#  Bohr recurrence and density of non-lacunary semigroups of ${\mathbb{N}}$

Nikos Frantzikinakis  University of Crete, Department of mathematics and
applied mathematics, Voutes University Campus, Heraklion 71003, Greece
<EMAIL_ADDRESS>,  Bernard Host  LAMA, Univ Gustave Eiffel, CNRS,
F-77447 Marne-la-Vallée, France<EMAIL_ADDRESS>and  Bryna Kra
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
<EMAIL_ADDRESS>
###### Abstract.

A subset $R$ of integers is a set of Bohr recurrence if every rotation on
${\mathbb{T}}^{d}$ returns arbitrarily close to zero under some non-zero
multiple of $R$. We show that the set $\\{k!\,2^{m}3^{n}\colon
k,m,n\in{\mathbb{N}}\\}$ is a set of Bohr recurrence. This is a particular
case of a more general statement about images of such sets under any integer
polynomial with zero constant term. We also show that if $P$ is a real
polynomial with at least one non-constant irrational coefficient, then the set
$\\{P(2^{m}3^{n})\colon m,n\in{\mathbb{N}}\\}$ is dense in ${\mathbb{T}}$,
thus providing a joint generalization of two well-known results, one of
Furstenberg and one of Weyl.

###### Key words and phrases:

Bohr recurrence, non-lacunary semigroups, Furstenberg’s theorem, Weyl’s
theorem.

###### 2020 Mathematics Subject Classification:

Primary: 37B20; Secondary: 37A44, 37C85, 11J71, 05B10.

NF was supported by the research grant ELIDEK-HFRI-FM17-1684 and BK was
partially supported by the National Science Foundation grant DMS-2348315.

##  1\. Main results

The objective of this note is to derive recurrence and density properties of
sets generated by thin non-lacunary semigroups of the integers, a prototypical
example being the set $\\{2^{m}3^{n}\colon m,n\in{\mathbb{N}}\\}.$

###  1.1. Bohr recurrence

A well-known question in dynamics is whether the set

$\\{k!\,2^{m}3^{n}\colon k,m,n\in{\mathbb{N}}\\}.$

is good for topological or measurable (multiple) recurrence. Although we are
unable to answer these questions, we derive some recurrence results of
intermediate strength, which follow from Furstenberg’s Diophantine Theorem
(see Section 1.2), using elementary manipulations.

Let ${\mathbb{T}}^{d}=({\mathbb{R}}/{\mathbb{Z}})^{d}$ denote the
$d$-dimensional torus and for $x\in{\mathbb{R}}^{d}$, write
$\left\|x\right\|_{{\mathbb{T}}^{d}}:=d(x,{\mathbb{Z}}^{d})$.

######  Definition.

A subset $R$ of the integers is a _set of Bohr recurrence_ if for every
integer $d\in{\mathbb{N}}$, all $\alpha_{1},\dots,\alpha_{d}\in{\mathbb{R}}$,
and every $\varepsilon>0$, there exists non-zero $r\in R$ such that

$\left\|r\alpha_{j}\right\|_{\mathbb{T}}\leq\varepsilon\quad\text{for
}j=1,\dots,d.$

Equivalently, the set $R$ is a set of Bohr recurrence if for every integer
$d\in{\mathbb{N}}$ and every $x\in{\mathbb{T}}^{d}$, the point $0$ belongs to
the closure of the set $\\{rx\colon r\in R\setminus\\{0\\}\\}$ in
${\mathbb{T}}^{d}$. An important open problem, popularized by Katznelson [6],
is whether a set of Bohr recurrence is necessarily a set of topological
recurrence.

Two positive integers are _multiplicatively independent_ if their only common
power is $1$. The first main goal of this note is to establish the following
result:

######  Theorem 1.

Let $p,q\in{\mathbb{N}}$ be multiplicatively independent integers. Then the
set $\\{k!\,p^{m}q^{n}\colon k,m,n\in{\mathbb{N}}\\}$ is a set of Bohr
recurrence.

More generally, it follows from our argument that $k!$ can be replaced by any
sequence that contains multiples of every positive integer.

On the other hand, considering rational multiples of elements of this set
shows that the multiplicative factor $k!$ cannot be removed, and, in fact, it
is necessary for our argument to work even if the real numbers
$\alpha_{1},\ldots,\alpha_{d}$ have no rational relations. As shown in [7,
Theorem 2], for $d\geq 4$, there exists $x\in{\mathbb{T}}^{d}$ with rationally
independent coordinates for which the set $\\{2^{m}3^{n}x\colon
m,n\in{\mathbb{N}}\\}$ is not dense in ${\mathbb{T}}^{d}$. Although for fixed
$d\in{\mathbb{N}}$ this particular obstruction can be removed if more powers
are included in the set (see [8] for a density statement on ${\mathbb{T}}^{2}$
that uses three powers), [7, Theorem 2] shows that the problem persists even
if one uses $\ell$ powers instead of two and take $d$ large enough; in this
case we have non-density in ${\mathbb{T}}^{d}$ for some $x\in{\mathbb{T}}^{d}$
with rationally independent coordinates whenever $d\geq 2\ell$. Furthermore,
by slightly modifying the example in [7, Lemma 4.2], we obtain a point
$x\in{\mathbb{T}}^{4}$ with rationally independent coordinates, such that for
some $\varepsilon>0$ we have
$\left\|2^{m}3^{n}x\right\|_{{\mathbb{T}}^{4}}\geq\varepsilon$ for every
$m,n\in{\mathbb{N}}$ (see Section 3.2).

We also remark that combining Theorem 1 with [5, Theorem 4.1] and [4, Theorem
A], we deduce that the sets in Theorem 1 (and in Theorem 2 below) are also
good for topological recurrence for all nilsystems and for a large class of
skew product systems.

