# K-Divergent lattices

Guy Lachman, Anurag Rao, Uri Shapira, Yuval Yifrach<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce a novel concept in topological dynamics, referred to as
$k$-divergence, which extends the notion of divergent orbits. Motivated by
questions in the theory of inhomogeneous Diophantine approximations, we
investigate this notion in the dynamical system given by a certain flow on the
space of unimodular lattices in $\mathbb{R}^{d}$. Our main result is the
existence of $k$-divergent lattices for any $k\geq 0$. In fact, we utilize the
emerging theory of parametric geometry of numbers and calculate the Hausdorff
dimension of the set of $k$-divergent lattices.

##  1\. Introduction

###  1.1. $k$-divergence

In this article we investigate a novel concept in topological dynamics. We are
motivated by a particular example of a diagonal flow acting on the space of
lattices with specific implications to questions in Diophantine approximation.
As a consequence, our analysis in this paper is restricted to this particular
example. Nevertheless, we start the introduction abstractly.

Let $X$ be a topological space and $H$ a topological (semi) group acting
continuously on $X$. The pair $(X,H)$ is referred to as a topological
dynamical system. Given $x\in X$, the set

$Hx:=\\{hx\in X:h\in H\\}$

is called the orbit of $x$. Broadly speaking, topological dynamics concerns
itself with investigating the topological properties of orbits. A basic notion
which is highly relevant to our discussion is that of divergence: An orbit
$Hx$ is said to be divergent if the map $h\mapsto hx$ is a proper map from $H$
to $X$; that is, the preimage of any compact set in $X$ is compact in $H$.
Clearly, it is of interest to discuss divergent orbits only when both $H$ and
$X$ are non-compact. It will be convenient to use the following
terminology111For the sake of simplicity all topological spaces are assumed to
be metric with the Heine-Borel property: compact sets are those which are
closed and bounded. Examples are manifolds equipped with a complete Riemannian
metric.: If $Y$ is a topological space, we say that a sequence
$(y_{n})_{n\in{\mathbb{N}}}\subset Y$ is divergent and write $y_{n}\to\infty$
if $y_{n}$ eventually leaves any compact set in $Y$. Using this terminology an
orbit $Hx$ is divergent if and only if

$h_{n}\to\infty\implies h_{n}x\to\infty.$

It is convenient to use the following terminology.

######  Definition 1.1 (Asymptotic accumulation points).

Given a topological dynamical system $(X,H)$ and a point $x\in X$, we have its
set of asymptotic accumulation points:

$D(x):=\\{y\in X:\text{ There exists }(h_{n})_{n\in{\mathbb{N}}}\subset
H\text{ with }h_{n}\to\infty\text{ and }h_{n}x\to y\\}.$

Since the acting (semi)-group $H$ is fixed we do not record it in the
notation, although strictly speaking this set should have been denoted
$D_{H}(x).$ For example, $Hx$ is divergent if and only if $D(x)=\varnothing$.
The following is the main concept which will interest us in the rest of this
paper.

######  Definition 1.2 ($k$-divergent orbits, or points).

Given a point $x\in X$, a finite sequence

$(y_{0}:=x,y_{1},y_{2},\dots,y_{k})\in X^{k+1}\ \text{ with }\ y_{i+1}\in
D(y_{i})\text{ for }0\leq i<k$

is called an accumulation sequence of length $k$ for $x$. The point $x$ is
said to be $k$-divergent if $k$ is the supremum over all lengths of
accumulation sequences for $x$. Finally, the orbit $Hx$ is said to be
$k$-divergent if any of its points are such (note that this is well defined).

In the above definition, $k$ takes values in ${\mathbb{Z}}_{\geq
0}\cup\\{\infty\\}$. Let

$\mathbf{Div}(k):=\\{x\in X:\text{$x$ is $k$-divergent}\\}$

so that these sets give a partition of $X$.

######  Remark 1.3.

We note that from the statistical point of view the sets $\mathbf{Div}(k)$ for
$k$ finite are very small in the sense that they are null sets with respect to
any $H$-invariant probability measure. This is because: if $x$ is
$k$-divergent for finite $k$, then $x\notin D(x)$, and so $x$ is not a
recurrent point. (cf. the proof of Theorem 6.2 below.)

As mentioned above, any point $x\in\mathbf{Div}(0)$ only has accumulation
sequences of length $0$ which implies that $Hx$ is divergent (and conversely).
Any point $x\in\mathbf{Div}(1)$ has an accumulation sequence of length $1$ and
none which are longer. This shows that it is not divergent but only has
asymptotic accumulation points which are themselves divergent. Indeed,
Definition 1.2 is recursive in nature and the reader can check that, for any
$k\in{\mathbb{N}}$,

$\mathbf{Div}(k)=\left\\{x\in
X:D(x)\subset\cup_{j=0}^{k-1}\mathbf{Div}(j)\right\\}\smallsetminus\cup_{j=0}^{k-1}\mathbf{Div}(j).$
(1.1)

In words, $x$ is $k$-divergent if its asymptotic accumulation points are all
$j$-divergent for $j<k$ and if $x$ itself is not $j$-divergent for any $j<k.$

###  1.2. Results in the space of lattices

Let $m$ and $n$ be positive integers and let $d=m+n$. Let $G$ denote the Lie
group $\operatorname{SL}_{d}({\mathbb{R}})$, let $\Gamma$ denote
$\operatorname{SL}_{d}({\mathbb{Z}})$, and consider the noncompact homogeneous
space $X_{d}:=G/\Gamma$. The space $X_{d}$ is the moduli space of lattices in
${\mathbb{R}}^{d}$ of covolume 1 (with $g\Gamma$ identified with the
${\mathbb{Z}}$-span of the columns of $g$). For the acting semi-group we take
the one-parameter diagonal semi-group $\\{a_{t}\in G:t\in{\mathbb{R}}_{\geq
0}\\}$ where

$a_{t}:=\text{diag}(e^{t/m},\dots,e^{t/m},e^{-t/n},\dots,e^{-t/n})$ (1.2)

(where the number of positive eigenvalues is $m$ and the number of negative
eigenvalues is $n$). Henceforth, whenever we refer to Definition 1.2 we keep
$(X_{d},(a_{t})_{t\geq 0})$ as the underlying dynamical system.

Understanding this dynamical system is tightly related to Diophantine
approximation via the so-called Dani correspondence. See for example [Dan85,
Theorem 2.14] where divergent orbits under this flow are shown to be in
correspondence with singular matrices. See [Dan85, Definition 2.13] for the
definition of singular matrices or see (1.4) where singular vectors are
defined. Thus, the set $\mathbf{Div}(0)$ has received a lot of attention from
both the dynamical and number theoretic perspectives. There are certain
obvious lattices in $\mathbf{Div}(0)$; the so called degenerate divergent
orbits which correspond to lattices having a rational subspace whose volume
form goes to zero under $a_{t}$ (see [Dan85, Definition 2.8]). These lattices
are easy to construct. On the other hand, the existence of nondegenerate
divergent lattices was first demonstrated by Khintchine in 1926 [Khi26]. Much
more recent work involves the computation of the exact Hausdorff dimension of
$\mathbf{Div}(0)$ (see [Che11, CC16, KKLM17, DFSU, Sol]) which turns out to be
$\dim X_{d}-\frac{nm}{n+m}$, and as this number is strictly bigger than the
dimension of the degenerate divergent trajectories, one obtains a refinement
of Khintchine’s result showing the abundance of non-degenerate divergent
lattices.

