phi}_{rsq_{r-1}}\right)\right)$

Further, these inequalities are reversed under the duality $\phi\mapsto-\phi$,
and are equalities for constant $\phi$ (and trivially also for $b_{r}=0$).

###### Proof.

All terms are linear functionals, so that results for $-\phi$ follow by
duality. The equalities follow the same argument as Lemma 89.

Since $\phi\in\Phi$, $\phi$ is descending and we have from Lemma 89 for
$q_{r}>2$

(7.13) $\displaystyle B_{r}^{\alpha}(\phi)$ $\displaystyle=\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}\left(P_{q_{r}}-\overline{\phi}(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
E_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\overline{\phi}\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\phi_{rsq_{r-1}}\right)\right)$
(7.14) $\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}(\phi)$
$\displaystyle=\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}\left(P_{q_{r}}-\phi(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\phi\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\phi_{rsq_{r-1}}\right)\right)$

Also since $\phi$ is descending, for $q_{r}>1$ we have
$O_{r}\phi_{rsq_{r}}\leq O_{r}\overline{\phi}(\frac{1}{q_{r}})$, and for $s>0$
or $r=n$ we also have $E_{r}\phi_{rsq_{r-1}}\leq
E_{r}\overline{\phi}(\frac{1}{q_{r}})$. However for $s=0\&r<n$ we may have
$\alpha_{r0q_{r-1}}$ pulled back to its alternate interval.

However in this case, since $q_{r}>2$ we have $E_{r}\phi_{r0q_{r-1}}\leq
E_{r}\overline{\phi}(\frac{2}{q_{r}})$ which then gives us
$\phi_{rsq_{r}}+\phi_{rsq_{r-1}}-\overline{\phi}(\frac{1}{q_{r}})\leq
E_{r}\phi_{rsq_{r}}+O_{r}\phi_{rsq_{r-1}}+\left\llbracket
s=0\&r<n\right\rrbracket
E_{r}(\overline{\phi}(\frac{2}{q_{r}})-\overline{\phi}(\frac{1}{q_{r}}))$. For
$b_{r}>0$ we can now substitute this result into (7.13) to obtain (7.12). The
result for $B_{r}(\overline{\phi})$ in (7.12)) follows by analogous argument
using $\overline{\phi}$ ascending.

∎

######  Lemma 94.

Let $\phi\in\Phi$. Then for $q_{r}=2$ we have:

(7.15) $\displaystyle B_{r}^{\alpha}\phi$ $\displaystyle\leq\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}P_{q_{r}}+\left\llbracket
2=q_{r}<q_{n}\right\rrbracket
E_{r}(\phi_{r0q_{r-1}}^{\alpha}-\overline{\phi}(\frac{1}{q_{r}}))+\sum_{s=0}^{b_{r}-1}\left(E_{r}\phi_{rsq_{r}}^{\alpha}+O_{r}\phi_{rsq_{r-1}}^{\alpha}\right)\right)$
(7.16)
$\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}\overline{\phi}$
$\displaystyle\leq\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}P_{q_{r}}+\left\llbracket
2=q_{r}<q_{n}\right\rrbracket
O_{r}(\overline{\phi}_{r0q_{r-1}}^{\alpha}-\overline{\phi}(\frac{1}{q_{r}}))+\sum_{s=0}^{b_{r}-1}\left(O_{r}\overline{\phi}_{rsq_{r}}^{\alpha}+E_{r}\overline{\phi}_{rsq_{r-1}}^{\alpha}\right)\right)$

Writing $X_{r}^{\alpha}(\phi)=\left\llbracket 2=q_{r}<q_{n}\right\rrbracket
E_{r}\left(\phi_{r0q_{r-1}}^{\alpha}-\overline{\phi}(\frac{1}{q_{r}})\right)$
for $q_{r}=2$ we have
$\left|X_{r}(\phi)\right|<\phi(\frac{1}{3})-\phi(\frac{1}{2})$. If
$X_{r}\phi\neq 0$ then
$-\operatorname{sgn}X_{r}\overline{\phi}=\left\llbracket\left\\{\alpha\right\\}<\frac{1}{2}\right\rrbracket=\operatorname{sgn}X_{r}\phi$,
so that the $X_{r}$ term can be ignored in 7.16 for
$\left\llbracket\left\\{\alpha\right\\}<\frac{1}{2}\right\rrbracket$, and in
7.16 for $\left\llbracket\left\\{\alpha\right\\}>\frac{1}{2}\right\rrbracket$.

Further, these inequalities are reversed under the duality $\phi\mapsto-\phi$,
and are equalities for constant $\phi$ (and also trivially for $b_{r}=0$).

###### Proof.

All terms are linear functionals, so that results for $-\phi$ follow by
duality. The equalities again follow the same argument as Lemma 89 above.

