Weak mixing and non-recurrence along finitely generated multiplicative semigroups of $\mathbb N$

Introduction

Poincaré’s recurrence theorem guarantees that a set of positive measure in a measure-preserving system returns to itself. One may ask whether the return time can be required to belong to a prescribed set of integers. Write $\mathbb{N}=\{1,2,\ldots\}$ and $\mathbb{N}_0=\{0,1,2,\ldots\}$. A set $R\subset\mathbb{N}$ is a set of measurable recurrence if, for every invertible measure-preserving system $(X,\mathcal{B},\mu,T)$ and every $A\in\mathcal{B}$ with $\mu(A)>0$, there is an $r\in R$ such that $\mu(A\cap T^{-r}A)>0$.

Consider the multiplicative semigroup $R_{2,3}=\{2^m3^n:m,n\ge1\}$. Since none of its elements is divisible by 5, rotation by 1 on $\mathbb{Z}/5\mathbb{Z}$ shows that $R_{2,3}$ is not a set of measurable recurrence. However, weakly mixing systems have no nontrivial measurable rotation factors, so this example does not answer the question for weakly mixing systems. In 1996, Bergelson [3 Question 3] asked whether every invertible weakly mixing system and every set $A$ of positive measure satisfy

Frantzikinakis, Host, and Kra recorded this question as open in their 2025 paper [7 Section 3.2].¹ By modifying a construction of Maucourant [10 Lemma 4.2], they showed that $R_{2,3}$ can fail to be recurrent even for a totally ergodic torus rotation. They also proved that the larger set $\{k!2^m3^n:k,m,n\ge1\}$ is Bohr recurrent, that is, recurrent for every finite-dimensional torus rotation [7 Theorem 1]. We give a negative answer to Bergelson’s question. The result holds for multiplicative semigroups generated by any finite nonempty set of primes.

There exist a standard nonatomic probability space $(X,\mathcal{B},\mu)$, an invertible weakly mixing probability-preserving transformation $T:X\to X$, and a Borel set $A\subset X$ such that $\mu(A)>0$ and $A\cap T^{-r}A=\varnothing$ for every $r\in S_{\mathcal{P}}$.

The system can be chosen to be a finite product of two-sided complex Gaussian shifts.

Every finitely generated multiplicative semigroup of positive integers is contained in some $S_{\mathcal{P}}$, since only finitely many primes divide its generators. Thus the theorem applies to all such semigroups, including the time 1. Taking $\mathcal{P}=\{2,3\}$ answers Bergelson’s question. Coding the set $A$ also gives a compact topologically weakly mixing subshift with a nonempty clopen set having no return at any time in $S_{\mathcal{P}}$; see Proposition 4.7.

Our argument also applies to sets involving infinitely many distinct primes. Write $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and $e(t)=\exp(2\pi it)$. A set $R\subset\mathbb{N}$ is Bohr recurrent if, for every $m\ge1$, $\alpha_1,\ldots,\alpha_m\in\mathbb{T}$, and $\varepsilon>0$, there exists $r\in R$ such that $\max_{1\le a\le m}|e(r\alpha_a)-1|<\varepsilon$. For a nonempty set $H\subset\mathbb{N}$, $\gcd H$ denotes the greatest positive integer dividing every member of $H$. We first prove the following general theorem.

Condition (1) says that arbitrarily large integers divide all but finitely many members of each $E_i$. Condition (2) says that $R$ is not Bohr recurrent; compare the definition in [7 Section 1.1]. The divisibility condition alone does not suffice, as the example in Remark 3.2 shows. For $S_{\mathcal{P}}$, assign each integer to a prime with maximal exponent. The resulting sets satisfy (1), and a prime outside $\mathcal{P}$ gives (2). Section 4 also gives an infinitely generated example and proves non-recurrence for $(S_{\mathcal{P}+h})\cap\mathbb{N}$ for every fixed $h\in\mathbb{Z}$.

