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Vitaly Bergelson

Publications and source records attributed to Vitaly Bergelson.

At least 19 recordsLinked to original sources
arXiv2026

Product sets in sets of returns and positivity of symmetric ergodic averages

We study sets of (measurable) returns in countable groups $G$, namely sets of the form $\{g\in G:\mu(A\cap T_gA)>0\}$ arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in $G\times G$ contain subsets of the form $B\times B$, where $B$ is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if $G$ is amenable, then every sufficiently large subset $A\subseteq G\times G$ satisfies $B\times B\subseteq AA^{-1}$ for some large set $B\subseteq G$. We also investigate when sets of returns in $G$ contain product sets $BB$ with $B$ large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form $g\mapsto \mu(T_g^{-1}A\cap T_gA)$. We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set $A\subseteq G$ contains a large subset $B$ satisfying $BB\subseteq AA^{-1}$. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.

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arXiv2026

Sets of large values of polynomial multi-correlation functions

Let $p_1,...,p_L\in Z[x_1,...,x_d]$ be non-constant polynomials with zero constant term. The ergodic theoretical proofs of the polynomial and the IP-polynomial Szemeredi theorems as well as some of the ergodic-theoretical and combinatorial consequences of the Density Polynomial Hales-Jewett conjecture (DPHJ) naturally lead to the study of sets of large returns which are defined as $$ R_ε^{p_1,...,p_L}(A):=\{n\in Z^d\,|\,μ(A\cap T_1^{-p_1( n)}A\cap\cdots\cap T_L^{-p_L(n)}A)>μ^{L+1}(A)-ε\}, $$ where the $T_j$'s are commuting and invertible $μ$-preserving transformations, $A$ is measurable, and $ε>0$. We obtain new results dealing with the sets of the form $R_ε^{p_1,...,p_L}(A)$. Among other things, we show that every set of the form $R_ε^{p_1,...,p_L}(A)$ is syndetic if and only if $p_1,...,p_L$ are linearly independent, answering a question asked by Frantzikinakis-Kuca. Moreover, the linear independence of $p_1,...,p_L$ implies that every set of the form $R_ε^{p_1,...,p_L}(A)$ has the A-IP$^*$ property (="almost" IP$^*$ property), which is stronger than syndeticity. The following is one of the new combinatorial results obtained in this paper. Suppose that $p_1,...,p_L$ are linearly independent. For any set $E\subseteq Z^D$ with upper Banach density $d^*(E)>0$, any non-zero $v_1,..., v_L\in Z^D$, and any $ε>0$, the set $$ S_ε^{p_1,...,p_L}(E):=\{ n\in Z^d\,|\,d^*(E\cap (E-p_1(n)v_1)\cap \cdots\cap (E-p_L(n)v_L))>(d^*(E))^{L+1}-ε\} $$ is A-IP$^*$. Furthermore, we prove that when $D>L>1$, this result is sharp: the A-IP$^*$ property cannot be upgraded to IP$^*$. The techniques developed in this paper lead to some additional applications. For example, we show that an amplified form of the IP-polynomial Szemeredi theorem conjectured by Bergelson- McCutcheon follows from the DPHJ.

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arXiv2026

Weighted averages of arithmetic functions and applications to equidistribution and ergodic theory

For a wide range of functions $W\colon\mathbb{N}\to\mathbb{N}$, we establish a general result for estimating weighted averages of the form\[\mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)),\]where $f\colon \{1,\ldots,N\}\to\mathbb{C}$ is an arbitrary function, and $\vartheta(n)$ is any arithmetic function that adheres to a certain Gaussian distribution condition. (For instance, one may take $\vartheta(n)=\Omega(n)$, where $\Omega(n)$ counts the number of prime factors of $n$ with multiplicity, or $\vartheta(n)=s_q(p_n)$, where $s_q$ is the sum-of-digits function in base $q$ and $p_n$ denotes the $n$-th prime. Additional natural examples are discussed in the paper.) Building on our main theorem, we show that if $h(n)$ is a function from a Hardy field with polynomial growth then $(h(\vartheta(n)))_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if one of the following (mutually exclusive) conditions is satisfied: (i) $\lim_{x\to\infty} \frac{|h(x)-p(x)|}{x \log x}=\infty$ for all $p(x)\in \mathbb{Q}[x]$; (ii) $\lim_{x\to\infty}\frac{|h(x)-p(x)|}{\sqrt{x}}=\infty$ for each $p(x)\in \mathbb{Q}[x]$ and there exists $q(x)\in \mathbb{Q}[x]$ such that $\lim_{x\to\infty}\frac{|h(x)-q(x)|}{x}<\infty$. This leads to several novel applications. For example, it follows that $(\Omega(n)^c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.