Since $\phi\in\Phi$, $\phi$ is descending and we have from Lemma 89 for
$q_{r}=2$

(7.17) $\displaystyle B_{r}(\phi)$ $\displaystyle=\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}\left(P_{q_{r}}-\overline{\phi}(\frac{1}{q_{r}})\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\phi_{rsq_{r-1}}\right)\right)$
(7.18) $\displaystyle\overline{B}_{r}(\phi)$ $\displaystyle=\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}\left(P_{q_{r}}-\phi(\frac{1}{q_{r}})\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\phi_{rsq_{r-1}}\right)\right)$

Also since $\phi$ is descending, for $q_{r}>1$ we have
$O_{r}\phi_{rsq_{r}}\leq O_{r}\overline{\phi}(\frac{1}{q_{r}})$, and for $s>0$
or $r=n$ we also have $E_{r}\phi_{rsq_{r-1}}\leq
E_{r}\overline{\phi}(\frac{1}{q_{r}})$. However for $s=0\&r<n$ we may have
$\alpha_{r0q_{r-1}}$ pulled back to its alternate interval which gives us
$\phi_{rsq_{r}}+\phi_{rsq_{r-1}}-\overline{\phi}(\frac{1}{q_{r}})\leq
E_{r}\phi_{rsq_{r}}+O_{r}\phi_{rsq_{r-1}}+\left\llbracket
s=0\&r<n\right\rrbracket
E_{r}(\phi_{r0q_{r-1}}-\overline{\phi}(\frac{1}{q_{r}}))$. This establishes
7.15 and then 7.16 follows as its qp-dual.

If $\phi$ is a constant function, the mid-term is zero. We now investigate
requirements for the mid-terms to be either zero or positive (non-negative).
Note that if $q_{r}=2$ we have $q_{r-1}=1$ and $r\in\\{1,2\\}$.

For $r=1$ we have $E_{r}=0$ and so the mid-term vanishes in $B_{1}\phi$.
However $O_{r}=1$ and so the mid-term in $B_{1}^{\alpha}\overline{\phi}$ is
$\overline{\phi}_{101}^{\alpha}-\overline{\phi}(\frac{1}{2})=\overline{X}_{1}^{\alpha}\overline{\phi}$
which is non-negative for $\alpha_{101}\geq\frac{1}{2}$. But
$\alpha_{101}<1-\left(\frac{1}{2}+\frac{1}{q_{r+2}^{/}}-\frac{1}{2q_{r+1}^{/}}\right)<\frac{1}{2}+\frac{1}{2q_{2}^{/}}$and
$q_{2}^{/}>3$ so $\alpha_{101}<\frac{1}{2}+\frac{1}{2.3}=\frac{2}{3}$. Also
for $\alpha_{101}\geq\frac{1}{2}$ we must have
$\frac{1}{q_{r+2}^{/}}\leq\frac{1}{2q_{r+1}^{/}}$ giving $a_{r+2}^{/}\geq 2$,
requiring $a_{3}\geq 2$. Finally $q_{1}=q_{r}=2$ which means $a_{1}=2$ and
hence $\frac{1}{3}<\alpha<\frac{1}{2}$.

For $r=2$ we have $O_{r}=0$ and so the mid-term vanishes in
$B_{2}\overline{\phi}$. However $E_{r}=1$ and so the mid-term in $B_{2}\phi$
is $\phi_{201}-\overline{\phi}(\frac{1}{2})$. We now use the identity
$\phi_{rst}^{\alpha}=\overline{\phi}_{\overline{r}st}^{\overline{\alpha}}$ to
get
$\phi_{201}^{\alpha}-\overline{\phi}(\frac{1}{2})=\overline{\phi}_{101}^{\overline{\alpha}}-\overline{\phi}(\frac{1}{2})$.
But this is the mid term of $B_{1}^{\overline{\alpha}}\overline{\phi}$ and so
by the previous result, non-negative requires
$\frac{1}{3}<\overline{\alpha}<\frac{1}{2},\frac{1}{2}<\overline{\alpha}_{101}<\frac{2}{3}$
and
$a_{1}^{\alpha}=1,a_{2}^{\alpha}=a_{1}^{\overline{\alpha}}-1=1,a_{4}^{\alpha}=a_{3}^{\overline{\alpha}f}\geq
2$ which establishes the result. ∎

Note that the constant $X_{r}\phi$ in (7.16) is generally a small constant
which will be insignificant for most purposes, and is often negative in which
case it can be ignored entirely in these inequalities. The conditions for it
to be positive are necessary but not sufficient. In the following example
$X_{2}\phi>0$ for strictly descending $\phi$, showing that this term cannot be
eliminated in the inequality.

######  Example 95.

$\alpha=[2,1,3,100,...]\approx
0.36,N=8=2*3+1*2+0*1,(101)=7,\alpha_{101}=\\{7\alpha\\}\approx
0.54>\frac{1}{2}$.

Analysis of Upper Bound Components

(7.19) $\displaystyle B_{r}^{\alpha}\phi$ $\displaystyle\leq\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}P_{q_{r}}+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
E_{r}\left(\phi_{r0,q_{r}-q_{r-1}}^{\alpha}-\overline{\phi}\left(\frac{1}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(E_{r}\phi_{rsq_{r}}^{\alpha}+\left\llbracket
q_{r}>1\right\rrbracket O_{r}\phi_{rsq_{r-1}}^{\alpha}\right)\right)$ (7.20)
$\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}\overline{\phi}$
$\displaystyle\leq\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}P_{q_{r}}+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\overline{\phi}_{r0,q_{r}-q_{r-1}}^{\alpha}-\overline{\phi}\left(\frac{1}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(O_{r}\overline{\phi}_{rsq_{r}}^{\alpha}+\left\llbracket
q_{r}>1\right\rrbracket
E_{r}\overline{\phi}_{rsq_{r-1}}^{\alpha}\right)\right)$

We now analyse further the third terms above. Using values of $\alpha_{rst}$
We now exploit bounds for $\alpha_{rst}$ developed in Section 5.

First we need to treat the case $q_{r}=1$ separately.