In the proof of Theorem 1, we consider a more general situation where sets of
the form $\\{p^{m}q^{n}\colon m,n\in{\mathbb{N}}\\}$ are replaced by more
general non-lacunary semigroups of ${\mathbb{N}}$.

######  Definition.

A _semigroup of ${\mathbb{N}}$_ is a sub-semigroup of $({\mathbb{N}},\cdot)$
and it is _non-lacunary_ if it contains two multiplicatively independent
integers.

Equivalently, a semigroup is non-lacunary if it is not contained in the set of
powers of a single integer. It is shown in [3, Lemma IV.1] that a semigroup
$\Sigma=\\{s_{1}<s_{2}<\cdots\\}$ of ${\mathbb{N}}$ is non-lacunary if and
only if $s_{n+1}/s_{n}\to 1$ as $n\to\infty$.

We say that $P$ is an _integer polynomial_ if it takes integer values on the
integers. We prove the following result, generalizing Theorem 1.

######  Theorem 2.

Let $\Sigma$ be a non-lacunary semigroup of ${\mathbb{N}}$,
$K\subset{\mathbb{N}}$ be a set that contains multiples of every positive
integer, and $P$ be a non-constant integer polynomial with $P(0)=0$. Then the
set $\\{P(ks)\colon k\in K,s\in\Sigma\bigr{\\}}$ is a set of Bohr recurrence.

In particular, if $p,q\in{\mathbb{N}}$ are multiplicatively independent and
$P$ is a non-constant integer polynomial with zero constant term, then the set

$\\{P(k!\,p^{m}q^{n})\colon k,m,n\in{\mathbb{N}}\bigr{\\}}$

is a set of Bohr recurrence. By taking $P(n)=n$, Theorem 1 follows.

The argument used to prove Theorem 2 can also be used, essentially without
change, to show that if $P_{1},\ldots,P_{\ell}$ are integer polynomials with
zero constant terms, then for every
$\alpha_{1},\ldots,\alpha_{\ell}\in{\mathbb{R}}$ and $\varepsilon>0$, there
exist $k\in K$ and $s\in\Sigma$ such that $\left\|P_{j}(k\cdot
s)\,\alpha_{j}\right\|_{\mathbb{T}}\leq\varepsilon$ for $j=1,\ldots,\ell$.

We prove Theorem 2 in Section 3. The main ingredient is Proposition 4, which
shows the existence of a point with rational coordinates in the orbit closure
of non-lacunary semigroup actions of ${\mathbb{T}}^{d}$.

###  1.2. A Weyl-type extension of Furstenberg’s result

We recall a celebrated theorem of Furstenberg [3, Theorem IV.1] (see also [2]
for an elementary proof):

######  Theorem (Furstenberg).

Let $\Sigma$ be a non-lacunary semigroup of ${\mathbb{N}}$ and $\alpha$ be
irrational. Then the set $\\{s\alpha\colon s\in\Sigma\\}$ is dense in
${\mathbb{T}}$.

This immediately implies that a closed $\Sigma$-invariant subset of
${\mathbb{T}}$ is either dense or finite. In the latter case, it consists
entirely of rational points.

Another celebrated theorem, this time of Weyl [9], is as follows:

######  Theorem (Weyl).

Let $P\in{\mathbb{R}}[t]$ be a polynomial with at least one non-constant
irrational coefficient. Then the set $\\{P(n)\colon n\in{\mathbb{N}}\\}$ is
dense in ${\mathbb{T}}$.

We prove the following common generalization of these two density statements:

######  Theorem 3.

Let $\Sigma$ be a non-lacunary semigroup and $P\in{\mathbb{R}}[t]$ be a
polynomial with at least one non-constant irrational coefficient. Then the set

$\\{P(s)\colon s\in\Sigma\\}$

is dense in ${\mathbb{T}}$.

Taking $P(n):=n$ gives Furstenberg’s result and taking $\Sigma:={\mathbb{N}}$
gives Weyl’s result. On the other hand, even for the simplest non-linear
cases, such as $P(n)=(n^{2}+n)\alpha$, where $\alpha$ is irrational, it is not
clear how to proceed and prove density, since the set $\\{P(s)\colon
s\in\Sigma\\}$ does not seem to satisfy any useful $\Sigma$-invariance
property. A key maneuver in our argument is to instead work with the subset
$\\{(s\alpha,s^{2}\alpha)\colon s\in\Sigma\\}$ of ${\mathbb{T}}^{2}$, which is
invariant under multiplication by all elements of the set $\\{(s,s^{2})\colon
s\in\Sigma\\}$. In Theorem 5, we show that this invariance implies that the
closure of this set contains a line segment parallel to some coordinate axis.
Theorem 3 follows easily from this fact.

Finally, we note that more general $d$-dimensional versions of Theorem 3 fail.
For example, if $P_{1},\ldots,P_{d}\in{\mathbb{R}}[t]$ are such that every
non-trivial integer combination of the polynomials is a polynomial with at
least one irrational non-constant coefficient, and $\Sigma$ is a non-lacunary
semigroup of ${\mathbb{N}}$, it is not always true that the set

$\\{(P_{1}(s),\ldots,P_{d}(s))\colon s\in\Sigma\\}$

is dense in ${\mathbb{T}}^{d}$ (this holds for $\Sigma:={\mathbb{N}}$ since
equidistribution follows from Weyl’s criterion). To see this, take $d=4$ and
$\Sigma:=\\{2^{m}3^{n}\colon m,n\in{\mathbb{N}}\\}$, $P_{j}(n)=n\alpha_{j}$,
$j=1,\ldots,4$, for the rationally independent reals
$\alpha_{1},\ldots,\alpha_{4}$ constructed in [7, Lemma 4.2], so that the
above set is not dense.