As we will explain shortly (see §1.3), our motivation is to prove existence of
$k$-divergent lattices for $k>0$ finite. The situation seems to differ from
the $\mathbf{Div}(0)$ case because there are seem to be no lattices which are
$k$-divergent for obvious reasons. When $n=m=1$ and continued fractions are
available, it is not hard to construct explicit examples of $k$-divergent
lattices in the plane. In an unpublished work, the third named author was able
to construct examples of $1$-divergent lattices building on the results in
[Sha17]. However, given the sophisticated theory of parametric geometry of
numbers, one can establish existence in a far more systematic manner.

This theory was developed in [SS09, Roy15, DFSU, Sol] and guarantees existence
of lattices whose orbit’s behaviour is controlled by certain combinatorial
graphs called templates. The following, which is the main result of this
paper, establishes that indeed, for any $k\in{\mathbb{N}}$, $\mathbf{Div}(k)$
is nonempty. Moreover, we show that (unless $n=m=1$) their dimensions coincide
with that of $\mathbf{Div}(0)$. This result should be regarded as a
demonstration of the strength and flexibility of the emerging tools from the
theory of parametric geometry of numbers.

######  Theorem 1.4.

For any $k\geq 1$,

$\dim(\mathbf{Div}(k))=\dim(X_{d})-\frac{mn}{m+n}$ (1.3)

where $\dim$ refers to the Hausdorff dimension.

Note, we prove Theorem 1.4 by establishing tight upper and lower bounds on the
dimension of $\mathbf{Div}(k)$, for each $k\in{\mathbb{N}}$, with the lower
bound coming from the formulas of [DFSU] and the upper bound arising as an
easy corollary of the main result in [KKLM17]. (See Corollaries 5.3, 6.3
below.)

###  1.3. Inhomogeneous Diophantine approximation

This subsection is not needed for the rest of the paper and is included for
motivational purposes and to put the discussion in context. Our original
motivation for proving the existence of $k$-divergent lattices for $k\geq 1$
stems from an attempt to construct a counterexample to a claim in [Sha13]. In
order to explain things in more detail we need to fix some terminology. Since
this is a motivational discussion, we do not strive to describe things in
utmost generality and restrict our attention to the case $n=1$ and $d=m+1$ so
that the sets $\mathbf{Div}(k)$ are defined with respect to the semiflow

$a_{t}=\operatorname{diag}(e^{t/m},\dots,e^{t/m},e^{-t})\text{ for
}t\in{\mathbb{R}}_{\geq 0}.$

One of the basic questions in inhomogeneous Diophantine approximation is to
understand, for given vectors $\theta,\eta\in\mathbb{R}^{m}$, the rate at
which the quantity $\langle q\theta-\eta\rangle$ decays as $q\to\infty$ in the
integers. Here, we have used the notation $\langle\cdot\rangle$ for the
distance to ${\mathbb{Z}}^{m}$ in the supremum norm. One set of particular
interest is the set of badly approximable targets for
$\theta\in{\mathbb{R}}^{m}$:

$\operatorname{Bad}_{\theta}:=\left\\{\eta\in\mathbb{R}^{m}:\liminf_{q\to\infty}q^{1/m}\langle
q\theta-\eta\rangle>0\right\\}.$

Note that since $\langle\cdot\rangle$ is defined modulo $\mathbb{Z}^{m}$, we
can (and will) treat $\theta$ and $\eta$ as points on the $m$-torus
$\mathbb{T}^{m}:=\mathbb{R}^{m}/\mathbb{Z}^{m}$ and, as a result, treat
$\operatorname{Bad}_{\theta}$ as a subset of $\mathbb{T}^{m}$.

Recall that $\theta$ is said to be singular if

$\lim_{Q\to\infty}Q^{1/m}\min\left\\{\langle q\theta\rangle:1\leq q\leq
Q\right\\}=0.$ (1.4)

Dani’s correspondence then says that if we let $x_{\theta}$ denote the
$d$-dimensional lattice

$x_{\theta}:=\left(\begin{matrix}I_{m}&\theta\\\
0&1\end{matrix}\right)\mathbb{Z}^{d},$

then $\theta$ is singular if and only if $\left\\{a_{t}x_{\theta}\in X:t\geq
0\right\\}$ is divergent. Alternatively, in our current terminology, $\theta$
is singular if and only if $x_{\theta}$ is $0$-divergent. We extend this
terminology as follows:

######  Definition 1.5.

A vector $\theta\in\mathbb{R}^{m}$ is said to be $k$-divergent if $x_{\theta}$
is a $k$-divergent lattice.

Let us note here that at the moment we do not have a reasonable
characterization of $k$-divergence of $\theta$ in terms of Diophantine
inequalities for $k\geq 1$ but we expect that such a characterization should
exist.

In [ET11] it was shown for every $\theta$, that $\operatorname{Bad}_{\theta}$
has full Hausdorff dimension and is in fact a winning set for Schmidt’s game.
The line of thought for the current discussion is to complement this fact
about $\operatorname{Bad}_{\theta}$ by establishing that although this set is
big from the dimension point of view, it is small in the sense that it is a
null set with respect to a large explicit family of probability measures on
$\mathbb{T}^{d}$. Clearly, we have in mind some Diophantine restrictions on
$\theta$: For if $\theta\in{\mathbb{R}}^{m}$ is rational, then
$\operatorname{Bad}_{\theta}$ is cofinite and is consequently of full measure
with respect to any non-atomic measure. In [Sha13] it was claimed that, once
$\theta$ is non-singular, then $\operatorname{Bad}_{\theta}$ should be a null-
set with respect to any non-atomic algebraic measure on $\mathbb{T}^{d}$. An
algebraic measure is by definition a translate of a Haar measure on a closed
subgroup of $\mathbb{T}^{d}$. David Simmons found a gap in the proof presented
in [Sha13] and since then the third named author has been trying to determine
whether the claim is true or not.

We mention that there is a subtle distinction between the full Haar measure
$\lambda$ on $\mathbb{T}^{m}$ and algebraic measures of smaller dimension. For
the full Haar measure, the argument in [Sha13] goes through with minor
modifications to show that: if $\theta$ is non-singular, then
$\lambda(\operatorname{Bad}_{\theta})=0$. For complete proofs, see [Kim23,
Mos23, MRS23]. Further, the recent paper [Kim23] also shows that for many222It
is not clear at the moment if $\lambda(\operatorname{Bad}_{\theta})=1$
characterizes singularity (although it seems unlikely to be the case). It is
easy to see that
$\lambda(\operatorname{Bad}_{\theta})\in\left\\{0,1\right\\}$. singular
$\theta$, $\lambda(\operatorname{Bad}_{\theta})=1$. See also the paper
[LdSS19] where related results on $\operatorname{Bad}_{\theta}$ are obtained.

As is clear from the technique used in [Sha13], the richer the dynamics of the
orbit of $x_{\theta}$ the better the chances to verify the aforementioned
claim. Indeed, in an attempt to close the aforementioned gap, recent work of
the second and third authors with N. Moshchevitin shows that once $\theta$ is
not $k$-divergent for small $k$, $\operatorname{Bad}_{\theta}$ is null with
respect to non-atomic algebraic measures.

######  Theorem 1.6 ([MRS23]).

Let $\theta\in\mathbb{R}^{m}$ be a vector which is not $k$-divergent for any
$k\leq m-1$. Then for any 1-dimensional algebraic measure $\mu$ on
$\mathbb{T}^{m}$, $\mu(\operatorname{Bad}_{\theta})=0$.