######  Lemma 96.

For $q_{r}=1$ we have $P_{q_{r}}=0$ and then
$B_{r}^{\alpha}\phi=\sum_{s=0}^{b_{r}-1}E_{r}\phi_{rsq_{r}}^{\alpha}$

(7.21)
$\sum_{s=0}^{b_{r}-1}\phi\left(\frac{s+1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)\leq\overline{B}_{r}(\phi)=\sum_{s=0}^{b_{r}-1}\phi_{rsq_{r}}=B_{r}(\phi)\leq\sum_{s=0}^{b_{r}-1}\phi\left(\frac{s}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)$

where the inequalities are reversed under the dualities
$\phi\mapsto-\phi,\phi\mapsto\overline{\phi}$, and become equalities for
constant $\phi$. The upper bounds also give
$\sum_{s=0}^{b_{r}-1}\phi_{rsq_{r}}=B_{r}\phi\leq\phi\left(\frac{1}{q_{r+2}^{/}}\right)+\sum_{s=1}^{b_{r}-1}\phi\left(\frac{s}{q_{r+1}^{/}}\right)$
and
$\sum_{s=0}^{b_{r}-1}\overline{\phi}_{rsq_{r}}=B_{r}\overline{\phi}\leq\sum_{s=1}^{b_{r}}\overline{\phi}\left(\frac{a_{r+1}^{/}-s}{q_{r+1}^{/}}\right)$

###### Proof.

Since $\phi\in\Phi$, $\phi$ is monotonic descending and we have by definition
(see 7.5) for any monotonic descending $\phi$ and $q_{r}=1$:

(7.22) $\displaystyle B_{r}(\phi)$
$\displaystyle=\overline{B}_{r}(\phi)=\left\llbracket
b_{r}>0\right\rrbracket\sum_{s=0}^{b_{r}-1}\phi_{rsq_{r}}$

If $b_{r}=0$ the sum is empty so the condition $\left\llbracket
b_{r}>0\right\rrbracket$ becomes redundant which establishes the central
equalities in (7.21) The rest of (7.21) then follows from (48). The duality
result follows from (2.4).

The second upper bound for $\sum_{s=0}^{b_{r}-1}\phi_{rsq_{r}}$ is just a
rewrite of (7.21). Using duality we have
$\sum_{s=0}^{b_{r}-1}\overline{\phi}_{rsq_{r}}\leq\sum_{s=0}^{b_{r}-1}\overline{\phi}\left(\frac{s+1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)$

Now using
$\frac{1}{q_{r+2}^{/}}+\frac{a_{r+1}-s}{q_{r+1}^{/}}=\frac{a_{r+1}^{/}-s}{q_{r+1}^{/}}$
from (16) we get
$\frac{s+1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}=\frac{a_{r+1}^{/}-(a_{r+1}-s-1)}{q_{r+1}^{/}}=\frac{a_{r+1}^{/}-u}{q_{r+1}^{/}}$
where $u=a_{r+1}-s-1$ so
$\sum_{s=0}^{b_{r}-1}\overline{\phi}\left(\frac{s+1}{q_{r+1}^{/}}+\frac{1}{q_{r+2}^{/}}\right)=\sum_{u=a_{r+1}-b_{r}}^{a_{r+1}-1}\overline{\phi}\left(\frac{a_{r+1}^{/}-u}{q_{r+1}^{/}}\right)\leq\sum_{u=a_{r+1}-b_{r}-t}^{a_{r+1}-1-t}\overline{\phi}\left(\frac{a_{r+1}^{/}-u}{q_{r+1}^{/}}\right)$
for $t\geq 0$. We put $t=a_{r+1}-b_{r}-1$ to get
$\sum_{u=1}^{b_{r}}\overline{\phi}\left(\frac{a_{r+1}^{/}-u}{q_{r+1}^{/}}\right)$.

∎

We now look at the general case where $q_{r}>1$. For convenience we define
some further shorthand notation:

######  Definition 97.

Define $C_{rs}^{\alpha}\phi\coloneqq
E_{r}\phi_{rsq_{r}}^{\alpha}+\left\llbracket q_{r}>1\right\rrbracket
O_{r}\phi_{rsq_{r-1}}^{\alpha}$ (from (7.20)and its double -dual
$\overline{C}_{rs}^{\alpha}\overline{\phi}\coloneqq
O_{r}\overline{\phi}_{rsq_{r}}^{\alpha}+\left\llbracket
q_{r}>1\right\rrbracket E_{r}\overline{\phi}_{rsq_{r-1}}^{\alpha}$

For $\phi\in\Phi$ (ie monotone decreasing, bounded below), we now investigate
upper bounds for the $q_{r}>1$ terms in $C_{rs}\phi$.

Using the bounds on $\alpha_{rst}$ from 44 (**Need a clear summary of
$\alpha_{rst}$ results for here!!) gives:

(7.23) $\displaystyle
C_{rs}\phi=\left(\phi_{rsq_{r}}^{E}+\phi_{rsq_{r-1}}^{O}\right)$
$\displaystyle=\phi^{E}\left(\frac{s+1}{q_{r+1}^{/}}\right)+\phi^{O}\left(\frac{1}{q_{r+2}^{/}}+\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$
$\displaystyle r=n$
$\displaystyle\geq\phi^{E}\left(\frac{1}{q_{r+2}^{/}}+\frac{s+1}{q_{r+1}^{/}}\right)+\phi^{O}\left(\frac{a_{r+1}-\left(s-1\right)}{q_{r+1}^{/}}\right)$
$\displaystyle r<n$
$\displaystyle\leq\phi^{E}\left(\frac{1}{q_{r+2}^{/}}+\frac{s}{q_{r+1}^{/}}\right)+\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$
$\displaystyle r<n$

and

(7.24)
$\displaystyle\overline{C}_{rs}\overline{\phi}=\left(\overline{\phi}_{rsq_{r}}^{O}+\overline{\phi}_{rsq_{r-1}}^{E}\right)$
$\displaystyle=\phi^{O}\left(\frac{s+1}{q_{r+1}^{/}}\right)+\phi^{E}\left(\frac{1}{q_{r+2}^{/}}+\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$
$\displaystyle r=n$
$\displaystyle\geq\phi^{O}\left(\frac{1}{q_{r+2}^{/}}+\frac{s+1}{q_{r+1}^{/}}\right)+\phi^{E}\left(\frac{a_{r+1}-\left(s-1\right)}{q_{r+1}^{/}}\right)$
$\displaystyle r<n$
$\displaystyle\leq\phi^{O}\left(\frac{1}{q_{r+2}^{/}}+\frac{s}{q_{r+1}^{/}}\right)+\phi^{E}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$
$\displaystyle r<n$

Grouping terms by quasiperiod

Note in the sums (7.23) that the denominator of each fraction is of order
$q_{r+1}^{/}$ with the exceptions of the upper bounds for
$\phi_{rsq_{r}}^{E},\overline{\phi}_{rsq_{r}}^{O}$ for $s=0,r<n$ (**check
r=0???) which are of order $q_{r+2}^{/}$.

But the size of $\phi\left(\frac{1}{q_{r}^{/}}\right)$ is sensitively
dependent on the size of $q_{r}^{/}$ when $\phi$ is unbounded at $0^{+}$, and
so we will now group terms by denominator, as follows.

######  Definition 98.

Let $C_{r}$ be the sum of the terms in the double sum
$DS\phi=\sum_{r=0}^{n}\sum_{s=1}^{b_{r}-1}C_{rs}\phi$ with denominator
$q_{r+1}^{/}$, so that $DS(\phi)=\sum_{r=0}^{n}C_{r}(\phi)$.

For $q_{r}=1$, we have from (7.21)
$C_{r}(\phi)=\sum_{s=1}^{b_{r}-1}\phi\left(\frac{s}{q_{r+1}^{/}}\right)$ and
$\overline{C}_{r}(\phi)=\sum_{s=1}^{b_{r}}\overline{\phi}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$.

Then by inspection of (7.23), promoting the terms with denominators
$q_{r+2}^{/}$ we have for $r\geq 1$:

(7.25) $\displaystyle C_{n}(\phi)$
$\displaystyle=\sum_{s=0}^{b_{n}-1}\phi^{E}\left(\frac{s+1}{q_{n+1}^{/}}\right)+\sum_{s=0}^{b_{n}-1}\phi^{O}\left(\frac{a_{n+1}^{/}-s}{q_{n+1}^{/}}\right)\,\,+\left\llbracket
b_{n-1}>0\right\rrbracket O_{r}\phi\left(\frac{1}{q_{n+1}^{/}}\right)$
$\displaystyle r=n$ $\displaystyle C_{r}(\phi)$
$\displaystyle\leq\sum_{s=1}^{b_{r}-1}\phi^{E}\left(\frac{1}{q_{r+2}^{/}}+\frac{s}{q_{r+1}^{/}}\right)+\sum_{s=0}^{b_{r}-1}\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)\,\,+\left\llbracket
b_{r-1}>0\right\rrbracket O_{r}\phi\left(\frac{1}{q_{r+1}^{/}}\right)$
$\displaystyle r<n$

Note that at this point we have simply rearranged terms using the originating
inequalities (7.23). However it is difficult to make use of all the
information in (7.25) except in special circumstances such as periodic partial
quotients. Our focus in this paper is on the general case, and so we now
develop some relaxations of (7.25) which are less precise but more tractable
for our purposes. Since $\phi$ is decreasing we have
$\phi^{E}\left(\frac{1}{q_{r+2}^{/}}+\frac{s}{q_{r+1}^{/}}\right)\leq\phi^{E}\left(\frac{s}{q_{r+1}^{/}}\right)$
and
$\phi^{O}\left(\frac{a_{r+1}^{/}-s}{q_{r+1}^{/}}\right)\leq\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$,
which with some rewriting results in the slightly less precise but more
uniform inequalities:

(7.26) $\displaystyle C_{n}(\phi)$
$\displaystyle\leq\sum_{s=1}^{b_{n}}\phi^{E}\left(\frac{s}{q_{n+1}^{/}}\right)+\left\llbracket
b_{n-1}>0\right\rrbracket\phi^{O}\left(\frac{1}{q_{n+1}^{/}}\right)+\sum_{s=0}^{b_{n}-1}\phi^{O}\left(\frac{a_{n+1}-s}{q_{n+1}^{/}}\right)$
$\displaystyle C_{r}(\phi)$
$\displaystyle\leq\sum_{s=1}^{b_{r}-1}\phi^{E}\left(\frac{s}{q_{r+1}^{/}}\right)+\left\llbracket
b_{r-1}>0\right\rrbracket\phi^{O}\left(\frac{1}{q_{r+1}^{/}}\right)+\sum_{s=0}^{b_{r}-1}\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$

Note that for $r$ even $C_{r}$ has $b_{r}-1$ or $b_{r}$ (for $r=n$ only) non-
zero terms, whereas for $r$ odd $C_{r}$ has $b_{r}$ or $b_{r}+1$ (for
$b_{r-1}>0$) non-zero terms. We will use this form for one particular
application in the next section, but for many purposes we can afford to relax
the inequalities still further to the point where we can eliminate the
asymmetry.