###  1.3. Notation

We denote the set of positive integers by ${\mathbb{N}}$. We let
${\mathbb{T}}$ denote the one dimensional torus ${\mathbb{R}}/{\mathbb{Z}}$,
and we often identify it with $[0,1)$. We denote elements of ${\mathbb{T}}$ by
real numbers and we implicitly assume that these real numbers are taken modulo
$1$.

If $(t_{1},\ldots,t_{d})$ and $(x_{1},\ldots,x_{d})$ are two vectors in
${\mathbb{R}}^{d}$, we define their product by coordinate-wise multiplication
as follows

$(t_{1},\ldots,t_{d})\cdot(x_{1},\ldots,x_{d}):=(t_{1}x_{1},\ldots,t_{d}x_{d}).$

Likewise, if $(k_{1},\ldots,k_{d})\in{\mathbb{Z}}^{d}$ and
$(x_{1},\ldots,x_{d})\in{\mathbb{T}}^{d}$, we write

$(k_{1},\ldots,k_{d})\cdot(x_{1},\ldots,x_{d}):=(k_{1}x_{1},\ldots,k_{d}x_{d}).$

If $S$ is a subset of ${\mathbb{Z}}^{d}$ and $A$ a subset of
${\mathbb{T}}^{d}$, we say that $A$ is $S$-invariant or invariant under $S$ if
$s\cdot x\in A$ for every $s\in S$, $x\in A$.

###  1.4. Acknowledgement

The authors would like to thank J. Griesmer and R. Alweiss whose questions
related to the proof of Theorem 1 motivated the authors to write this
manuscript.

##  2\. Existence of rational points

Furstenberg’s Theorem implies that if $\Sigma$ is a non-lacunary semigroup of
${\mathbb{N}}$, then a closed $\Sigma$-invariant subset of ${\mathbb{T}}$
always contains a rational point. A natural question to consider is whether
there are generalizations of this statement in higher dimensions. We prove
that this is the case, and use this result in the proofs of both Theorems 2
and 3. To give the precise statement, we need another definition.

######  Definition.

A point $x\in{\mathbb{T}}^{d}$ is _rational_ if its coordinates are rational.
If $\Sigma$ is a subset of ${\mathbb{N}}$, we say that the _denominators of a
rational point $x$ are relatively prime to $\Sigma$_ if its coordinates are
rationals with denominators relatively prime to each element of $\Sigma$. By
convention, the denominator of $0$ is $1$.

######  Proposition 4.

Let $\Sigma$ be a non-lacunary semigroup, $d\in{\mathbb{N}}$, and
$\ell_{1},\ldots,\ell_{d}\in{\mathbb{N}}$ (not necessarily distinct). Let $A$
be a non-empty closed subset of ${\mathbb{T}}^{d}$, invariant under

$\Sigma^{\ell_{1},\ldots,\ell_{d}}:=\\{(s^{\ell_{1}},\ldots,s^{\ell_{d}}):s\in\Sigma\\}.$

Then $A$ contains a rational point with denominators relatively prime to
$\Sigma$.

######  Remark.

For the proof of Theorem 1, only the case $\ell_{1}=\cdots=\ell_{d}=1$ is
needed. The more general case is needed in the proofs of Theorems 2 and 3.

###### Proof.

We proceed by induction on $d\in{\mathbb{N}}$. Throughout, for
$\ell\in{\mathbb{N}}$ we let

$\Sigma^{\ell}:=\\{s^{\ell}:s\in\Sigma\\}.$

#### The case $d=1$

Since $\Sigma^{\ell_{1}}$ is a non-lacunary semigroup of ${\mathbb{N}}$, by
Furstenberg’s Theorem (see Section 1.2) we have that $A$ contains a rational
point $x=a/b$ with $a\in{\mathbb{Z}}$ and $b\in{\mathbb{N}}$. Write
$b=b^{\prime}c$, where $b^{\prime}$ divides some element $k$ of $\Sigma$ and
$c$ is relatively prime to every element of $\Sigma$. Then
$kx=\frac{a(k/b^{\prime})}{c}$ and thus this point of $A$ is rational with
denominator relatively prime to $\Sigma$.

#### Suppose that the result holds for some $d\in{\mathbb{N}}$

We show that it holds for $d+1$. Let $A$ be a non-empty closed subset of
${\mathbb{T}}^{d+1}$, invariant under $\Sigma^{\ell_{1},\ldots,\ell_{d+1}}$,
where $\ell_{1},\ldots,\ell_{d+1}\in{\mathbb{N}}$. Let $B$ be the image of $A$
under the coordinate projection

$(x_{1},\dots,x_{d},x_{d+1})\mapsto x_{d+1}$

of ${\mathbb{T}}^{d+1}$ onto ${\mathbb{T}}$. Then $B$ is a non-empty closed
subset of ${\mathbb{T}}$, invariant under the non-lacunary semigroup
$\Sigma^{\ell_{d+1}}$, and hence by the case $d=1$, it contains a rational
point $y$ with denominator relatively prime to $\Sigma$.

Write $y=a/m$, where $a\in{\mathbb{Z}}$ and $m$ is a positive integer
relatively prime to every element of $\Sigma$, and let

$\Sigma_{m}:=\Sigma\cap(m{\mathbb{Z}}+1).$

Then $\Sigma_{m}$ is a semigroup of ${\mathbb{N}}$ with the property that

$sy=y\quad\text{for every }s\in\Sigma_{m}.$

We claim that $\Sigma_{m}$ is non-lacunary. Indeed, if $p$ and $q$ are two
multiplicatively independent integers belonging to $\Sigma$, then $p$ and $q$
are relatively prime to $m$ (because the denominator of $y=a/m$ is relatively
prime to $\Sigma$). Hence, by Euler’s Theorem, there exist integers $t$ and
$n$ such that $p^{t}\equiv 1\pmod{m}$ and $q^{n}\equiv 1\pmod{m}$. Then
$p^{t}$ and $q^{n}$ are multiplicatively independent and belong to
$\Sigma_{m}$ and thus the semigroup $\Sigma_{m}$ is non-lacunary.