More explicitly, given any $\eta\in{\mathbb{R}}^{m}$ and ${\bf
b}\in{\mathbb{Z}}^{m}$, we have for Lebesgue almost every $t\in{\mathbb{R}}$,

$\liminf_{q\to\infty}\ q^{1/m}\langle q\theta-(\eta+t\textbf{b})\rangle=0.$

To complement the story, we mention that in [MRS23] the authors also give a
counterexample to the aforementioned claim from [Sha13] by constructing a non-
singular vector $\theta\in\mathbb{R}^{3}$ along with a $\eta\in\mathbb{R}^{3}$
such that for any $t\in\mathbb{R}$, one has

$\liminf_{q\to\infty}\ q^{1/3}\langle q\theta-(\eta+t{\bf e}_{3})\rangle\geq
c$

for some fixed $c>0$. Here ${\bf e}_{3}\in{\mathbb{R}}^{3}$ denotes the third
vector in the standard coordinates. Thus, the algebraic measure corresponding
to the translation by $\eta$ of the 1-dimensional subtorus of the third
coordinate is fully supported inside $\operatorname{Bad}_{\theta}$. By Theorem
1.6 we must have that $\theta$ is either $1$-divergent or $2$-divergent. For
more information we refer the reader to [MRS23].

##  2\. Invariance properties of $k$-divergent lattices

In this section we prove certain invariance properties of $k$-divergent
lattices. Let $\mathfrak{g}$ denote the Lie algebra of $G$ and consider the
diagonal element ${\bf a}\in\mathfrak{g}$ for which $\exp({\bf a})=a_{1}\in
G$. Consider the derivation $\text{ad}_{\bf a}:\mathfrak{g}\to\mathfrak{g}$.
Clearly, this map is diagonalizable. We define

$\mathfrak{h}^{+}:=\text{span}_{\mathbb{R}}\left\\{{\bf
b}\in\mathfrak{g}:\text{ad}_{\bf a}({\bf b})=\lambda{\bf b}\text{ with
}\lambda>0\right\\},$ (2.1)
$\mathfrak{h}^{-}:=\text{span}_{\mathbb{R}}\left\\{{\bf
b}\in\mathfrak{g}:\text{ad}_{\bf a}({\bf b})=\lambda{\bf b}\text{ with
}\lambda<0\right\\},$ (2.2) $\mathfrak{h}^{0}:=\left\\{{\bf
b}\in\mathfrak{g}:\text{ad}_{\bf a}({\bf b})=0\right\\}.$ (2.3)

We thus have a vector space direct sum

$\mathfrak{g}=\mathfrak{h}^{+}\oplus\mathfrak{h}^{-}\oplus\mathfrak{h}^{0}.$

Moreover, each of the above summands is a Lie subalgebra of $\mathfrak{g}$.
Let $H^{+},H^{-},H^{0}$ be the corresponding connected Lie subgroups of $G$.

######  Proposition 2.1.

Let $k\in{\mathbb{Z}}_{\geq 0}$. Let $x\in\mathbf{Div}(k)$ and let $g^{-}$ and
$g^{0}$ belong to $H^{-}$ and $H^{0}$ respectively. We then have that

$\left(x\in\mathbf{Div}(k)\right)\iff\left(g^{-}g^{0}x\in\mathbf{Div}(k)\right).$
(2.4)

###### Proof.

We work with elements $g^{-}=\exp({\bf b})$ and $g^{0}=\exp({\bf c})$ along
with the assumptions $\operatorname{ad}_{\bf a}{\bf b}=\lambda{\bf b}$ for
$\lambda<0$ and $\operatorname{ad}_{\bf a}{\bf c}=0$. The reader is left to
deduce the general case from this. We then have, for $t>0$,

$\begin{split}a_{t}\exp({\bf b})\exp({\bf c})x&=\left(a_{t}\exp({\bf
b})a_{-t}\right)\left(a_{t}\exp({\bf c})a_{-t}\right)a_{t}x\\\
&=\exp\left(\operatorname{Ad}(\exp(t{\bf a})){\bf
b}\right)\exp\left(\operatorname{Ad}(\exp(t{\bf a})){\bf c}\right)a_{t}x\\\
&=\exp\left(e^{t\operatorname{ad}_{\bf a}}{\bf
b}\right)\exp\left(e^{t\operatorname{ad}_{{\bf a}}}{\bf c}\right)a_{t}x\\\
&=\exp\left(e^{t\lambda}{\bf b}\right)\exp\left({\bf
c}\right)a_{t}x.\end{split}$ (2.5)

The key observation is that $\lim_{t\to\infty}e^{t\lambda}=0$. Thus, for a
divergent positive sequence $({t_{n}})$,

$\left(\lim_{n\to\infty}a_{t_{n}}g^{-}g^{0}x=y\right)\iff\left(\lim_{n\to\infty}a_{t_{n}}x=\left(\exp(-{\bf
c}\right)y\right).$ (2.6)

This immediately proves the proposition for the $\mathbf{Div}(0)$ case. For
the $\mathbf{Div}(k)$ case, we proceed by induction, assuming that the
proposition has been proved for all $\mathbf{Div}(j)$ with $j<k$. Then, we can
use the equivalence (2.6) along with the induction hypothesis that

$\exp({\bf c})\mathbf{Div}(j)=\mathbf{Div}(j)\text{ for }j<k.$

This completes the proof and shows the so-called weak stable invariance
property of $\mathbf{Div}(k)$. ∎

Using the fact that $a_{t}$ has the form of (1.2), one can check that the
groups defined above are actually given by

$H^{-}=\left\\{\left[{\begin{array}[]{cc}I_{m}&0\\\
A&I_{n}\end{array}}\right]:A\in\text{Mat}_{n\times m}({\mathbb{R}})\right\\},$
$H^{0}=\left\\{\left[{\begin{array}[]{cc}A&0\\\
0&B\end{array}}\right]:A\in\text{Mat}_{m\times
m}({\mathbb{R}}),B\in\text{Mat}_{n\times n}({\mathbb{R}})\right\\}$

and

$H^{+}=\left\\{\left[{\begin{array}[]{cc}I_{m}&A\\\
0&I_{n}\end{array}}\right]:A\in\text{Mat}_{m\times n}({\mathbb{R}})\right\\}.$

For future use, we use the notation

$u_{A}:=\left[{\begin{array}[]{cc}I_{m}&A\\\ 0&I_{n}\end{array}}\right]\text{
and }x_{A}:=u_{A}{\mathbb{Z}}^{d}.$ (2.7)

Our second invariance property arises from looking at the log-minima functions
of a lattice. We consider the supremum norm $\|\cdot\|$ on ${\mathbb{R}}^{d}$.
Let $B(r)$ denote the supremum norm ball with radius $r$ and center at the
origin. If $x$ is a lattice, the $d$ successive minima functions of $x$ with
respect to the supremum norm are defined as

$\lambda_{i}(x):=\inf\left\\{r\in{\mathbb{R}}:x\cap B(r)\text{ contains $i$
independent vectors}\right\\}.$

The functions $\lambda_{i}:X_{d}\to{\mathbb{R}}$ are in fact continuous. We
consider, for a fixed lattice $x\in X_{d}$ and the flow $(a_{t})_{t\geq 0}$,
the log-minima functions

$f_{x}(t):{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}^{d};\
t\mapsto\left(\log(\lambda_{1}(a_{t}x)),\dots,\log(\lambda_{d}(a_{t}x))\right).$
(2.8)

We denote the components of $f_{x}$ by $f_{1},\dots,f_{d}$. We also need to
use the supremum norm to compare such functions on subsets of ${\mathbb{R}}$.
We use the convention that, for $E\subset{\mathbb{R}}_{\geq 0}$ and
$f,g:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}^{d}$,

$\|f-g\|_{E}:=\sup_{t\in E}\|f(t)-g(t)\|=\sup_{t\in
E}\left(\max_{i=1,\dots,d}|f_{i}(t)-g_{i}(t)|\right).$ (2.9)

For repeated implicit use in the future, we note the following reformulation
of Mahler’s compactness criterion:

######  Lemma 2.2.

Let $x\in X_{d}$ be a lattice with log-minima function
$f_{x}=(f_{i})_{i=1,\dots,d}$. Then, for any sequence of positive times
$(t_{i})_{i\in{\mathbb{N}}}$, the sequence
$(a_{t_{i}}x)_{i\in{\mathbb{N}}}\subset X_{d}$ has a convergent subsequence if
and only if

$\limsup_{i\to\infty}f_{1}(t_{i})>-\infty.$

###### Proof.

We simply note that, for any $M\in{\mathbb{R}}$, Mahler’s compactness
criterion gives that

$\left\\{y\in X_{d}:\log(\lambda_{1}(y))\geq M\right\\}$ (2.10)

is a compact set. Moreover, every compact subset of $X_{d}$ is contained in a
set of the form (2.10). ∎

######  Definition 2.3.

We define an equivalence relation on the set of continuous functions
$C({\mathbb{R}}_{\geq 0},{\mathbb{R}}^{d})$ by writing $f\sim g$ if

$\|f-g\|_{{\mathbb{R}}_{\geq 0}}<\infty.$ (2.11)

######  Proposition 2.4.

Let $k\in{\mathbb{Z}}_{\geq 0}$, let $x\in\mathbf{Div}(k)$ and let $y$ be
another lattice in $X_{d}$. Further, let $f_{x}$ and $f_{y}$ denote the log-
minima functions for $x$ and $y$ respectively. Then,

$\left(f_{x}\sim f_{y}\right)\implies\left(y\in\mathbf{Div}(k)\right).$ (2.12)

###### Proof.

The proposition consists of ${\mathbb{N}}$-many assertions and we use
induction (on $k$) to prove them all.

Assume $k=0$. By the definition of $\mathbf{Div}(0)$ and Lemma 2.2,

$\lim_{t\to\infty}\log(\lambda_{1}(a_{t}x))=-\infty.$

Since $f_{x}\sim f_{y}$, we see that
$\lim_{t\to\infty}\log(\lambda_{1}(a_{t}y))=-\infty$ as well. Again, an
application of Lemma 2.2 shows that $y\in\mathbf{Div}(0)$.

Now assume $k>0$ and suppose our claim holds for every $j<k$. Let $C$ be a
constant such that

$\|f_{x}-f_{y}\|_{{\mathbb{R}}_{\geq 0}}<C.$ (2.13)

Given that $x\in\mathbf{Div}(k)$, we are to show that $y\in\mathbf{Div}(k)$.
Suppose $(t_{i})_{i\in{\mathbb{N}}}$ is an unbounded positive sequence such
that

$\lim_{i\to\infty}a_{t_{i}}y=w$ (2.14)

for some $w\in X_{d}$. Using the joint continuity of the action of
$(a_{t})_{t\geq 0}$ on $X_{d}$, we may assume (perhaps after passing to a
subsequence) that

$\|f_{a_{t_{i}}y}-f_{w}\|_{[0,i]}\leq 1\ \text{ for every }i\in{\mathbb{N}}.$
(2.15)

Since $\\{a_{t_{i}}y\in X_{d}:i\in{\mathbb{N}}\\}$ is precompact and since
$f_{x}\sim f_{y}$, Lemma 2.2 shows that a subsequence of
$(a_{t_{i}}x)_{i\in{\mathbb{N}}}$ converges to some $z\in\mathbf{Div}(j)$ (for
$j<k$). Again, passing to a subsequence, we can assume in addition to (2.15)
that

$\|f_{a_{t_{i}}x}-f_{z}\|_{[0,i]}\leq 1.$ (2.16)

Applying (2.13), (2.15) and (2.16), we get

$\begin{split}\|f_{w}-f_{z}\|_{[0,i]}&=\|f_{w}-f_{a_{t_{i}}y}+f_{a_{t_{i}}y}-f_{a_{t_{i}}x}+f_{a_{t_{i}}x}-f_{z}\|_{[0,i]}\\\
&\leq 1+C+1\end{split}$ (2.17)

for every $i\in{\mathbb{N}}$. Thus $\|f_{z}-f_{w}\|_{{\mathbb{R}}_{\geq
0}}<\infty$ and the induction hypothesis gives us that $w\in\mathbf{Div}(j)$.
Thus,

$D(y)\subset\bigcup_{j<k}\mathbf{Div}(j).$

The argument to show that $D(y)\cap\mathbf{Div}(k-1)\neq\varnothing$ is
similar. We are then finished in light of the characterization of
$\mathbf{Div}(k)$ as in (1.1). ∎

##  3\. Templates

Our proof of the non-emptiness and, furthermore, of a lower bound on the
Hausdorff dimension of the set of $k$-divergent lattices is based on the
results appearing in [DFSU]. In order to present and then use their results,
we first need to define the notion of templates.

######  Definition 3.1 (Templates).

For any real interval $I$, we use $I_{\mathbb{Z}}$ to denote its intersection
with ${\mathbb{Z}}$. An $m\times n$ template is a continuous piecewise linear
function333$f$ is piecewise linear means its domain is divided into a locally
finite collection of closed intervals with disjoint interiors such that $f$ is
linear on each one. $f:[0,\infty)\xrightarrow[]{}{\mathbb{R}}^{d}$ (where
$d=m+n$) that satisfies the following properties:

  1. (a)

$f_{1}\leq\dots\leq f_{d}$.

  2. (b)

For all $i=1,\dots,d$, we have $-\frac{1}{n}\leq
f^{\prime}_{i}\leq\frac{1}{m}$.

  3. (c)

For all $j=0,\dots,d$ and for every interval $I$ such that $f_{j}<f_{j+1}$ we
have that the function

$\sum_{0<i\leq j}f_{i}$

is convex and piecewise linear on $I$ with slopes in the set

$Z(j):=\left\\{\frac{L_{+}}{m}-\frac{L_{-}}{n}:L_{+}\in[0,m]_{{\mathbb{Z}}},\
L_{-}\in[0,n]_{\mathbb{Z}},\ L_{+}+L_{-}=j\right\\}.$

As a convention we set $f_{0}=-\infty,f_{d+1}=\infty$.

Templates are meant to model the log-minima functions of lattices. See [DFSU,
Lemma 32.7] where a standard exterior algebra argument explains the relevance
of condition (c) above. We have the following theorem of fundamental
importance:

######  Theorem 3.2 (Theorem 4.2 [DFSU]).

For $A\in\text{Mat}_{m\times n}({\mathbb{R}})$ let $f_{A}$ denote the log-
minima function defined for the lattice $x_{A}$ defined in (2.7). Then there
exists a template $f$ such that $f\sim f_{A}$. Conversely, given a template
$f$, there exists an $A\in\text{Mat}_{m\times n}({\mathbb{R}})$ for which
$f_{A}\sim f$.