We first use the fact that $\phi$ is descending so that for $b_{r}<a_{r+1}$ we
have
$\sum_{s=0}^{b_{r}-1}\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)=\sum_{s=a_{r+1}-(b_{r}-1)}^{a_{r+1}}\phi^{O}\left(\frac{s}{q_{r+1}^{/}}\right)\leq\sum_{s=2}^{b_{r}+1}\phi^{O}\left(\frac{s}{q_{r+1}^{/}}\right)$.
Using $\left\llbracket b_{r-1}>0\right\rrbracket=1-\left\llbracket
b_{r-1}=0\right\rrbracket$ and $\phi$ descending gives

$\left\llbracket
b_{r-1}>0\right\rrbracket\phi^{O}\left(\frac{1}{q_{r+1}^{/}}\right)+\sum_{s=0}^{b_{r}-1}\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)\leq\sum_{s=1}^{b_{r}+1}\phi^{O}\left(\frac{s}{q_{r+1}^{/}}\right)-\left\llbracket
b_{r-1}=0\right\rrbracket\phi^{O}\left(\frac{1}{q_{r+1}^{/}}\right)$

If $b_{r}=a_{r+1}$ we have
$\sum_{s=0}^{b_{r}-1}\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)=\sum_{s=1}^{a_{r+1}}\phi^{O}\left(\frac{s}{q_{r+1}^{/}}\right)$,
and we also have from(18) that $b_{r-1}=0$ so that
$\sum_{s=0}^{b_{r}-1}\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)=\sum_{s=1}^{a_{r+1}}\phi^{O}\left(\frac{s}{q_{r+1}^{/}}\right)$.

In addition we have
$\phi^{E}\left(\frac{1}{q_{r+2}^{/}}+\frac{s}{q_{r+1}^{/}}\right)\leq\phi^{E}\left(\frac{s}{q_{r+1}^{/}}\right)$
and
$\phi^{O}\left(\frac{a_{r+1}^{/}-s}{q_{r+1}^{/}}\right)\leq\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)$.
Hence, writing
$c_{r}^{\alpha}=c(N,\alpha,r)=E_{r}\left(b_{r}-1+\left\llbracket
r=n\right\rrbracket\right)+O_{r}\min(b_{r}+1,a_{r+1})$, we can combine all
these results for $2\leq r\leq n$ into:

(7.27) $\displaystyle C_{r}(\phi)$
$\displaystyle\leq\sum_{s=1}^{c_{r}^{\alpha}}\phi\left(\frac{s}{q_{r+1}^{/}}\right)\,\,-\left\llbracket
b_{r-1}=0\right\rrbracket\phi^{O}\left(\frac{1}{q_{r+1}^{/}}\right)$

Finally, if $\phi\left(\frac{1}{q_{r+1}^{/}}\right)\geq 0$ we can reduce this
to $C_{r}\leq\sum_{s=1}^{c_{r}}\phi\left(\frac{s}{q_{r+1}^{/}}\right)$. Note
that for $r$ odd, this adds significant imprecision if $b_{r-1}=0$, but then
we cannot use this information if we do not know $b_{r-1}=0$, and in general
we will not.

The argument for $\overline{\phi}$ is precisely analogous giving for
$\overline{c}_{r}^{\alpha}=c_{\overline{r}}^{\overline{\alpha}}=O_{r}\left(b_{r}-1+\left\llbracket
r=n\right\rrbracket\right)+E_{r}\min(b_{r}+1,a_{r+1})$, and $1\leq r\leq n$:

(7.28) $\displaystyle\overline{C}_{r}^{\alpha}(\overline{\phi})$
$\displaystyle\leq\sum_{s=1}^{\overline{c}_{r}^{\alpha}}\phi\left(\frac{s}{q_{r+1}^{/}}\right)\,\,-\left\llbracket
b_{r-1}=0\right\rrbracket\phi^{E}\left(\frac{1}{q_{r+1}^{/}}\right)$

###  7.6. Summary of Results

For convenience we summarise here the results of this Section.

Recall $S_{N}\phi=\sum_{r=0}^{n}\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}\phi$ where
$S_{rs}^{\alpha}\phi=S_{q_{r}}^{\alpha}(\phi,\alpha_{rs})=\sum_{t=1}^{q_{r}}\phi(\alpha_{rs}+t\alpha)=\sum_{t=1}^{q_{r}}\phi_{rst}^{\alpha}$.

######  Lemma 99 (Bounds for monotonic functions).

Let $\phi$ be a monotonic decreasing observable. Then for $r\geq 0$ we have
$\overline{B_{\overline{r}}^{\overline{\alpha}}}\phi\leq\sum_{s=0}^{b_{r}-1}S_{rs}^{\alpha}\phi\leq
B_{r}^{\alpha}\phi$ where
$B_{r}^{\alpha},\overline{B_{\overline{r}}^{\overline{\alpha}}}$ are the Bound
Functionals:

(7.29) $\displaystyle B_{r}^{\alpha}\phi$ $\displaystyle=\left\llbracket
b_{r}>0\right\rrbracket\left(\left\llbracket q_{r}>1\right\rrbracket
b_{r}\left(P_{q_{r}}-\overline{\phi}(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
E_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\overline{\phi}\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\phi_{rsq_{r-1}}\right)\right)$ (7.30)
$\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}\phi$
$\displaystyle=B_{\overline{r}}^{\overline{\alpha}}\overline{\phi}=\left\llbracket
b_{r}>0\right\rrbracket\left(\left\llbracket q_{r}>1\right\rrbracket
b_{r}\left(P_{q_{r}}-\phi(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\phi\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\phi_{rsq_{r-1}}\right)\right)$