We define the set

$C:=\bigl{\\{}(x_{1},\dots,x_{d})\in{\mathbb{T}}^{d}\colon(x_{1},\dots,x_{d},y)\in
A\bigr{\\}}.$

Then $C$ is a closed subset of ${\mathbb{T}}^{d}$, and is non-empty since
$y\in B$. We claim that $C$ is invariant under the semigroup

$\Sigma_{m}^{\ell_{1},\ldots,\ell_{d}}:=\\{(s^{\ell_{1}},\ldots,s^{\ell_{d}}):s\in\Sigma_{m}\\}.$

Indeed, if $(x_{1},\dots,x_{d})\in C$, using that $s^{\ell_{d+1}}y=y$ for
every $s\in\Sigma_{m}$ and the invariance of $A$ under
$\Sigma^{\ell_{1},\ldots,\ell_{d+1}}$, we have that

$(s^{\ell_{1}}x_{1},\dots,s^{\ell_{d}}x_{d},y)=(s^{\ell_{1}}x_{1},\dots,s^{\ell_{d}}x_{d},s^{\ell_{d+1}}y)\in
A\quad\text{for every }s\in\Sigma_{m}.$

Using the defining property of $C$, we deduce that
$(s^{\ell_{1}}x_{1},\dots,s^{\ell_{d}}x_{d})\in C$.

Using the induction hypothesis with $C$ substituted for $A$ and
$\Sigma_{m}^{\ell_{1},\ldots,\ell_{d}}$ substituted for
$\Sigma^{\ell_{1},\ldots,\ell_{d}}$, we deduce that $C$ contains a rational
point $(y_{1},\dots,y_{d})$ with denominators relatively prime to $\Sigma$.
Then the point $(y_{1},\dots,y_{d},y)$ is rational with denominators
relatively prime to $\Sigma$, and belongs to $A$, as desired. ∎

##  3\. Proof of the Bohr-recurrence result

###  3.1. Proof of Theorem 2

Suppose that

$P(n)=c_{1}n+c_{2}n^{2}+\cdots+c_{r}n^{r}$

for some $r\in{\mathbb{N}}$ and $c_{1},\ldots,c_{r}\in{\mathbb{Z}}$, not all
of them zero. Let $\varepsilon>0$, $d\in{\mathbb{N}}$, and
$x=(x_{1},\ldots,x_{d})\in{\mathbb{T}}^{d}$. Let $A$ be the closure of the set

$\\{(sc_{1}x_{1},\dots,sc_{1}x_{d},s^{2}c_{2}x_{1},\dots,s^{2}c_{2}x_{d},\dots,s^{r}c_{r}x_{1},\dots,s^{r}c_{r}x_{d}):s\in\Sigma\\}.$

Then $A$ is a non-empty closed subset of ${\mathbb{T}}^{rd}$ that is invariant
under

$\\{(s,\ldots,s,s^{2},\ldots,s^{2},\ldots,s^{r},\ldots,s^{r}):s\in\Sigma\\},$

where each power of $s$ is repeated $d$ times. Therefore, by Proposition 4,
$A$ contains a rational point $y\in{\mathbb{T}}^{rd}$. Since $K$ contains
multiples of every positive integer, there exists $k\in K$ such that all the
coordinates of $ky$ are integers. It follows that $0$ (as an element of
${\mathbb{T}}^{rd}$) belongs to the closure of the set

$\\{(ksc_{1}x,(ks)^{2}c_{2}x,\ldots,(ks)^{r}c_{r}x):s\in\Sigma\\}.$

Since $P(ks)\,x=ksc_{1}x+(ks)^{2}c_{2}x+\cdots+(ks)^{r}c_{r}x,$ we have that

$\left\|P(ks)\,x\right\|_{{\mathbb{T}}^{d}}\leq\left\|ksc_{1}x\right\|_{{\mathbb{T}}^{d}}+\left\|(ks)^{2}c_{2}x\right\|_{{\mathbb{T}}^{d}}+\cdots+\left\|(ks)^{r}c_{r}x\right\|_{{\mathbb{T}}^{d}}.$

It follows that $0$ (as an element of ${\mathbb{T}}^{d}$) belongs to the
closure of the set

$\\{P(ks)\,x\colon k\in K:s\in\Sigma\bigr{\\}}$

and so

(1) $\left\|P(ks)\,x\right\|_{{\mathbb{T}}^{d}}\leq\varepsilon,\quad\text{for
some }k\in K,s\in\Sigma.$

The polynomial $P$ has finitely many zeros, all bounded by some integer
$N\in{\mathbb{N}}$. Estimate (1) also holds for the semigroup
$\Sigma\cap(N,+\infty)$ instead of $\Sigma$, and thus replacing $\Sigma$ with
this semigroup we deduce that there exist $k\in K$ and $s\in\Sigma$ such that
$P(ks)\neq 0$ and $\left\|P(ks)\,x\right\|_{{\mathbb{T}}^{d}}\leq\varepsilon$.
It follows that the set $\\{P(ks)\colon k\in K,s\in\Sigma\bigr{\\}}$ is a set
of Bohr recurrence. ∎

###  3.2. Non Bohr recurrence of $2^{m}3^{n}$ for totally ergodic rotations

Although the set $\\{2^{m}3^{n}\colon m,n\in{\mathbb{N}}\\}$ is not good for
recurrence of rational rotations, it is reasonable to hope that it is good for
recurrence for all $x\in{\mathbb{T}}^{d}$ that have rationally independent
coordinates. An even more optimistic conjecture is that it is good for
measurable recurrence for all totally ergodic systems. Unfortunately, we show
that neither claim holds.