The Gaussian construction associates continuous measures on the circle with weakly mixing systems; see [8 Section 2]. Related constructions of perfect sets and continuous measures under divisibility assumptions appear in Eisner and Grivaux [6, Propositions 3.8 and 3.9]. Grivaux’s non-recurrence criterion [8 Theorems 1.1 and 2.6] combines separation by a rotation with uniform approximation: for an infinite set $Q\subset\mathbb{N}$ containing $1$, one requires unit complex numbers $\lambda_k\ne1$ such that $\sup_{r\in Q}|\lambda_k^r-1|\to0$. For a single prime, weakly mixing non-recurrence, including the corresponding result for integer translates, follows from [8 Theorem 1.2]. For multiplicative semigroups, Badea and Grivaux [2 Section 6.1] group the elements according to a maximal prime exponent and construct continuous measures whose Fourier coefficients tend to $1$ within each group. This convergence alone does not ensure that

is small enough for the Gaussian argument. It also does not give the uniform approximation required by Grivaux’s criterion.

Our proof starts with a finite partition of $S_{\mathcal{P}}$. Following Badea and Grivaux [2 Section 6.1], we assign each element to a prime with maximal exponent in its factorization. In each part, the exponent of the chosen prime tends to infinity as the elements tend to infinity. Thus every power of that prime divides all sufficiently large elements of the part.

Fix $\eta> 0$, to be chosen sufficiently small below, and one part $E$. Choose denominators $D_j$ and integers $1 = N_0 < N_1 < N_2 < \cdots$ in the order $D_1, N_1, D_2, N_2, \ldots$, so that $D_j$ divides every $r \in E$ with $r \ge N_j$. For $\epsilon= 0, 1$, let $Q_\epsilon$ consist of those $r \in E$ lying in an interval $[N_j, N_{j+1})$ with $j \equiv\epsilon\pmod{2}$. For each $Q_\epsilon$, consider the random series

where the digits $\omega_\ell$ are independent, each equal to 0 or 1 with probability $1/2$. We take the sums modulo one. Choose the denominators large enough and make them grow fast enough that, for each $\epsilon$, different choices of the digits used in $\Theta_\epsilon$ give different points of $\mathbb{T}$. Each such digit sequence has probability zero, so the distribution of each series has no atoms.

Now take $r \in Q_\epsilon\cap[N_j, N_{j+1})$, so that $j \equiv\epsilon\pmod{2}$. The denominators of the terms of $\Theta_\epsilon$ with indices at most $j$ divide $r$. After multiplication by $r$, these terms are integers. They therefore do not change the value of $e(r\Theta_\epsilon)$, which depends only on the remaining terms. Since $j+1$ has the opposite parity, the remaining terms have indices $j+2, j+4, \ldots$.

When we choose $D_{j+2}$, the endpoint $N_{j+1}$ is already fixed, and every $r$ in the interval satisfies $r < N_{j+1}$. We can therefore choose $D_{j+2}$ and the later denominators large enough to make $|e(r\Theta_\epsilon)-1|$ as small as needed, uniformly for all $r \in Q_\epsilon\cap[N_j, N_{j+1})$ and all choices of the digits. Since there are finitely many such $r$, we can also make the sum of the squared bounds over the interval as small as needed. We choose the bounds for these interval sums so that their total is less than $\eta$. Since the bounds do not depend on the digits, the same estimates hold on the supports of the measures. Grivaux [8 Fact 4.2 and Section 4] also uses alternating finite intervals to obtain summable errors. Here we choose each denominator to divide all sufficiently large elements of the corresponding part.

Thus, for every $\eta> 0$, we obtain a finite partition of $S_{\mathcal{P}}$ and, for each part $Q$, a continuous probability measure $\nu$ on $\mathbb{T}$ such that

The partition and measures may depend on $\eta$.

We now use this estimate to build a weakly mixing system with no returns along $Q$. Take a prime $q \notin\mathcal{P}$ and put $\zeta=e(1/q)$. No element of $S_{\mathcal{P}}$ is divisible by $q$. Hence there is a constant $\delta> 0$ such that $|\zeta^r-1| \ge\delta$ for all $r \in S_{\mathcal{P}}$. Translate $\nu$ by $1/q$ to obtain a continuous probability measure $\sigma$. Let $(X,\mu,T)$ be the complex Gaussian shift associated with $\sigma$, with coordinates $Z_k=Z_0\circ T^k$. The shift is invertible and weakly mixing. Its coordinates satisfy

Thus the sum of these mean square errors over $r \in Q$ is less than $\eta$.