######  Lemma 100 (Upper bounds for primitive functions).

Let $\phi\in\Phi$, the set of monotone descending positive functions on
$(0,1)$.. Then for $q_{r}>2$ we have:

(7.31) $\displaystyle B_{r}^{\alpha}\phi$ $\displaystyle\leq\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}P_{q_{r}}+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
E_{r}\left(\phi_{r0,q_{r}-q_{r-1}}^{\alpha}-\overline{\phi}\left(\frac{1}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(E_{r}\phi_{rsq_{r}}^{\alpha}+O_{r}\phi_{rsq_{r-1}}^{\alpha}\right)\right)$
(7.32)
$\displaystyle\overline{B_{\overline{r}}^{\overline{\alpha}}}\overline{\phi}$
$\displaystyle\leq\left\llbracket
b_{r}>0\right\rrbracket\left(b_{r}P_{q_{r}}+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\overline{\phi}_{r0,q_{r}-q_{r-1}}^{\alpha}-\overline{\phi}\left(\frac{1}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(O_{r}\overline{\phi}_{rsq_{r}}^{\alpha}+E_{r}\overline{\phi}_{rsq_{r-1}}^{\alpha}\right)\right)$

######  Definition 101.

We define $C_{rs}^{\alpha}\phi\coloneqq
E_{r}\phi_{rsq_{r}}^{\alpha}+O_{r}\phi_{rsq_{r-1}}^{\alpha}$ and its
quasiperiod dual $\overline{C}_{rs}^{\alpha}\overline{\phi}\coloneqq
O_{r}\overline{\phi}_{rsq_{r}}^{\alpha}+E_{r}\overline{\phi}_{rsq_{r-1}}^{\alpha}$.
We let $C_{r}$ be the sum of the terms in the double sum
$DS\phi=\sum_{r=0}^{n}\sum_{s=1}^{b_{r}-1}C_{rs}\phi$ with denominator
$q_{r+1}^{/}$, so that $DS(\phi)=\sum_{r=0}^{n}C_{r}(\phi)$. And
$S_{N}^{\alpha}\phi\leq\sum_{r}B_{r}^{\alpha}\phi\leq\sum_{r}SS_{r}^{\alpha}\phi+C_{r}^{\alpha}\phi$

(7.33) $\displaystyle C_{n}(\phi)$
$\displaystyle=\sum_{s=1}^{b_{n}}\phi^{E}\left(\frac{s}{q_{n+1}^{/}}\right)+\sum_{s=0}^{b_{n}-1}\phi^{O}\left(\frac{a_{n+1}^{/}-s}{q_{n+1}^{/}}\right)\,\,+\left\llbracket
b_{n-1}>0\right\rrbracket O_{r}\phi\left(\frac{1}{q_{n+1}^{/}}\right)$
$\displaystyle r=n$ $\displaystyle C_{r}(\phi)$
$\displaystyle\leq\sum_{s=1}^{b_{r}-1}\phi^{E}\left(\frac{1}{q_{r+2}^{/}}+\frac{s}{q_{r+1}^{/}}\right)+\sum_{s=0}^{b_{r}-1}\phi^{O}\left(\frac{a_{r+1}-s}{q_{r+1}^{/}}\right)\,\,+\left\llbracket
b_{r-1}>0\right\rrbracket O_{r}\phi\left(\frac{1}{q_{r+1}^{/}}\right)$
$\displaystyle r<n$

Writing $c_{r}^{\alpha}=c(N,\alpha,r)=E_{r}\left(b_{r}-1+\left\llbracket
r=n\right\rrbracket\right)+O_{r}\min(b_{r}+1,a_{r+1})$ this gives

(7.34) $\displaystyle C_{r}(\phi)$
$\displaystyle\leq\sum_{s=1}^{c_{r}^{\alpha}}\phi\left(\frac{s}{q_{r+1}^{/}}\right)\,\,-\left\llbracket
b_{r-1}=0\right\rrbracket\phi^{O}\left(\frac{1}{q_{r+1}^{/}}\right)$

##  8\. Application to the function family
$\theta^{\beta}(x)=x^{-\beta},\beta\geq 1$

Could also cover $0\leq\beta<1$ but it’s tedious - is it worth it???

In the previous section we established inequalities applicable to any suitable
decreasing observable $\phi$. In this section we will further develop the
inequalities for the special family of functions
$\theta^{\beta}(x)=x^{-\beta}$ for $\text{$\beta\geq$1}.$ (For $\beta\leq 1$
$S_{N}\theta^{\beta}$ can be estimated using the Denjoy-Koksma(**) result,
combining it with the technique of truncated forms of the observables for
$0<\beta<1$).