######  Definition 3.3 (Score of a template).

Let $f$ be a template and let $I$ be an interval on which it is linear. An
interval of equality for $f$ on $I$ is a set of integers $(p,q]_{\mathbb{Z}}$
where $0\leq p<q\leq d$ and where the components of $f$ satisfy

$f_{p}<f_{p+1}=\dots=f_{q}<f_{q+1}\text{ on }I.$

We define $M_{\pm}(p,q,I)$ to be the unique real numbers satisfying

$M_{+}(p,q,I)+M_{-}(p,q,I)=q-p\ \text{ and }\
\frac{M_{+}(p,q,I)}{m}-\frac{M_{-}(p,q,I)}{n}=\sum_{i=p+1}^{q}f^{\prime}_{i}.$
(3.1)

It follows from part (c) of Definition 3.1 that $M_{\pm}$ are positive
integers. (See also the footnote on page 24 of [DFSU].) Further define

$S_{+}(f,I):=\bigcup\left\\{\left(p,p+M_{+}(p,q,I)\right]_{\mathbb{Z}}\subset(p,q]_{\mathbb{Z}}:(p,q]_{\mathbb{Z}}\text{
is an interval of equality}\right\\}$ (3.2)

and

$S_{-}(f,I):=[1,d]_{\mathbb{Z}}\smallsetminus S_{+}(f,I).$ (3.3)

We also have

$\delta(f,I):=\\#\left\\{(i_{+},i_{-})\in S_{+}(f,I)\times
S_{-}(f,I):i_{+}<i_{-}\right\\}.$ (3.4)

We take $\delta(f,t)$ to be the piecewise constant function with value
$\delta(f,I)$ on $I$. Finally,

$\underline{\delta}(f):=\liminf_{T\to\infty}\frac{1}{T}\int_{0}^{T}\delta(f,t)dt.$
(3.5)

We aim to compute the Hausdorff dimension of the set of matrices
$A\in\text{Mat}_{m\times n}({\mathbb{R}})$ for which $x_{A}$ is $k$-divergent.
Given a template $f$, we define the set

$\mathcal{D}(f):=\left\\{A\in\text{Mat}_{m\times n}({\mathbb{R}}):f_{A}\sim
f\right\\}.$ (3.6)

We say that a collection of templates $\mathcal{F}$ is closed under finite
perturbation if

$\left(f\in\mathcal{F}\text{ and }f^{\prime}\text{ is a template with
}f^{\prime}\sim f\right)\implies\left(f^{\prime}\in\mathcal{F}\right).$ (3.7)

For such a $\mathcal{F}$, we define

$\mathcal{D}(\mathcal{F}):=\bigcup_{f\in\mathcal{F}}\mathcal{D}(f)\subset\operatorname{Mat}_{m\times
n}({\mathbb{R}}).$ (3.8)

A tool to compute dimension is:

######  Theorem 3.4 (Theorem 4.3 [DFSU]).

Let $\mathcal{F}$ be a Borel444Under the compact-open topology. collection of
templates closed under finite perturbation. Then we have that the Hausdorff
dimension of $\mathcal{D}(\mathcal{F})$ is

$\sup\left\\{\underline{\delta}(f):f\in\mathcal{F}\right\\}.$ (3.9)

##  4\. Existence of $k$-divergent lattices

Our goal is to build, for each $k\in{\mathbb{N}}$, an example of a template
$f^{(k)}$ giving rise to $k$-divergent lattices via Theorem 3.2. Following
this, we apply Theorem 3.4 to the equivalence class of templates

$\mathcal{F}^{k}:=\left\\{f\in C({\mathbb{R}}_{\geq
0},{\mathbb{R}}^{d}):f\text{ is a template with }f\sim f^{(k)}\right\\}.$
(4.1)

We have the following definition which is to be the input for the inductive
procedure giving rise to the templates $f^{(k)}$.

######  Definition 4.1 (Linked Templates).

A linked template is a pair $(f,\mathcal{I})$ where $f$ is a template and

$\mathcal{I}=\\{I_{p}:=[b_{p},b_{p+1}]:p\in{\mathbb{N}}\\}$

is a collection of intervals satisfying

$b_{1}=0,\ b_{p}\nearrow\infty\ \text{ and }\
f(b_{p})=0\in{\mathbb{R}}^{d}\text{ for all }p\in{\mathbb{N}}.$

Our first example of a linked template is built with the following block:

######  Definition 4.2.

We define a piecewise linear function $g:[0,m+n]\to{\mathbb{R}}^{d}$ by
specifying its coordinates. We have

$g_{1}(t)=\begin{cases}-\frac{t}{n}&\text{ if }\ 0\leq t\leq n\\\
\frac{t-(m+n)}{m}&\text{ if }\ n\leq t\leq m+n\end{cases}$ (4.2)

and

$g_{2}(t)=\begin{cases}\frac{t}{n(m+n-1)}&\text{ if }\ 0\leq t\leq n\\\
\frac{m+n-t}{m(m+n-1)}&\text{ if }\ n\leq t\leq m+n.\end{cases}$ (4.3)

We set $g_{3}=\dots=g_{m+n}=g_{2}$ (if $d>2$). We leave to the reader the
straightforward check that $g:=(g_{i})_{i=1,\dots,d}$ satisfies the template
axioms of Definition 3.1 on its domain.

This example of a linked template gives rise to a template associated to
$1$-divergent lattices.

######  Example 4.3 ($1$-divergent lattice template).

We define a family of closed intervals

$\mathcal{I}=\\{I_{p}\subset[0,\infty):p\in{\mathbb{N}}\\}$

inductively. Let $I_{1}=[0,d]$ and let $I_{p}$ have length $pd$ and be
contiguous to $I_{p-1}$.

For each $p\in{\mathbb{N}}$, let $\psi_{p}$ be the orientation preserving
linear bijection $I_{p}\to[0,d]$ and let
$\phi_{p}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be the linear map which scales
vectors by the factor $p$. Define

$f_{p}:=\phi_{p}\circ g\circ\psi_{p}:I_{p}\to{\mathbb{R}}$ (4.4)

where $g$ is as in Definition 4.2. Since $f_{p}$ takes the value
$0\in{\mathbb{R}}^{d}$ on both endpoints of $I_{p}$, we define
$f:[0,\infty)\to{\mathbb{R}}^{d}$ to be the unique continuous function which
restricts to $f_{p}$ on each $I_{p}$. Since the derivatives of $\psi_{p}$ and
$\phi_{p}$ are $p^{-1}$ and $pI$ respectively, we see that $f$ is a template.
Note that with respect to part (c) of Definition 3.1, the components of $f$
satisfy inequalities only on intervals avoiding the endpoints of the $I_{p}$.

Moreover, by construction, $(f,\mathcal{I})$ is a linked template. In the case
when $m=2$ and $n=1$ we have the following graph of $f$.

The purple graph is of $f_{1}$ and the orange graph is of $f_{2}=f_{3}$.

######  Theorem 4.4.

Fix the template $f:[0,\infty)\to{\mathbb{R}}^{d}$ as in Example 4.3. Let
$x\in X$ be a lattice with log-minima function $f_{x}$. Assume further that
$f_{x}\sim f$. Then $x\in\mathbf{Div}(1)$.

###### Proof.

Fix a constant $C$ such that $\|f_{x}-f\|_{{\mathbb{R}}_{\geq 0}}\leq C$. This
and Lemma 2.2 show that $D(x)$ is nonempty. In order to show
$x\in\mathbf{Div}(1)$, we show that any $y\in D(x)$ has a divergent orbit. So
let $y\in D(x)$ and let $f_{y}$ be the log-minima function of $y$. Again,
using the continuity of the action, we can find a sequence of times
$(t_{i})_{i\in{\mathbb{N}}}$ going to infinity such that

$\|f_{a_{t_{i}}x}-f_{y}\|_{[0,i]}\leq 1.$ (4.5)

Here $f_{a_{t_{i}}x}$ denotes the log-minima function of the lattice
$a_{t_{i}}x$. Recall now that $d=m+n$ is the dimension of the ambient space of
$x$.