We claim that there exists $x\in{\mathbb{T}}^{4}$ with rationally independent
coordinates and $\varepsilon>0$ such that
$\left\|2^{m}3^{n}x\right\|_{{\mathbb{T}}^{4}}\geq\varepsilon$ for every
$m,n\in{\mathbb{N}}$. Indeed, using [7, Lemma 4.2], we have the existence of
$y=(y_{1},y_{2},y_{3},y_{4})\in{\mathbb{R}}^{4}$ with rationally independent
coordinates such that for all but finitely many $m,n\in{\mathbb{N}}$ we have
$\\{2^{m}3^{n}y_{j}\\}\leq 1/10$ for some $j\in\\{1,2,3,4\\}$, where $\\{t\\}$
denotes the fractional part of $t\in{\mathbb{R}}$. Let $x_{j}:=y_{j}+1/5$,
$j\in\\{1,2,3,4\\}$, and $x=(x_{1},x_{2},x_{3},x_{4})$; then obviously $x$
also has rationally independent coordinates. Since
$\\{2^{m}3^{n}/5\\}\in[1/5,4/5]$ for all $m,n\in{\mathbb{N}}$, it follows that
for all but finitely many $m,n\in{\mathbb{N}}$ we have
$\\{2^{m}3^{n}x_{j}\\}\in[1/5,9/10]$ for some $j\in\\{1,2,3,4\\}$. Then
$\left\|2^{m}3^{n}x\right\|_{{\mathbb{T}}^{4}}\geq 1/10$ for all but finitely
many $m,n\in{\mathbb{N}}$. This immediately implies the claim.

The situation does not improve much if we consider sets of the form

$\Sigma:=\\{p_{1}^{n_{1}}\cdots p_{\ell}^{n_{\ell}}\colon
n_{1},\ldots,n_{\ell}\in{\mathbb{N}}\\}.$

Using [7, Lemma 4.2], we can show in a similar fashion that there exists
$x\in{\mathbb{T}}^{2\ell}$ with rationally independent coordinates and
$\varepsilon>0$ such that
$\left\|sx\right\|_{{\mathbb{T}}^{2\ell}}\geq\varepsilon$ for every
$s\in\Sigma$.

However, the possibility that the set $\\{2^{m}3^{n}\colon
m,n\in{\mathbb{N}}\\}$ is good for measurable or topological recurrence for
systems without rotational factors (i.e. for weakly mixing systems) remains
open. For measurable recurrence this problem is explicitly stated in [1,
Question 3].

##  4\. Proof of the density result

Our goal in this section is to prove Theorem 3.

###  4.1. A reduction

Theorem 3 is a direct consequence of the following higher dimensional density
result. This higher dimensional setting has the advantage of some dilation
invariance that the $1$-dimensional setting lacks.

Let $e_{1},\ldots,e_{d}$ denote the standard unit vectors of
${\mathbb{T}}^{d}$ or ${\mathbb{R}}^{d}$, depending on the context.

######  Theorem 5.

Let $d\in{\mathbb{N}}$, $\Sigma$ be a non-lacunary semigroup of
${\mathbb{N}}$, and $A$ be a closed infinite subset of ${\mathbb{T}}^{d}$ that
is invariant under

$\\{(s,s^{2},\ldots,s^{d})\colon s\in\Sigma\\}.$

Then $\\{x+te_{j}\colon t\in{\mathbb{T}}\\}\subset A$ for some rational point
$x\in{\mathbb{T}}^{d}$ and $j\in\\{1,\ldots,d\\}$.

###### Proof of Theorem 3 assuming Theorem 5.

Let $\varepsilon>0$ and $\alpha\in{\mathbb{T}}$ be arbitrary. It suffices to
show that there exists $s\in\Sigma$ such that
$\left\|P(s)-\alpha\right\|_{\mathbb{T}}\leq\varepsilon$.

We can assume that $P(0)=0$ and write $P(n)=c_{1}n+\cdots+c_{d}n^{d}$ for some
$d\in{\mathbb{N}}$ and real numbers $c_{1},\ldots,c_{d}$, at least one of
which is irrational. Let $A$ be the closure of the set

$\\{(c_{1}s,c_{2}s^{2},\ldots,c_{d}s^{d})\colon n\in\Sigma\\},$

considered as a subset of ${\mathbb{T}}^{d}$. Clearly $A$ is invariant under
the map $x\mapsto(s,s^{2},\ldots,s^{d})\cdot x$ for every $s\in\Sigma$, and
since at least one of the $c_{1},\ldots,c_{d}$ is irrational the set $A$ is
infinite. It follows from Theorem 5 that there exist some rational point
$x=(x_{1},\ldots,x_{d})\in{\mathbb{T}}^{d}$ and $j_{0}\in\\{1,\ldots,d\\}$
such that $\\{x+te_{j_{0}}\colon t\in{\mathbb{T}}\\}\subset A$. Hence, for
$t:=\alpha-\sum_{1\leq j\leq d}x_{j}\pmod{1}$ there exists $s\in\Sigma$ such
that

$\left\|c_{j}s^{j}-x_{j}\right\|_{\mathbb{T}}\leq\frac{\varepsilon}{d}\,\text{
for }\,j\neq j_{0}\quad\text{ and
}\quad\left\|c_{j_{0}}s^{j_{0}}-\alpha+\sum_{1\leq j\leq d,\,j\neq
j_{0}}x_{j}\right\|_{\mathbb{T}}\leq\frac{\varepsilon}{d}.$