Set $a=\delta/8$ and let $C=\{|Z_0-1|<a\}$. Since $Z_0$ is standard complex Gaussian, $\mu(C)>0$ depends only on $a$, not on $\nu$. On $C\cap T^{-r}C$, we have $|Z_0-1|<a,|Z_r-1|<a$. Together with $|\zeta^r-1|\ge\delta$, this gives $|Z_r-\zeta^r Z_0|>\delta-2a>\delta/2$. By Markov's inequality,

Since $\mu(C)$ depends only on $a$, we apply the preceding construction with $\eta$ small enough that $4\eta/\delta^2<\mu(C)/2$.

Remove from $C$ all points that return to $C$ at a time in $\mathbb{Q}$, and set $A=C\setminus\bigcup_{r\in\mathbb{Q}}T^{-r}C$. The sum bound gives $\mu(A)>\mu(C)/2>0$. Also, $A\subset C$ and $A\cap T^{-r}C=\varnothing$ for every $r\in\mathbb{Q}$. Therefore $A\cap T^{-r}A=\varnothing$ for all $r\in\mathbb{Q}$. The bound on the sum is needed because we remove the return events for all $r\in\mathbb{Q}$.

Apply this construction to each part of the partition. Take the product of the systems and the product of the sets. The finite product is weakly mixing, and the product set has positive measure. Every $r\in S_{\mathcal{P}}$ belongs to one part. The set in that coordinate has no return at time $r$, so neither does the product set.

The arithmetic construction only uses arbitrarily large integers that divide all sufficiently large elements of each part. It therefore applies beyond finitely generated multiplicative semigroups. If finitely many rotations give a uniform separation from $1$, we further partition each part so that one rotation gives this separation on each new part. The same Gaussian construction and finite product argument then prove Theorem 1.2. The choice of denominators also gives the uniform approximation used in Corollary 4.5 to prove fixed integer translates via Grivaux’s criterion.

Organization of the paper. Section 2 recalls the necessary preliminaries and establishes a Gaussian criterion for non-recurrence. Section 3 constructs continuous measures with summable Fourier errors and proves the general theorem. Section 4 applies this result to multiplicative semigroups, integer translates, and symbolic systems.

Preliminaries and Gaussian non-recurrence

Probability measures and weak mixing

A probability space $(X,\mathcal{B},\mu)$ consists of a set $X$, a $\sigma$-algebra $\mathcal{B}$, and a countably additive measure with $\mu(X)=1$. We work with standard Borel spaces: these are measurable spaces isomorphic to a complete separable metric space with its Borel $\sigma$-algebra. A measure has no atoms if every measurable set of positive measure contains a measurable subset of strictly smaller positive measure. On a standard Borel space this is equivalent to every singleton having measure zero. A Borel measure on $\mathbb{T}$ is called continuous when it has no atoms.

For a measurable map $f:X\to Y$, its pushforward is $(f_*\mu)(B)=\mu(f^{-1}(B))$. For every nonnegative measurable function $h$, and every integrable $h$,

For a Borel probability measure on a separable metric space, its support is

It is closed and has full measure: its complement is a countable union of measure-zero basic open sets. The measure has full support on $X$ if every nonempty open subset of $X$ has positive measure. We write $\mathbf{1}_A$ for the indicator of $A$.

An invertible measure-preserving system $(X,\mathcal{B},\mu,T)$ consists of a probability space and a measurable bijection $T:X\to X$ with measurable inverse and $T_*\mu=\mu$. For $r\in\mathbb{Z}$, we use $T^{-r}A=\{x:T^r x\in A\}$. The transformation is ergodic if $\mu(T^{-1}A\mathbin{\triangle}A)=0$ implies $\mu(A)\in\{0,1\}$, where $\triangle$ denotes symmetric difference. It is weakly mixing if, for all bounded measurable complex functions $f,g$,

Equivalently, $T\times T$ is ergodic for $\mu\times\mu$; see [5]. Taking indicators gives the equivalent set formulation with $\mu(A\cap T^{-n}B)-\mu(A)\mu(B)$.