######  Claim 4.4.1.

Fix any $M>0$. Given any $T_{0}>0$ we have, for all sufficiently large $t$,

$\left(\log(\lambda_{1}(a_{t}x))>-M\right)\implies\left(\sup\left\\{\log(\lambda_{1}(a_{t+s}x)):s\in[Md,Md+T_{0}]\right\\}\leq-M\right)$
(4.6)

###### Proof.

Let $f_{1}$ denote the first component of $f$. By the very definition of $f$,
we see that the claim (with $M$ replaced by $M+C$) holds for $f_{1}$. Since
$\|f_{x}-f\|_{{\mathbb{R}}_{\geq 0}}<C$, we see that the claim holds for the
function $t\mapsto\log(\lambda_{1}(a_{t}x))$ as well. ∎

We now show that

$\lim_{t\to\infty}\log(\lambda_{1}(a_{t}y))=-\infty,$

that is, $y$ has divergent orbit. Let $M>0$ be any positive constant large
enough so that $\log(\lambda_{1}(y))>-(M-1)$ and let $T_{0}>0$ be arbitrary.
By (4.5) we see that, for every $i\in{\mathbb{N}}$,

$\log(\lambda_{1}(a_{t_{i}}x))>-M.$

We apply claim 4.4.1 and see that, for all sufficiently large
$i\in{\mathbb{N}}$,

$\log(\lambda_{1}(a_{t_{i}+s})x)\leq-M\text{ for every }s\in[Md,Md+T_{0}].$

Another application of (4.5) gives that

$\log(\lambda_{1}(a_{s}y))\leq-(M-1)\text{ for all }s\in[Md,Md+T_{0}].$

Since $T_{0}$ was arbitrary, this proves the required divergence property of
the function $t\mapsto\log(\lambda_{1}(a_{t}y))$. ∎

We define an operator $\Phi$ from the set of linked templates to itself, the
iterates of which will help us in our inductive construction. The following
definition should be read as a guided exercise and the reader should fill in
the details.

######  Definition 4.5.

We set

$\Phi(f,\mathcal{I}):=(\widetilde{f},\widetilde{\mathcal{I}})$

where $\widetilde{f}$ and $\widetilde{\mathcal{I}}$ are defined as follows:
Let $\widetilde{I}_{1}:=I_{1}$ and, assuming we have defined

$\widetilde{I}_{1}=[c_{1},c_{2}],\widetilde{I}_{2}=[c_{2},c_{3}],\dots,\widetilde{I}_{q}=[c_{q},c_{q+1}],$

we set

$\widetilde{I}_{q+1}:=c_{q+1}+\left(I_{1}\cup I_{2}\cup\dots\cup I_{q}\cup
I_{q+1}\right).$

Here is a representation:

Can you guess the colors that go in the empty bin?

We now define $\widetilde{f}:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}^{d}$ to be
the unique function which satisfies the property that, for every
$q\in{\mathbb{N}}$ and $t\in\widetilde{I}_{q}$,

$\widetilde{f}(t)=f(t-c_{q}).$ (4.7)

This function $\widetilde{f}$ is continuous by the hypothesis that $f$ takes
value $0\in{\mathbb{R}}^{d}$ on the endpoints of $I_{p}$. The template axioms
can be easily checked by using the fact that $f$ itself is a template. Note
that with respect to part $(c)$ of Definition 3.1, if for some $j$,
$\widetilde{f}_{j}<\widetilde{f}_{j+1}$ on an interval $I$, the linked
template definition implies that $I\subset\widetilde{I}_{q}$ for some
$q\in{\mathbb{N}}$. Moreover, the same property holds for $f$.

The pair $(\widetilde{f},\widetilde{I})$ is another linked template.

We now construct putative examples of templates associated to $k$-divergent
lattices.

######  Example 4.6 ($k$-divergent lattice template).

Let $(f^{(1)},\mathcal{I}^{(1)})$ be our linked template from example 4.3.
Define

$(f^{(k)},\mathcal{I}^{(k)}):=\Phi^{k}(f^{(1)},\mathcal{I}^{(1)}).$

######  Theorem 4.7.

Fix $k\in{\mathbb{N}}$. Let $x\in X_{d}$ be a lattice with log-minima function
$f_{x}$. Assume further the equivalence $f_{x}\sim f^{(k)}$ where $f^{(k)}$ is
constructed as in Example 4.6. Then $x\in\mathbf{Div}(k)$.

###### Proof.

We proceed by induction on $k$ with the base step having already been proved
in Theorem 4.4. Thus we can assume $k>1$. Let $C$ be a constant witnessing
$f_{x}\sim f^{(k)}$. Let $y$ belong to $D(x)$. As before, we can find a
sequence of positive unbounded times $(t_{i})_{i\in{\mathbb{N}}}$ with

$\|f_{a_{t_{i}}x}-f_{y}\|_{[0,i]}<1.$ (4.8)

Again, $f_{a_{t_{i}}x}$ denotes the log-minima function of the lattice
$a_{t_{i}}x$.

Denote $(f^{(k-1)},\mathcal{I}^{(k-1)})$ by

$(f,\\{I_{p}:=[b_{p},b_{p+1}]:p\in{\mathbb{N}}\\})$

and $(f^{(k)},\mathcal{I}^{(k)})$ by

$(\widetilde{f},\\{\widetilde{I}_{q}:=[c_{q},c_{q+1}]:q\in{\mathbb{N}}\\}).$

For each $i\in{\mathbb{N}}$, let $q_{i}$ be the unique integer such that
$t_{i}\in[c_{q_{i}},c_{q_{i}+1})$. Assume for now that

$[t_{i},t_{i}+i]\subset[c_{q_{i}},c_{q_{i}+1}].$ (4.9)

Consider the differences $(t_{i}-c_{q_{i}})$ and the two possibilities:

  1. (a)

The sequence $(t_{i}-c_{q_{i}})_{i\in{\mathbb{N}}}$ has a bounded subsequence.
Without loss of generality, assume the sequence itself is bounded. We compute
that for each $i\in{\mathbb{N}}$ and $t\in[0,i]$,

$\begin{split}\|f_{y}(t)-f(t)\|&\leq\|f_{y}(t)-f_{a_{t_{i}}x}(t)\|+\|f_{x}(t_{i}+t)-f(t)\|\\\
&=1+\|f_{x}(t_{i}+t)-\widetilde{f}(t_{i}+t)\|+\|\widetilde{f}(t_{i}+t)-f(t)\|\\\
&\leq 1+C+\|\widetilde{f}(t_{i}+t)-f(t)\|\\\
&=1+C+\|f(t_{i}-c_{q_{i}}+t)-f(t)\|,\end{split}$ (4.10)

where the last equality follows from using the assumption of (4.9) and the
defining property of $\Phi$ as in (4.7). Since $f$ is continuous and piecewise
linear with derivatives of its components bounded in absolute value by
$\max\left\\{\frac{1}{m},\frac{1}{n}\right\\}$, and since
$(t_{i}-c_{q_{i}})_{i\in{\mathbb{N}}}$ is a bounded sequence, we see that the
last term in (4.10) has a uniform bound over all $i\in{\mathbb{N}}$. This
shows that $f_{y}\sim f^{(k-1)}$ with the induction hypothesis kicking in to
give

$y\in\mathbf{Div}(k-1).$

  2. (b)

The sequence $(t_{i}-c_{q_{i}})_{i\in{\mathbb{N}}}$ diverges to infinity. For
each $i$ and $t\in[0,i]$, we get the following chain of inequalities

$\begin{split}\|f_{y}(t)-f(t_{i}-c_{q_{i}}+t)\|&\leq
1+\|f_{x}(t_{i}+t)-f(t_{i}-c_{q_{i}}+t)\|\\\ &\leq
1+C+\|\widetilde{f}(t_{i}+t)-f(t_{i}-c_{q_{i}}+t)\|\\\ &=1+C,\end{split}$
(4.11)

where the last equality follows from the assumption in (4.9) and the defining
property in (4.7). Since $f$ is a template, Theorem 3.2 gives the existence of
a lattice $z$ for which $f_{z}\sim f(=f^{(k-1)})$. The induction hypothesis
implies that $z\in\mathbf{Div}(k-1)$. The computation in (4.11) shows that

$\|f(t_{i}-c_{q_{i}})\|\geq\|f_{y}(0)\|-(1+C)\text{ for all
}i\in{\mathbb{N}}.$

Thus, the equivalence $f_{z}\sim f$ and Lemma 2.2 shows that the sequence of
lattices $a_{(t_{i}-d_{q_{i}-1})}z$ has a convergent subsequence. Let $w\in
D(z)$ be the limit. The computation (4.11) shows that that $f_{y}\sim f_{w}$.
Theorem 2.4 then shows that

$y\in\bigcup_{j<k-1}\mathbf{Div}(j),$ (4.12)

since $w$ lies in the same set.