Then

$\left\|P(s)-\alpha\right\|_{\mathbb{T}}\leq\sum_{1\leq j\leq d,\,j\neq
j_{0}}\left\|c_{j}s^{j}-x_{j}\right\|_{\mathbb{T}}+\left\|c_{j_{0}}s^{j_{0}}-\alpha+\sum_{1\leq
j\leq d,\,j\neq j_{0}}x_{j}\right\|_{\mathbb{T}}\leq\varepsilon,$

completing the proof. ∎

###  4.2. A key density property on ${\mathbb{T}}^{d}$

The main goal in this subsection is to prove Proposition 7. In its proof we
use the following fact.:

######  Lemma 6.

For every $d\in{\mathbb{N}}$ and
$u_{1},\ldots,u_{d}\in{\mathbb{R}}\setminus\\{0\\}$, the set

$\\{(tu_{1}+k_{1},t^{2}u_{2},\ldots,t^{d}u_{d}+k_{d})\colon
t\in(0,+\infty),k_{1},\ldots,k_{d}\in{\mathbb{Z}}\\}$

is dense in ${\mathbb{R}}^{d}$.

###### Proof.

We can assume that $u_{1}>0$. First, note that if we let $t^{\prime}:=tu_{1}$
and $u_{j}^{\prime}:=u_{j}/u_{1}^{j}$, for $j=2,\ldots,d$, we see that it is
sufficient to verify the stated property when $u_{1}=1$. We work with this
assumption from now on.

Let $x:=(x_{1},\ldots,x_{d})\in{\mathbb{R}}^{d}$. It suffices to show that for
every $\varepsilon>0$ there exists $t>0$ such that

(2)
$\left\|t-x_{1}\right\|_{\mathbb{T}}\leq\varepsilon,\,\left\|t^{2}u_{2}-x_{2}\right\|_{\mathbb{T}}\leq\varepsilon,\ldots,\,\left\|t^{d}u_{d}-x_{d}\right\|_{\mathbb{T}}\leq\varepsilon.$

So let $\varepsilon>0$. First, we claim that there exists
$\delta_{0}\in(0,\varepsilon)$ such that every non-trivial integer combination
of the polynomials (in $k$)

$(k+x_{1}+\delta_{0})^{2}u_{2},\ldots,\,(k+x_{1}+\delta_{0})^{d}u_{d}$

has at least one irrational non-constant coefficient. Indeed, note that modulo
rational multiples, the coefficients of the (degree $1$) monomial $k$ in these
polynomials are respectively

$(x_{1}+\delta_{0})u_{2},\,(x_{1}+\delta_{0})^{2}u_{3},\ldots,\,(x_{1}+\delta_{0})^{d-1}u_{d}.$

So it suffices to choose $\delta_{0}$ such that all non-trivial integer
combinations of these numbers are irrational. Such a choice of $\delta_{0}$
exists, since for any choice of $\ell_{1},\ldots,\ell_{d}\in{\mathbb{Z}}$, not
all of them $0$, the equation

$\ell_{1}u_{2}\delta+\ell_{2}u_{3}\delta^{2}+\cdots+\ell_{d}u_{d}\delta^{d-1}=0$

has finitely many solutions in the variable $\delta$, so there exists
$\delta_{0}\in(0,\varepsilon)$ such that $x_{1}+\delta_{0}$ avoids all these
(countably many) solutions. This proves the claim.

For this choice of $\delta_{0}$, Weyl’s criterion [9] gives that the sequence

$\big{(}(k+x_{1}+\delta_{0})^{2}u_{2},\ldots,(k+x_{1}+\delta_{0})^{d}u_{d}\big{)}_{k\in{\mathbb{N}}}$

is equidistributed in ${\mathbb{T}}^{d}$ and thus dense in ${\mathbb{T}}^{d}$.
It follows that there exists an integer $k_{0}>|x_{1}|+\delta_{0}$ such that

$\left\|(k_{0}+x_{1}+\delta_{0})^{2}-x_{2}\right\|_{\mathbb{T}}\leq\varepsilon,\ldots,\left\|(k_{0}+x_{1}+\delta_{0})^{d}-x_{d}\right\|_{\mathbb{T}}\leq\varepsilon.$

Letting $t:=k_{0}+x_{1}+\delta_{0}$, and recalling that
$\delta_{0}\in(0,\varepsilon)$, we deduce that (2) holds, completing the
proof. ∎

######  Proposition 7.

Let $d\in{\mathbb{N}}$, $\Sigma$ be a non-lacunary semigroup of
${\mathbb{N}}$, and $A$ be a closed subset of ${\mathbb{T}}^{d}$, invariant
under

$\\{(s,s^{2},\ldots,s^{d})\colon s\in\Sigma\\}.$

If $0$ is a non-isolated point of $A$, then $\\{te_{j}\colon
t\in{\mathbb{T}}\\}\subset A$ for some $j\in\\{1,\ldots,d\\}$.

###### Proof.

For $d=1$, this follows from Furstenberg’s Theorem (see Section 1.2). For
general $d\in{\mathbb{N}}$, we argue as follows using ideas from [3] and [8].,
with the caveat that we are not working with the Euclidean norm but with the
quantity defined in (3).

Let
$\pi\colon{\mathbb{R}}^{d}\to{\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$
be the natural projection and let $B=\pi^{-1}(A)=A+{\mathbb{Z}}^{d}$. Then for
every $s\in\Sigma$, the closed subset $B$ of ${\mathbb{R}}^{d}$ is invariant
under the map $x\mapsto(s,s^{2},\ldots,s^{d})\cdot x$, and the point $0$ of
${\mathbb{R}}^{d}$ is a non-isolated point of $B$.