For a Borel probability measure $\sigma$ on $\mathbb{T}$, write

Then $\widehat{\sigma}(0)=1$ and $\widehat{\sigma}(-n)=\overline{\widehat{\sigma}(n)}$. For a topological space $X$, we write $\mathcal{B}(X)$ for its Borel $\sigma$-algebra. All sequence spaces below carry the product topology, with $\mathbb{C}$ carrying its usual topology and $\{0,1\}$ the discrete topology.

Random variables and Gaussian shifts

Let $(\Omega,\mathcal{F},\rho)$ be a probability space. A complex random variable is a measurable map $W:\Omega\to\mathbb{C}$. For integrable $W$, write $\mathbb{E}_{\rho}W=\int_{\Omega}W\,d\rho$, omitting the subscript when the measure is clear. The distribution of a random vector $W:\Omega\to\mathbb{C}^{m}$ is the pushforward measure $W_{*}\rho$.

A standard complex Gaussian random variable is a random variable of the form

where $X$ and $Y$ are independent real random variables with density $(2\pi)^{-1/2}e^{-t^{2}/2}$ on $\mathbb{R}$. The density of $Z$ with respect to Lebesgue area measure on $\mathbb{C}$ is $\pi^{-1}e^{-|z|^{2}}$. In particular,

The following Gaussian construction is classical; see [9 Appendix C]. We use its complex-valued formulation from [1 Theorem 2.3 and Proposition 2.5].

Proof. Set $\varphi(n)=\widehat{\sigma}(n)$, for any $n\in\mathbb{Z}$. Since $\sigma$ is a probability measure, $\varphi(0)=1$. Moreover, for any $n_1,\ldots,n_m\in\mathbb{Z}$ and $c_1,\ldots,c_m\in\mathbb{C}$,

Thus $\varphi$ is positive definite, and [1 Theorem 2.3] applies.

That theorem gives a Borel probability measure $\gamma_{\sigma}$ on $\mathbb{C}^{\mathbb{Z}}$, invariant under the shift $(Sz)_k=z_{k-1}$. The coordinate functions $Z_k(z)=z_k$ satisfy

Since these variables are Gaussian and $\widehat{\sigma}(0)=1$, each $Z_k$ is standard complex Gaussian. Define $T=S^{-1}$. Then $(Tz)_k=z_{k+1}$. Since $S$ is invertible and preserves $\gamma_{\sigma}$, so does $T$.

Since $Z_0$ is standard complex Gaussian, it has density $\pi^{-1}e^{-|w|^{2}}$ on $\mathbb{C}$. Thus $\gamma_{\sigma}(\{Z_0=a\})=0$ for every $a\in\mathbb{C}$. For every $z=(z_k)_{k\in\mathbb{Z}}\in\mathbb{C}^{\mathbb{Z}}$, we have $\{z\}\subset\{w\in\mathbb{C}^{\mathbb{Z}}:Z_0(w)=z_0\}$ and hence $\gamma_{\sigma}(\{z\})\leq\gamma_{\sigma}(\{Z_0=z_0\})=0$. Therefore $\gamma_{\sigma}$ has no atoms.

Finally, suppose that $\sigma$ has no atoms. By [1 Proposition 2.5], the Gaussian system associated with $\sigma$, namely $(\mathbb{C}^{\mathbb{Z}},\gamma_{\sigma},S)$, is weakly mixing. Since weak mixing is preserved under inversion, $(\mathbb{C}^{\mathbb{Z}},\gamma_{\sigma},T)$ is also weakly mixing. □

We call $(\mathbb{C}^{\mathbb{Z}},\gamma_{\sigma},T)$ the complex Gaussian shift associated with $\sigma$. For later use, note that $Z_{0}\circ T^{n}=Z_{n}$, so

A Gaussian non-recurrence criterion

The following estimate uses the Gaussian non-recurrence argument of Grivaux [8, Proposition 4.5]; see also [4 Section 4]. It combines separation by a rotation with a summable bound on return events. We formulate the argument for coordinate Gaussian shifts and include explicit constants for later use.