We recall from the previous section that the lower bound for $S_{N}\phi$ can
be studied in a straightforward way, but the upper bound is more complex and
we will it in parts. In particular $S_{N}$ splits naturally into a sum of 3
parts: $S_{N}\phi\leq\left(S_{N}^{1}+S_{N}^{2}+S_{N}^{3}\right)\phi$ where

(8.1) $\displaystyle S_{N}^{1}\phi$
$\displaystyle=\sum_{r=0}^{n}\left\llbracket b_{r}>0\right\rrbracket
b_{r}P_{q_{r}}(\phi)$ (8.2) $\displaystyle S_{N}^{2}\phi$
$\displaystyle=\sum_{r=0}^{n}\left\llbracket
b_{r}>0\right\rrbracket\left\llbracket 2<q_{r}<q_{n}\right\rrbracket
E_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\overline{\phi}\left(\frac{1}{q_{r}}\right)\right)$
(8.3) $\displaystyle S_{N}^{3}\phi$
$\displaystyle=\sum_{r=0}^{n}\left\llbracket
b_{r}>0\right\rrbracket\sum_{s=0}^{b_{r}-1}\left(E_{r}\phi_{rsq_{r}}+O_{r}\phi_{rsq_{r-1}}\right)=\sum_{r=0}^{n}C_{r}\phi$

Note that the first two sums are single sums whilst the third is a double sum.
Also $S_{N}^{1}\phi$ vanishes for $\phi$ anti-symmetric (**).

From this point we assume $\alpha$ is fixed and can be dropped from the
notation, and that the canonical Ostrowski representation of $N$ is
$N=\sum_{r=0}^{n}b_{r}q_{r}$ .

###  8.1. Some ancillary lemmas

The following function will play a large role in this section:

######  Definition 102 (Generalised Harmonic Function).

For $y>0$ we define $H_{k}^{\beta}(y)\coloneqq\sum_{s=0}^{k-1}(y+s)^{-\beta}$
(with $H_{k}^{\beta}(y)\coloneqq 0$ for $k\leq 0$). We will also write
$H_{k}^{\beta}\coloneqq H_{k}^{\beta}(1)$.

Note that this is a positive function decreasing in $y$ and increasing in $k$.
It follows that for $y\geq 1$ that $H_{k}^{\beta}(y)\leq
H_{k}^{\beta}<H_{\infty}^{\beta}=\zeta(\beta)$. So for $\beta>1$
$H_{k}^{\beta}(y)$ is $O(1)$, but the constant becomes arbitrarily large as
$\beta\downarrow 1$.

######  Proposition 103.

$N\geq b_{n}q_{n}>\frac{b_{n}}{b_{n}+1}N\geq\frac{1}{2}N$

###### Proof.

By definition $N<\left(b_{n}+1\right)q_{n}$ and the left hand inequality
follows immediately. The right hand follows since $b_{n}\geq 1$. ∎

This gives yet less sharp, but again more digestible alternatives.

######  Lemma 104.

Suppose $\beta\geq 1$ then $\sum_{r=0}^{n-1}\left\llbracket
b_{r}>0\right\rrbracket
q_{r}^{\beta}\leq\sum_{r=0}^{n-1}b_{r}q_{r}^{\beta}<q_{n}^{\beta}$

###### Proof.

The first inequality is trivial. For $\beta\geq 1$,
$\sum_{r=0}^{n-1}b_{r}q_{r}^{\beta}\leq\left(\sum_{r=0}^{n-1}b_{r}q_{r}\right)^{\beta}$
and $\sum_{r=0}^{n-1}b_{r}q_{r}=N-b_{n}q_{n}<q_{n}$. ∎

######  Corollary 105.

$\sum_{r=0}^{n}\left\llbracket b_{r}>0\right\rrbracket
q_{r}^{\beta}\leq\min\left(2q_{n}^{\beta},N^{\beta}\right)$ and
$\sum_{r=0}^{n-1}\left\llbracket b_{r}>0\right\rrbracket
q_{r}^{\beta}<\min\left(2q_{n-1}^{\beta},q_{n}^{\beta}\right)$

######  Corollary 106.

For $\phi x=(1-x)^{-\beta},\beta\geq 1$ From (48), for $r>0$ we have both
$\phi_{r0,q_{r}-q_{r-1}}^{E}\leq\overline{\phi}(\frac{1}{2q_{r}})=(2q_{r})^{\beta}$
and also $\phi_{r0,q_{r}-q_{r-1}}^{O}\leq\overline{\phi}(\frac{1}{2q_{r}})$
and hence also $\sum_{r=0}^{n-1}\left\llbracket
b_{r}>0\right\rrbracket\left(\phi_{r0,q_{r}-q_{r-1}}^{E}+\phi_{r0,q_{r}-q_{r-1}}^{O}\right)\leq\sum_{r=0}^{n-1}\left\llbracket
b_{r}>0\right\rrbracket\overline{\phi}(\frac{1}{2q_{r}})\leq
2^{\beta}q_{n}^{\beta}.$ By the lemma
$\sum_{r=1}^{n-1}\phi_{r0,q_{r}-q_{r-1}}^{E}+\phi_{r0,q_{r}-q_{r-1}}^{O}\leq
2^{\beta}q_{n}^{\beta}$.

Similarly
$\sum_{r=1}^{n-1}\overline{\phi}^{E}\left(\frac{2}{q_{r}}\right)+\overline{\phi}^{O}\left(\frac{2}{q_{r}}\right)\leq\left(\frac{1}{2}\right)^{\beta}q_{n}^{\beta}$

######  Lemma 107 (Generalised Harmonic Function).

The following results hold:

$P_{q_{r}}\theta^{\beta}\leq\sum_{t=1}^{q_{r}-1}\frac{q_{r}^{\beta}}{t^{\beta}}=q_{r}^{\beta}H_{q_{r}-1}^{\beta}$.