For $x=(x_{1},\ldots,x_{d})\in{\mathbb{R}}^{d}$ let

(3)
$\lvert\\!|\\!|x|\\!|\\!\rvert:=|x_{1}|+|x_{2}|^{\frac{1}{2}}+\cdots+|x_{d}|^{\frac{1}{d}}.$

Note that $\lvert\\!|\\!|\cdot|\\!|\\!\rvert$ satisfies the triangle
inequality and we have the following identity

(4)
$\lvert\\!|\\!|(tx_{1},t^{2}x_{2},\ldots,t^{d}x_{d})|\\!|\\!\rvert=|t|\lvert\\!|\\!|(x_{1},x_{2},\ldots,x_{d})|\\!|\\!\rvert.$

Note also that a sequence of vectors in ${\mathbb{R}}^{d}$ converges to a
vector $x$ with respect to the Euclidean norm if and only if it converges to
$x$ with respect to the distance associated with
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert$; we use this fact without further
reference.

For every $\varepsilon>0$, we let

(5)
$D_{\varepsilon}:=\Bigl{\\{}\Big{(}\frac{x_{1}}{\lvert\\!|\\!|x|\\!|\\!\rvert},\frac{x_{2}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{2}},\ldots,\frac{x_{d}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{d}}\Big{)}\colon
x\in B,\ 0<\lvert\\!|\\!|x|\\!|\\!\rvert\leq\varepsilon\Bigr{\\}}$

and

(6) $D:=\bigcap_{\varepsilon>0}\overline{D_{\varepsilon}}.$

Then $D$ is a closed subset of the compact set
$\\{x\in{\mathbb{R}}^{d}\colon\lvert\\!|\\!|x|\\!|\\!\rvert=1\\}$. Since, by
assumption, $0$ is a non-isolated point of $A$ (this is the only point where
this assumption is used), for every $\varepsilon>0$ the compact set
$\overline{D_{\varepsilon}}$ is non-empty, so the set $D$ is non-empty.

We claim that for every $u\in D$ and every $t>0$ we have
$(t,t^{2},\ldots,t^{d})\cdot u\in B$ (a similar statment holds for every
$t\in{\mathbb{R}}$ but we do not need this). So let $u=(u_{1},\ldots,u_{d})\in
D$, $t>0$, and $\varepsilon>0$. It suffices to show that there exists $b\in B$
with

(7) $\lvert\\!|\\!|(t,t^{2},\ldots,t^{d})\cdot
u-b|\\!|\\!\rvert\leq\varepsilon.$

Let $\Sigma=\\{s_{1}<s_{2}<\cdots\\}$. Since $\Sigma$ is a non-lacunary
semigroup of ${\mathbb{N}}$, we have $\lim_{n\to\infty}s_{n}=\infty$ and
$\lim_{n\to\infty}\frac{s_{n+1}}{s_{n}}=1$. Hence, for every $\delta>0$ (to be
chosen later, depending on $\varepsilon$ and $t$ only) there exists
$M=M(\delta)\geq s_{1}$ such that if $s_{n+1}>M$ we have

$|s_{n+1}^{j}-s_{n}^{j}|\leq\delta s_{n}^{j},\quad\text{for }j=1,\ldots,d.$

It follows that for every real $\sigma\geq M$, there exists $s\in\Sigma$ with

(8) $|s^{j}-\sigma^{j}|\leq\delta\cdot\sigma^{j},\quad\text{ for
}j=1,\ldots,d.$

Indeed, we can choose $s:=s_{n_{0}}$ where $n_{0}\in{\mathbb{N}}$ satisfies
$s_{n_{0}}\leq\sigma<s_{n_{0}+1}$, then $s_{n_{0}+1}>M$ and we have
$|\sigma^{j}-s_{n_{0}}^{j}|\leq|s_{n_{0}+1}^{j}-s_{n_{0}}^{j}|\leq\delta
s_{n_{0}}^{j}\leq\delta\sigma^{j}$.

Since $u$ belongs to $D$, by (5) and (6) there exists
$x=(x_{1},\ldots,x_{d})\in B$, depending on $\varepsilon,t,u,M$, with

(9)
$0<\lvert\\!|\\!|x|\\!|\\!\rvert\leq\frac{t}{M}\quad\text{and}\quad\lvert\\!|\\!|\big{(}\frac{x_{1}}{\lvert\\!|\\!|x|\\!|\\!\rvert},\frac{x_{2}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{2}},\ldots,\frac{x_{d}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{d}}\big{)}-u|\\!|\\!\rvert\leq\frac{\varepsilon}{2t}.$

Applying (8) with $\delta:=\frac{\varepsilon^{d}}{2^{d}(1+t^{d})}$ and
$\sigma:=\frac{t}{\lvert\\!|\\!|x|\\!|\\!\rvert}\geq M$ (by (9)), it follows
that there exists $s\in\Sigma$ (depending on $\varepsilon,t,x$) such that

(10)
$\Bigl{|}s^{j}-\frac{t^{j}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{j}}\Bigr{|}\leq\delta\cdot\frac{t^{j}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{j}}=\frac{\varepsilon^{d}}{2^{d}\lvert\\!|\\!|x|\\!|\\!\rvert^{j}},\quad\text{
for }j=1,\ldots,d.$

If we add and subtract
$\big{(}\frac{tx_{1}}{\lvert\\!|\\!|x|\\!|\\!\rvert},\frac{t^{2}x_{2}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{2}},\ldots,\frac{t^{d}x_{d}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{d}}\big{)}$
in the expression below and use the triangle inequality, we get