Proof. Choose $\alpha\in\mathbb{T}$ with $\zeta=e(\alpha)$. Define a Borel probability measure $\sigma$ on $\mathbb{T}$ by

for every Borel set $B\subset\mathbb{T}$. Since $\nu$ is continuous, so is $\sigma$. Moreover,

By Lemma 2.1, let

and define

Then $(X,\mathcal{B},\mu,T)$ is an invertible weakly mixing measure preserving system. For $k\in\mathbb{Z}$, let $Z_{k}:X\to\mathbb{C}$ be the coordinate function $Z_{k}(z)=z_{k}$. Each $Z_{k}$ is a standard complex Gaussian random variable. In particular,

Since $\mathbb{E}_{\mu}|Z_{r}|^{2}=\mathbb{E}_{\mu}|Z_{0}|^{2}=1$, we have

Let $C=\{x\in X:|Z_0(x)-1|<a\}$. Since $Z_0$ is standard complex Gaussian, it has density $\pi^{-1}e^{-|z|^2}$ on $\mathbb{C}$. Thus, we have

where $dA$ denotes Lebesgue area measure on $\mathbb{C}$.

Fix $r\in\mathbb{Q}$. If $x\in C\cap T^{-r}C$, then

Since $|\zeta^r-1|\geq\delta$, we have

Thus Markov’s inequality gives

Now define

By the assumption on $\nu$,

Finally, $A\subset C$ and $A\cap T^{-r}C=\varnothing$ for every $r\in\mathbb{Q}$. Since $T^{-r}A\subset T^{-r}C$, we obtain

Products and finite sets of times

We use the following finite product argument to combine the sets of times; compare [4 Section 4].

Proof. The product of finitely many weakly mixing systems is weakly mixing; see [4 Section 4]. Moreover,

For $r \in\bigcup_{j=1}^{\ell} Q_j$, there exists $j$ such that

Hence $A \cap T^{-r} A = \varnothing$.

Proof. Take the Gaussian shift in Lemma 2.1, with $\sigma$ equal to normalized Lebesgue measure on $\mathbb{T}$. Its coordinates are independent standard complex Gaussian variables, and the shift is weakly mixing. Put $L=\max(H\cup\{0\})$ and

By independence and the positivity of the Gaussian density, $\mu(B)>0$. For every $r\in H$, if $x\in B\cap T^{-r}B$, then

Hence

a contradiction. Thus $B\cap T^{-r}B=\varnothing$ for all $r\in H$.

Divisibility and spectral measures

We divide the given set of positive integers into two parts and construct a continuous probability measure for each part. For each measure, we bound $|e(rt)-1|$ uniformly over its support and make the sum of the squared bounds over all integers in the corresponding part as small as needed. We will also use this construction to prove integer translates in Corollary 4.5.

The reciprocal-denominator series below are closely related to the constructions of Eisner and Grivaux [6 Propositions 3.8 and 3.9] and Maucourant [10 Lemma 4.2]. Maucourant uses separate even- and odd-indexed series: divisibility makes the earlier terms integers after multiplication, while the remaining terms are small. Grivaux [8 Fact 4.2] also uses alternating finite intervals to obtain summable estimates. Here we use random digits and choose denominators dividing all sufficiently large elements of the relevant set, so that the resulting estimates are uniform on the supports of the measures.

Proof. For each $N$, the integer $\gcd\{r\in E:r\ge N\}$ divides all but finitely many members of $E$. Hence the hypothesis implies that

is unbounded. Set $D_0=N_0=1$. After $D_{j-1}$ and $N_{j-1}$ have been fixed, choose $D_j\in D$ so that

and then choose an integer $N_j>N_{j-1}$ such that

Such an $N_j$ exists because $D_j\in D$ and $E$ is unbounded. No divisibility between successive $D_j$ is required. Define

Every interval meets $E$, so $E=C_0\sqcup C_1$ and both sets are infinite. For $\epsilon\in\{0,1\}$, let $\Omega_\epsilon=\{0,1\}^{I_\epsilon}$ be equipped with the product probability measure $\mathbb{P}_\epsilon$ under which each coordinate takes the values 0 and 1 with equal probability. For $\omega=(\omega_\ell)_{\ell\in I_\epsilon}\in\Omega_\epsilon$, define

We define a probability measure $\nu_\epsilon$ on $\mathbb{T}$ by

for every Borel set $B\subset\mathbb{T}$. Since $D_{\ell+1}\ge4D_\ell$ and $D_1\ge4$, we have

The series converges uniformly on the compact space $\Omega_\epsilon$. Thus $\Theta_\epsilon$ is continuous on $\Omega_\epsilon$. Consequently,

is compact. By the definition of $\nu_\epsilon$, $\nu_\epsilon(K_\epsilon)=1$, and $\operatorname{supp}\nu_\epsilon\subset K_\epsilon$.