For $\beta=1,k\geq 1$ we have
$\log\left(1+\frac{k}{y}\right)<H_{k}(y)\leq\frac{1}{y}+\log\left(1+\frac{k-1}{y}\right)$
. For $y=1$ this gives $\log\left(1+k\right)<H_{k}\leq 1+\log k$

For $\beta>1,k\geq 1$ we have
$\frac{1}{\beta-1}\left(\frac{1}{y^{\beta-1}}-\frac{1}{(y+k)^{\beta-1}}\right)<H_{k}^{\beta}(y)\leq\frac{1}{y^{\beta}}+\frac{1}{\beta-1}\left(\frac{1}{y^{\beta-1}}-\frac{1}{(y+k-1)^{\beta-1}}\right)$.

Finally a useful trivial estimate is for $n\geq 0$
$H_{n}^{\beta}\leq\left\llbracket
n>0\right\rrbracket\left(1+\left(\frac{1}{2^{\beta}}\right)(n-1)\right)$
(equality for $n\in\\{1,2\\}$) which also gives $H_{n+1}\leq 1+\frac{1}{2}n$
and $H_{n-1}\leq\frac{1}{2}n$

By definition $H_{n}^{\beta}=0$ for $n\leq 0$.

For $i\geq j\geq 0$,

(8.4)
$H_{i-j}^{\beta}(j+1)=H_{i}^{\beta}-H_{j}^{\beta}=\sum_{k=j+1}^{i}\frac{1}{t^{\beta}}\leq\frac{1}{\left(j+1\right)^{\beta}}(i-j)$

and for $i\geq 1$, $H_{i}^{\beta}\leq 1+\sum_{t=2}^{i}\frac{1}{t^{\beta}}\leq
1+\frac{1}{2^{\beta}}(i-1)$. In ptic for $\beta=1$ we get for $n\geq 0$

(8.5) $H_{n}\leq\left\llbracket
n>0\right\rrbracket\frac{1}{2}\left(n+1)\right)$

(equality for $n\leq 2$) which also gives $H_{n+1}\leq 1+\frac{1}{2}n$ and
$H_{n-1}\leq\frac{1}{2}n$

$Q_{r}\leq
E_{r}\left(\frac{1}{2}b_{r}+1\right)+O_{r}\left(\frac{1}{2}b_{r}\right)=E_{r}+\frac{1}{2}b_{r}$

###  8.2. Lower bound for $S_{N}\theta^{\beta}$ using the generalised harmonic
function $H^{\beta}$

Most interest in the mathematical community has been on upper bounds for these
sums, and we will also focus primarily on these upper bounds. However first we
take a little time here to derive some simple results for a lower bound also.

Since $\theta^{\beta}>0$ we have the simple and crude lower bound
$S_{N}\theta^{\beta}=\sum_{r=0}^{n}S_{b_{r}q_{r}}\theta^{\beta}\geq
S_{b_{n}q_{n}}\theta^{\beta}$ (where $N=\sum_{r=0}^{n}b_{r}q_{r}$). Recall we
are regarding $\alpha$ as fixed and drop it from the notation, so that from
(7.8),(7.7) we get:

$\sum_{s=0}^{b_{r}-1}S_{rs}\phi\geq\overline{B}_{r}\phi=\\\ \left\llbracket
b_{r}>0\right\rrbracket\left(\left\llbracket q_{r}>1\right\rrbracket
b_{r}\left(P_{q_{r}}-\phi(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
O_{r}\left(\phi_{r0,q_{r}-q_{r-1}}-\phi\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\phi_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\phi_{rsq_{r-1}}\right)\right)$

$\sum_{s=0}^{b_{r}-1}S_{rs}\overline{\phi}\geq
B_{r}^{\alpha}\overline{\phi}=\\\ \left\llbracket
b_{r}>0\right\rrbracket\left(\left\llbracket q_{r}>1\right\rrbracket
b_{r}\left(P_{q_{r}}-\phi(\frac{1}{q_{r}})\right)+\left\llbracket
2<q_{r}<q_{n}\right\rrbracket
E_{r}\left(\overline{\phi}_{r0,q_{r}-q_{r-1}}-\phi\left(\frac{2}{q_{r}}\right)\right)+\sum_{s=0}^{b_{r}-1}\left(\overline{\phi}_{rsq_{r}}+\left\llbracket
q_{r}>1\right\rrbracket\overline{\phi}_{rsq_{r-1}}\right)\right)$

Hence $S_{N}\theta^{\beta}\geq
S_{b_{n}q_{n}}\theta^{\beta}=\sum_{s=0}^{b_{n}-1}S_{ns}\theta^{\beta}\geq\overline{B}_{n}\theta^{\beta}$.

Note for $q_{n}>1$,
$\overline{B}_{n}\theta^{\beta}=b_{n}\left(P_{q_{n}}-\theta^{\beta}(\frac{1}{q_{n}})\right)+\sum_{s=0}^{b_{n}-1}\left(\theta_{nsq_{n}}^{\beta}+\theta_{nsq_{n-1}}^{\beta}\right)$.
And for any $1<q_{r}\leq q_{n}$,
$P_{q_{r}}-\theta^{\beta}(\frac{1}{q_{r}})=q_{r}^{\beta}H_{q_{r}-1}^{\beta}-q_{r}^{\beta}=q_{r}^{\beta}\left(H_{q_{r}-1}^{\beta}-1\right)$.

For $n$ even
$\sum_{s=0}^{b_{n}-1}\theta_{nsq_{n}}^{\beta}=q_{n+1}^{/\beta}H_{b_{n}}^{\beta}$
and