(11)
$\lvert\\!|\\!|(tu_{1},t^{2}u_{2},\ldots,t^{d}u_{d})-(sx_{1},s^{2}x_{2},\ldots,s^{d}x_{d}))|\\!|\\!\rvert\leq\\\
\lvert\\!|\\!|\Big{(}t\big{(}\frac{x_{1}}{\lvert\\!|\\!|x|\\!|\\!\rvert}-u_{1}\big{)},t^{2}\big{(}\frac{x_{2}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{2}}-u_{2}\big{)},\ldots,t^{d}\big{(}\frac{x_{d}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{d}}-u_{d}\big{)}\Big{)}|\\!|\\!\rvert+\\\
\lvert\\!|\\!|\big{(}x_{1}\big{(}\frac{t}{\lvert\\!|\\!|x|\\!|\\!\rvert}-s\big{)},x_{2}\big{(}\frac{t^{2}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{2}}-s^{2}\big{)},\ldots,x_{d}\big{(}\frac{t^{d}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{d}}-s^{d}\big{)}\big{)}|\\!|\\!\rvert\leq\varepsilon,$

where the last estimate follows from (9) and (10) as follows: By (4) the first
term is equal to

$t\,\large\lvert\\!|\\!|\big{(}\frac{x_{1}}{\lvert\\!|\\!|x|\\!|\\!\rvert},\frac{x_{2}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{2}},\ldots,\frac{x_{d}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{d}}\big{)}-u|\\!|\\!\rvert\leq\frac{\varepsilon}{2},$

where the last estimate follows from (9). By (3) the second term is equal to

$|x_{1}|\Big{|}\frac{t}{\lvert\\!|\\!|x|\\!|\\!\rvert}-s\Big{|}+|x_{2}|^{\frac{1}{2}}\Big{|}\frac{t^{2}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{2}}-s^{2}\Big{|}^{\frac{1}{2}}+\cdots+|x_{d}|^{\frac{1}{d}}\Big{|}\frac{t^{d}}{\lvert\\!|\\!|x|\\!|\\!\rvert^{d}}-s^{d}\Big{|}^{\frac{1}{d}},$

and using (10) we can bound this by

$\frac{\varepsilon}{2\lvert\\!|\\!|x|\\!|\\!\rvert}(|x_{1}|+|x_{2}|^{\frac{1}{2}}+\cdots+|x_{d}|^{\frac{1}{d}})=\frac{\varepsilon}{2}.$

It follows from (11) that (7) holds for $b:=(s,s^{2},\ldots,s^{d})\cdot x$,
which is an element of $B$ since $x\in B$ and $B$ is invariant under the map
$x\mapsto(s,s^{2},\ldots,s^{d})\cdot x$. This completes the proof of the
claim.

Let $u$ now be any element in $D$. We have just shown that the set
$C:=\\{(t,t^{2},\ldots,t^{d})\cdot u\colon t>0\\}$ is contained in $B$ and
thus the closure of its image $\pi(C)$ under the natural projection
$\pi\colon{\mathbb{R}}^{d}\to{\mathbb{T}}^{d}$ is contained in $A=\pi(B)$.
Since $u\in D$ we have $\lvert\\!|\\!|u|\\!|\\!\rvert=1$, hence there exists
$j_{0}\in\\{1,\ldots,d\\}$ such that $u_{j_{0}}\neq 0$. We deduce from this
and Lemma 6 that $\\{te_{j_{0}}\colon t\in{\mathbb{R}}\\}\subset
C+{\mathbb{Z}}^{d}\subset B$, hence $\\{te_{j_{0}}\colon
t\in{\mathbb{T}}\\}\subset A$. This completes the proof. ∎

We deduce the following:

######  Corollary 8.

Let $d\in{\mathbb{N}}$, $\Sigma$ be a non-lacunary semigroup of
${\mathbb{N}}$, and $A$ be a closed subset of ${\mathbb{T}}^{d}$ that is
invariant under

$\\{(s,s^{2},\ldots,s^{d})\colon s\in\Sigma\\}.$

If $x$ is a non-isolated rational point of $A$, with denominators relatively
prime to $\Sigma$, then $\\{x+te_{j}\colon t\in{\mathbb{T}}\\}\subset A$ for
some $j\in\\{1,\ldots,d\\}$.

###### Proof.

As in the proof of Proposition 4 we let $m$ be an integer that is relatively
prime to every element of $\Sigma$ and such that the coordinates of $x$ remain
invariant when multiplied by any element of the non-lacunary semigroup
$\Sigma_{m}:=\Sigma\cap(m{\mathbb{Z}}+1)$. Then the set $A-x$ is closed,
invariant under

$\\{(s,s^{2},\ldots,s^{d})\colon s\in\Sigma_{m}\\},$

and has $0$ as a non-isolated point. Applying Proposition 7 with $\Sigma_{m}$
instead of $\Sigma$, we get that there exists $j\in\\{1,\ldots,d\\}$ such that
$\\{te_{j}\colon t\in{\mathbb{T}}\\}\subset A-x$, completing the proof. ∎

###  4.3. Proof of Theorem 5

By combining Proposition 4 and Corollary 8, we complete the proof.

###### Proof of Theorem 5.

Since $A$ is an infinite subset of ${\mathbb{T}}^{d}$, the set $A^{\prime}$ of
non-isolated points of $A$ is non-empty. It is also closed and invariant under
the set $\Sigma^{1,\ldots,d}$ of Proposition 4. Applying Proposition 4 for
$\ell_{1}:=1,\ldots,\ell_{d}:=d$, we get that $A^{\prime}$ contains a rational
point with denominators relatively prime to $\Sigma$. The result now follows
from Corollary 8 and the fact that $A^{\prime}\subset A$ (since $A$ is
closed). ∎

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