Let $\omega\ne\omega'$, and let $\ell\in I_\epsilon$ be the first index at which they differ. Then

Hence, for $\omega\ne\omega'$, $0<|\Theta_\epsilon(\omega)-\Theta_\epsilon(\omega')|\le\frac12<1$. In particular, $\Theta_\epsilon(\omega)-\Theta_\epsilon(\omega')\notin\mathbb{Z}$. Thus $\Theta_\epsilon(\omega)\not\equiv\Theta_\epsilon(\omega')\pmod1$, and then for every $t\in K_\epsilon$, there is a unique $\omega\in\Omega_\epsilon$ such that

Finally, we show that $\nu_\epsilon$ has no atoms. Fix $a=(a_\ell)_{\ell\in I_\epsilon}\in\Omega_\epsilon$. For every integer $M\ge1$, independence gives

Letting $M \to\infty$, we obtain $\mathbb{P}_{\epsilon}(\{\omega=a\})=0$. Since $\omega\mapsto\Theta_{\epsilon}(\omega)$ (mod 1) is injective, for every $t \in\mathbb{T}$ the set

contains at most one element. Since every singleton in $\Omega_{\epsilon}$ has probability zero, we obtain

Hence $\nu_{\epsilon}$ is a continuous probability measure.

Fix $r \in E \cap[N_j,N_{j+1})$, where $j \equiv\epsilon\pmod2$. For every $\ell\in I_{\epsilon}$ with $\ell\le j$, we have $r \ge N_j \ge N_{\ell}$. By the choice of $N_{\ell}$, $D_{\ell}\mid r$, $\frac{r\omega_{\ell}}{D_{\ell}}\in\mathbb{Z}$. Since $j+1 \notin I_{\epsilon}$, it follows that

When $j=0$, we have $\epsilon=0$ and $I_{\epsilon}=\{2,4,6,\ldots\}$, so the first sum is empty and the same identity holds.

Using $\lvert e^{2\pi i u}-1\rvert\le2\pi\lvert u\rvert$ and $0\le\omega_{\ell}\le1$, we obtain

where the last inequality uses $r<N_{j+1}$. Since this estimate holds for every $\omega\in\Omega_{\epsilon}$, the definition of $K_{\epsilon}$ gives

Since the interval contains at most $N_{j+1}$ integers,

Also,

Combining the estimates for $r\in C_{\epsilon}$ and $r=1$, we obtain

Finally, for each $J\ge2$ with $J\equiv\epsilon\pmod2$, define $\lambda_J=e(1/D_J)\ne1$. For $r\in C_{\epsilon}\cap[N_j,N_{j+1})$,

because $j<J$ implies $j\le J-2$. Hence, including $r=1$,

Since $D_J \to\infty$, we have $\lambda_J \to1$. Taking $J=2k+\varepsilon$ and

proves (4). □

Now, we prove our main result.

Proof of Theorem 1.2. Let $\alpha_1,\ldots,\alpha_m \in\mathbb{T}$ and $\delta>0$ satisfy (2), and set

Apply Lemma 3.1 to each $E_i$, obtaining $E_i=C_{i,0}\sqcup C_{i,1}$ and measures $\nu_{i,\varepsilon}$. For $r\in R$, put

On each nonempty $R_{i,\varepsilon,a}$, the number $\zeta_a=e(\alpha_a)$ and the measure $\nu_{i,\varepsilon}$ satisfy

Then

Write the nonempty sets among $F$ and $R_{i,\varepsilon,a}$ as $Q_1,\ldots,Q_L$. Then $L\le2dm+1$. By Proposition 2.2 and Lemma 2.4, for each $Q_j$ there is a weakly mixing Gaussian shift $(X_j,\mu_j,T_j)$ and a Borel set $A_j$ such that