If (4.9) fails to hold, we can assume without loss of generality that, there
exists $s\in{\mathbb{R}}$ such that

$|t_{i}-c_{q_{i}+1}|<s\text{ for all }i\in{\mathbb{N}}.$

But then we can replace $t_{i}$ by $t_{i}+s$ and $y$ by $a_{s}y$ to return to
the situation of (4.9). Note we use here that the divergence properties of $y$
are invariant under the flow.

Summing up, we have shown that

$D(x)\subset\bigcup_{j<k}\mathbf{Div}(j),$ (4.13)

and it remains to show that $D(x)\cap\mathbf{Div}(k-1)$ is nonempty.

And this is straightforward: let $(t_{i})_{i\in{\mathbb{N}}}$ be the sequence
of initial points of the intervals $\widetilde{I}_{q}$, that is
$(c_{q})_{q\in{\mathbb{N}}}$. Since $f^{(k)}$ takes value
$0\in{\mathbb{R}}^{d}$ at these points, Mahler’s compactness shows that
$(a_{t_{i}}x)_{i\in{\mathbb{N}}}$ has a convergent subsequence. Call the limit
$y$. By construction, the log-minima function $f_{y}$ is equivalent to
$f^{(k-1)}$. The induction hypothesis shows that $y\in\mathbf{Div}(k-1)$ and
we are done. ∎

##  5\. Computations for a lower bound on dimension

We can now turn to computing the Hausdorff dimension of $\mathbf{Div}(k)$. Let
$\mathcal{F}^{k}$ be the set of templates which are equivalent to $f^{(k)}$.
It is clearly closed under finite perturbations and Borel in the compact-open
topology on $C({\mathbb{R}}_{\geq 0},{\mathbb{R}}^{d})$. Each
$A\in\mathcal{D}(\mathcal{F}^{k})$ gives rise to a lattice $x_{A}$ which is
necessarily in $\mathbf{Div}(k)$, by Theorem 4.7. Thus, in light of
Proposition 2.1, computing the dimension of
$\mathcal{D}(\mathcal{F}^{k})\subset\text{Mat}_{m\times n}({\mathbb{R}})$ will
lead to a lower bound for the dimension of $\mathbf{Div}(k)\subset X_{d}$.

######  Lemma 5.1.

The score $\underline{\delta}(f^{(k)})$ is equal to $\delta_{m,n}$ which is
defined by

$\delta_{m,n}:=mn-\frac{mn}{m+n}.$ (5.1)

###### Proof.

We note that the first part of this computation has already been done in
[DFSU, Figure 4]. We proceed by induction and begin with the case $k=1$. We
must contend with the function $\delta(f^{(1)},t)$ and thus also the
quantities $S_{+},M_{+}$ and $M_{-}$ from Definition 3.3. Since $f^{(1)}$ is
defined by repeating the same pattern over larger and larger scales, it
suffices to compute $\delta(f^{(1)},t)$ over the interval $[0,d]$. First
consider the interval $K_{1}:=[0,n]$. There are two intervals of equality
here, $(0,1]_{\mathbb{Z}}$ and $(1,d]_{\mathbb{Z}}$. Solving the system of
equations

$M_{+}(0,1,K_{1})+M_{-}(0,1,K_{1})=1\ \text{ and }\
\frac{M_{+}(0,1,K_{1})}{m}-\frac{M_{-}(0,1,K_{1})}{n}=-\frac{1}{n},$

we see that $M_{+}(0,1,K_{1})=0$. Solving

$M_{+}(1,d,K_{1})+M_{-}(1,d,K_{1})=d-1\ \text{ and }\
\frac{M_{+}(1,d,K_{1})}{m}-\frac{M_{-}(1,d,K_{1})}{n}=\frac{1}{n},$

we see that $M_{+}(1,d,K_{1})=m$. We thus have

$S_{+}(f^{(1)},K_{1})=[2,m+1]_{\mathbb{Z}}\ \text{ and }\
S_{-}(f^{(1)},K_{1})=\\{1\\}\cup[m+2,m+n]_{\mathbb{Z}}.$

This shows that

$\delta(f^{(1)},t)=m(n-1)\ \text{ for }\ t\in[0,n].$ (5.2)

We pause here to note that the interval $[m+2,m+n]_{\mathbb{Z}}$ could be
empty; for example, in the case when $m=n=1$.

Now consider the interval $K_{2}:=[n,d]$. We again have two intervals of
equality, $(0,1]_{\mathbb{Z}}$ and $(1,d]_{\mathbb{Z}}$. One can perfom a
similar computation as above to see that $M_{+}(0,1,K_{2})=1$ and
$M_{+}(1,d,K_{2})=m-1$. We thus have

$S_{+}(f^{(1)},K_{2})=[1,m]_{\mathbb{Z}}\ \text{ and }\
S_{-}(f^{(1)},K_{2})=[m+1,m+n]_{\mathbb{Z}}.$

This shows that

$\delta(f^{(1)},t)=mn\ \text{ for }\ t\in[n,d].$ (5.3)

Recall that $f^{(1)}$ was defined over a specific collection of intervals
which we now call

$\left\\{I_{p}=[b_{p},b_{p+1}]:p\in{\mathbb{N}}\right\\}.$

Each $I_{p}$ has length $pd$. The definition in (4.4) and equations (5.2) and
(5.3) show that, for any $q\in{\mathbb{N}}$,

$\int_{I_{q}}\delta(f^{(1)},t)dt=qn(mn-m)+qm(mn).$ (5.4)

Thus, for any $T\in I_{p}$ with $p>1$, we see that

$\begin{split}\frac{1}{T}\int_{0}^{T}\delta(f^{(1)},t)dt&=\frac{1}{T}\left(\int_{0}^{b_{p}}\delta(f^{(1)},t)dt+\int^{T}_{b_{p}}\delta(f^{(1)},t)dt\right)\\\
&=\frac{1}{T}\int_{0}^{b_{p}}\delta(f^{(1)},t)dt+\frac{1}{T}\int^{T}_{b_{p}}\delta(f^{(1)},t)dt\\\
&=\frac{1}{T}\sum_{q=1}^{p-1}\left(qmn^{2}+qm^{2}n-qmn\right)+\frac{1}{T}\int_{b_{p}}^{T}\delta(f^{(1)},t)dt\\\
&=\frac{mn^{2}+m^{2}n-mn}{d}\cdot\frac{1}{T}\sum_{q=1}^{p-1}qd+\frac{1}{T}\int_{b_{p}}^{T}\delta(f^{(1)},t)dt\\\
&=\delta_{m,n}\cdot\frac{1}{T}\sum_{q=1}^{p-1}qd+\frac{1}{T}\int_{b_{p}}^{T}\delta(f^{(1)},t)dt.\end{split}$
(5.5)

Since

$\frac{\text{length}\left(\bigcup^{p-1}_{q=1}\
I_{q}\right)}{\text{length}\left(\bigcup^{p}_{q=1}\
I_{q}\right)}=\frac{p-1}{p+1}$ (5.6)

and since $\delta(f^{(1)},t)$ is a bounded function, computation (5.5) shows
that