Now set

By Lemma 2.3, $T$ is weakly mixing, and

This completes the proof. □

Remark 3.2. Condition (1) alone does not guarantee non-recurrence. Consider

We first show that $R_*$ is a set of recurrence. Let $(X,\mathcal{B},\mu,T)$ be an invertible probability-preserving system, and let $A\in\mathcal{B}$ satisfy $\mu(A)>0$. Choose $k\ge1$ such that $(k+1)\mu(A)>1$. Since

there exist $0\le i<j\le k$ such that

By invariance of $\mu$,

Since $(j-i)2^k\in R_*$, the set $R_*$ is recurrent; compare [8 Remark 3.1]. On the other hand, for every $K\ge1$, all elements of $R_*$ outside the finitely many sets

are divisible by $2^{K}$. Hence

Thus $R_*$ satisfies the divisibility condition (1) but cannot satisfy (2).

Applications

Finitely generated semigroups

We use the partition according to a maximal exponent from [2 Section 6.1].

Proof. For each $r \in R$, choose the smallest index $i$ for which $v_{p_i}(r)$ is maximal. More precisely, set $i(r) = \min\{i : v_{p_i}(r) = \max_{1 \le j \le d} v_{p_j}(r)\}$, $G_i = \{r \in R : i(r) = i\}$. Then

Fix $1 \le i \le d$. We show that if $G_i$ is infinite, then

Let $K \ge1$ be an integer. If $r \in G_i$ and $p_i^K \nmid r$, then

Consequently,

The set on the right is finite by assumption. Hence there exists $N_K \ge1$ such that for any $r \in G_i$, $r \ge N_K$, we have $p_i^K \mid r$.

Now suppose that $G_i$ is infinite. For every $N \ge N_K$, the set $\{r \in G_i : r \ge N\}$ is nonempty, and all its elements are divisible by $p_i^K$. Therefore

and in particular,

Since $K$ is arbitrary, we obtain

Finally, let $F$ be the union of the finite sets among $G_1,\ldots,G_d$, and denote the remaining sets by $E_1,\ldots,E_s$. Since $R$ is infinite, at least one $G_i$ is infinite, so $1 \le s \le d$. Thus

where $F$ is finite and each $E_i$ is infinite and satisfies

Proof of Theorem 1.1 and Corollary 4.2. Let $\mathcal{P}=\{p_1,\ldots,p_d\}$. Every $r\in S_{\mathcal{P}}$ has a unique representation

with $v_{p_i}(r)=n_i$ for $1\le i\le d$. For every integer $K\ge1$, we therefore have

By Lemma 4.1, we can write

where $F$ is finite, $1\le s\le d$, and each $E_i$ is infinite and satisfies

Thus the first condition of Theorem 1.2 holds.

Next, choose a prime $q\notin\mathcal{P}$. By the definition of $S_{\mathcal{P}}$, we have $q\nmid r$ for every $r\in S_{\mathcal{P}}$. Hence, for each $r\in S_{\mathcal{P}}$, there exists an integer $b$ with $1\le b\le q-1$ such that $r\equiv b\pmod q$. Consequently,

Therefore (2) holds with

Theorem 1.2 now gives the required system and set, proving Theorem 1.1.

To prove Corollary 4.2, choose a finite nonempty set of primes $\mathcal{Q}$ such that every $a_j$ can be written as

For any $n_1,\ldots,n_l\in\mathbb{N}_0$, we have

Thus $S\subset S_{\mathcal{Q}}$. By Theorem 1.1, there exist an invertible weakly mixing system $(X,\mathcal{B},\mu,T)$ and a Borel set $A$ such that

Since $S\subset S_{\mathcal{Q}}$, the same conclusion holds for every $r\in S$. $\square$

An infinitely generated semigroup

Integer translates

We use the following consequence of [8 Theorems 1.1 and 2.6, and Section 3]. The proof of Grivaux’s Theorem 1.1 uses the Hilbert-space operator construction of Eisner and Grivaux [6 Theorem 2.1].