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\delta(f^{(1)},t)dt=\delta_{m,n}.$
(5.7)

Now we come to $f^{(k)}$ where $k>1$. Assume the specified decomposition of
its domain is denoted by

$\\{J_{p}=[c_{p},c_{p+1}]:p\in{\mathbb{N}}\\}.$

First, by recalling the formula of (4.7), one proves inductively on $k$ that

$\int_{J_{p}}\delta(f^{(k)},t)dt=\text{length}(J_{p})\left(\frac{n}{m+n}(mn-m)+\frac{m}{m+n}(mn)\right).$
(5.8)

Second, we leave it to the reader as an exercise to prove the limit

$\lim_{p\to\infty}\ \frac{\text{length}\left(\bigcup^{p-1}_{q=1}\
J_{q}\right)}{\text{length}\left(\bigcup^{p}_{q=1}\ J_{q}\right)}=1$ (5.9)

using induction on $k$, with the base step being seen by (5.6). Hint: the
induction hypothesis guarantees, for any $i\in{\mathbb{N}}$, that

$\lim_{p\to\infty}\frac{\operatorname{length}(J_{p})}{\operatorname{length}(J_{p+i})}=1.$

We can now take $T\in J_{p}=[c_{p},c_{p+1}]$ and then compute, exactly as in
(5.5),

$\frac{1}{T}\int_{0}^{T}\delta(f^{(k)},t)dt=\delta_{m,n}\cdot\frac{\text{length}\left(\bigcup_{q=1}^{p-1}J_{q}\right)}{T}+\frac{1}{T}\int_{c_{p}}^{T}\delta(f^{(k)},t)dt.$
(5.10)

Equation (5.9) then shows that

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\delta(f^{(k)},t)dt=\delta_{m,n}.$

This completes the proof. ∎

Theorem 3.4 and the above score computation immediately yields:

######  Corollary 5.2.

The dimension of $\mathcal{D}(\mathcal{F}^{k})\subset\text{Mat}_{m\times
n}({\mathbb{R}})$ is bounded below by $\delta_{m,n}$.

######  Corollary 5.3.

Let $k\in{\mathbb{N}}$. The dimension of $\mathbf{Div}(k)\subset X_{d}$ is
bounded below by

$\dim(X_{d})-\frac{mn}{m+n}.$ (5.11)

###### Proof.

By Proposition 2.1 we have

$\mathbf{Div}(k)=H^{-}H^{0}\mathbf{Div}(k).$

As a consequence, since locally $G$ is a metric product of small
neighbourhoods of the identity in the groups $H^{-},H^{0},H^{+}$, it follows
that

$\dim\mathbf{Div}(k)=\dim(\mathbf{Div}(k)\cap
H^{+}{\mathbb{Z}}^{d})+\dim(H^{-}H^{0}).$

Corollary 5.2 says that $\delta_{m,n}$ is a lower bound for the dimension of
$\mathbf{Div}(k)\cap H^{+}{\mathbb{Z}}^{d}$ and so we see that
$\delta_{m,n}+\dim(H^{-}H^{0})$ is a lower bound for $\mathbf{Div}(k)$. Since

$\dim(X_{d})=d^{2}-1\ \text{ and }\ \dim(H^{-}H^{0})=d^{2}-1-mn,$

we are done. ∎

##  6\. Full escape of mass and an upper bound on dimension

Let $\mathcal{M}(X_{d})$ and $\mathcal{P}(X_{d})$ denote the set of Borel
complex measures on $X_{d}$ and the set of Borel probability measures on
$X_{d}$ respectively. Every open set in $X_{d}$ is $\sigma$-compact and so, if
we consider the $C^{*}$-algebra of continuous complex-valued functions
vanishing at infinity, $C_{0}(X_{d})$, the Riesz Representation theorem
[Rud87, Theorem 6.19] asserts that $\mathcal{M}(X_{d})$ is in bijection with
the continuous dual $C_{0}(X_{d})^{\vee}$, via

$\nu\mapsto\left(f\mapsto\int fd\nu\right).$ (6.1)

We induce the weak∗-topology on $\mathcal{M}(X_{d})$ and the corresponding
subset topology on $\mathcal{P}(X_{d})$. Given $x\in X_{d}$, we define the
family of probabilities $\\{\mu_{x,T}\in\mathcal{P}(X_{d}):T>0\\}$ by

$\int fd\mu_{x,T}:=\frac{1}{T}\int_{0}^{T}f(a_{t}x)dt$ (6.2)

where $f\in C_{0}(X_{d})$. The following notion of divergence on average is
crucial for us:

######  Definition 6.1 (cf. [KKLM17] Theorem 1.1).

We say $x\in X_{d}$ is divergent on average if

$\lim_{T\to\infty}\mu_{x,T}=0\ \text{ in }\mathcal{M}(X_{d}).$

######  Theorem 6.2.

Fix $k\in{\mathbb{N}}$. Let $x\in\mathbf{Div}(k)$. Then, $x$ is divergent on
average.

###### Proof.

Note that the Banach-Aloaglu theorem implies that every sequence
$(\mu_{x,T_{i}})_{i\in{\mathbb{N}}}$ has a limit point in
$\mathcal{M}(X_{d})$. Thus, to prove the theorem it suffices to show that for
any sequence $(T_{i})_{i\in{\mathbb{N}}}$ of divergent times,

$\left(\lim_{i\to\infty}\mu_{x,T_{i}}=\mu\ \text{ in
}\mathcal{M}(X_{d})\right)\implies\left(\mu=0\right).$ (6.3)

Note that the limit measure $\mu$ in the equation above is necessarily non-
negative, and also invariant under the action of $(a_{t})_{t\in{\mathbb{R}}}$.
Thus, if we have a limit $\mu$ which is nonzero, we may scale to obtain an
$(a_{t})_{t\in{\mathbb{R}}}$-invariant probability measure $\nu$. Assume we
have such a $\nu$ for sake of contradiction.

First observe that $\text{supp}(\nu)=\text{supp}(\mu)$ and so any
$y\in\text{supp}(\nu)$ must belong to $D(x)$. In particular, there must be
some $j<k$ for which

$y\in\mathbf{Div}(j).$

On the other hand, we claim that we can find a $y\in\text{supp}(\nu)$ for
which there exists a sequence of unbounded positive times
$(t_{i})_{i\in{\mathbb{N}}}$ with

$\lim_{t\to\infty}a_{t_{i}}y=y.$ (6.4)

To see this, consider a countable base for the topology of $X_{d}$,
$\\{U_{l}\subset X_{d}:l\in{\mathbb{N}}\\}$, and consider the sets

$A_{l}:=\left\\{z\in U_{l}:a_{j}z\notin U_{l}\text{ for all
}j\in{\mathbb{N}}\right\\}.$

Poincare recurrence shows that $\nu\left(\cup A_{l}\right)=0$.

Thus, we can choose any $y\in\text{supp}(\nu)\smallsetminus\cup A_{l}$ and get
the claim in (6.4). This contradicts the fact that $y\in\mathbf{Div}(j)$. ∎

Applying [KKLM17, Theorem 1.1], we immediately get:

######  Corollary 6.3.

For each $k\in{\mathbb{N}}$, the Hausdorff dimension of $\mathbf{Div}(k)$ is
bounded above by

$\dim(X_{d})-\frac{mn}{m+n}.$

### Acknowledgements

The second, third and fourth name authors acknowledge generous support from
the European Research Council (ERC) under the European Union’s Horizon 2020
Research and Innovation Program, Grant agreement no. 754475. The authors thank
the organizers of the 2022 conference ‘Group Actions, Geometry and Dynamics’
in Ohalo, Israel where this collaboration began. We also thank Tushar Das for
taking the time to answer some questions.

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