Proof. For $h=0$, the conclusion follows from Theorem 1.1. Suppose $h\ne0$. By Lemma 4.1, write $S_{\mathcal{P}}=F\cup E_1\cup\cdots\cup E_s$, where $F$ is finite and each $E_i$ is infinite and satisfies

For each $1\le i\le s$, choose integers $c_i>|h|$ and $L_i>|h|$ such that

Put $E_i'=E_i\cap[L_i,\infty)$. Each $E_i'$ is infinite and still satisfies the same condition on greatest common divisors. Apply Lemma 3.1 to $E_i'$ with $\eta=1$. We obtain $E_i'=C_{i,0}\sqcup C_{i,1}$ and unit complex numbers $\lambda_{i,\epsilon,k}\ne1$ such that

as $k\to\infty$, for $\epsilon=0,1$.

Fix $i$ and $\epsilon$, and define $Q_{i,\epsilon}=(C_{i,\epsilon}+h)\cup\{1\}$. Since $L_i>|h|$, all elements of $C_{i,\epsilon}+h$ are positive. Moreover, $Q_{i,\epsilon}$ is infinite and contains $1$. For every $r\in C_{i,\epsilon}$,

Consequently,

Next, let $\zeta_i=e(1/c_i)$. Since $c_i\mid r$ for $r\in C_{i,\epsilon}$, $\zeta_i^{r+h}=e(h/c_i)$. As $0<|h|<c_i$, we have

Thus $Q_{i,\epsilon}$ satisfies the assumptions of Lemma 4.4. We obtain an invertible weakly mixing system and a positive-measure Borel set with no returns at times in $Q_{i,\epsilon}$.

The remaining positive times belong to the finite set

If $H$ is nonempty, apply Lemma 2.4 to $H$. Since

Lemma 2.3 gives the required system and Borel set. □

Remark 4.6. The proof of Corollary 4.5 also gives a more general result. Suppose

where $F$ is finite, each $E_i$ is infinite, and

Then, for every $h \in\mathbb{Z} \setminus\{0\}$, there exist an invertible weakly mixing system $(X,\mathcal{B},\mu,T)$ and a Borel set $A$ such that

Indeed, for each $i$, we can choose $c_i>|h|$ and $L_i>|h|$ such that

Hence

The argument used to prove Corollary 4.5 also applies to these sets. We use an additional Gaussian shift to rule out returns at the remaining finitely many times, and then take the product of all the systems.

A compact symbolic realization

A homeomorphism $S$ of a compact metric space $Y$ is topologically transitive if for any nonempty open sets $U,V\subset Y$, some $n\ge0$ satisfies $U\cap S^{-n}V\ne\varnothing$. It is topologically weakly mixing if $S\times S$ is topologically transitive on $Y\times Y$. A set is clopen if it is both open and closed. Theorem 1.1 also gives the following result for symbolic systems.

Proof. Take $(X,\mathcal{B},\mu,T)$ and $A$ from Theorem 1.1. Define

Since $\Phi\circ T=S\circ\Phi$ and $T_*\mu=\mu$, we have $S_*m=m$ and $S(Y)=Y$. Moreover, applying (3) to $f\circ\Phi$ and $g\circ\Phi$ shows that $(Y,m,S)$ is weakly mixing. For the clopen set $U=\{y\in Y:y_0=1\}$,

Each $U\cap S^{-r}U$ is open in $Y$, so full support makes it empty. To check topological weak mixing, take nonempty open sets $U_1,U_2,V_1,V_2\subset Y$. Since $m$ has full support, we have $c_i:=m(U_i)m(V_i)>0$. By (3),

There is therefore a common $n$ for which both intersections have positive measure. Since open rectangles form a base for $Y\times Y$, this proves topological transitivity of $S\times S$. □

The same construction applies to Theorem 1.2, Corollaries 4.3 and 4.5, and Remark 4.6.

Acknowledgments

The author would like to thank his advisor, Professor Siming Tu, for his continued guidance, support, and encouragement. The author is also grateful to Professor Bryna Kra for sharing her earlier unpublished notes and for drawing attention to related results in the literature.

Statement on the Use of AI Tools in This Work: While preparing this work, the author used ChatGPT 6 for assistance with mathematical discussions and editorial revisions. All mathematical results, proofs, and interpretations were verified and approved by the author, who takes full responsibility for the content of this work.

References

(Z. Liu) School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong, 519000, P.R. China

Email address: [email protected]