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arXiv2026

Furstenberg--Sárközy theorem and partition regularity of polynomial equations over finite fields

We prove new combinatorial results about polynomial configurations in large subsets of finite fields. Bergelson--Leibman--McCutcheon (2005) showed that for any polynomial $P(x) \in \mathbb{Z}[x]$ with $P(0) = 0$, if $A \subseteq \mathbb{F}_q$ is a subset of a $q$-element finite field and $A$ does not contains distinct $a, b$ such that $b - a = P(x)$ for some $x$, then $|A| = o(q)$. In fields of sufficiently large characterstic, the bound $o(q)$ can be improved to $O(q^{1/2})$ by the Weil bound. We match this bound in the low characteristic setting and give a complete algebraic characterization of the class of polynomials for which the Furstenberg--Sárközy theorem holds over finite fields of fixed characteristic. Our next main result deals with an enhancement of the Furstenberg--Sárközy theorem over finite fields. Another consequence of the Weil bound is that if $P(x) \in \mathbb{Z}[x]$, $A, B \subseteq \mathbb{F}_q$, and there do not exist elements $a \in A$ and $b \in B$ with $b - a = P(x)$ for some $x$, then $|A| |B| = O(q)$, provided that the characteristic of $\mathbb{F}_q$ is sufficiently large depending on $P$. We provide a complete description of the family of polynomials for which this asymmetric enhancement holds over fields of fixed characteristic, achieving the same quantitative bounds that are available in the high characteristic setting. The exponential sum estimates that we produce in dealing with the above problems also allow us to establish partition regularity of families of polynomial equations over finite fields. As an example, we prove: if $P(x) \in \mathbb{Z}[x]$ with $P(0) = 0$, then for any $r \in \mathbb{N}$, there exists $N = N(P,r)$ and $c = c(P,r) > 0$ such that if $q > N$ and $\mathbb{F}_q = \bigcup_{i=1}^r{C_i}$, then there are at least $cq^2$ monochromatic solutions to the equation $P(x) + P(y) = P(z)$.

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arXiv2026

A note on polynomial equidistribution and recurrence in finite characteristic

This paper addresses the topic of equidistribution and recurrence for polynomial sequences over function fields. The main focus is to note and correct two small errors in [V. Bergelson and A. Leibman, A Weyl-type equidistribution theorem in finite characteristic, Adv. Math. 289 (2016) 928-950], contextualized within the broader developing literature on number theory and additive combinatorics in function fields. Connected with the resolution of these issues, we also prove new results characterizing intersective polynomials in finite characteristic in terms of various algebraic, combinatorial, and dynamical properties.

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arXiv2026

Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications

We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and non-polynomial) extensions of Szemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10; BMR17]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume $f_i(t)=a_{i,1}t^{c_{i,1}}+\cdots+a_{i,d}t^{c_{i,d}}$ for $c_{i,j}>0$ and $a_{i,j}\in\mathbb{R}$. Then $\bullet$ for any measure preserving system $(X,\mathcal{B},μ,T)$ and $h_1,\dots,h_k\in L^\infty(X)$, the limit $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{[f_1(n)]}h_1\cdots T^{[f_k(n)]}h_k$$ exists in $L^2$; $\bullet$ for any $E\subset \mathbb{N}$ with $\overline{\mathrm{d}}(E)>0$ there are $a,n\in\mathbb{N}$ such that $\{a,\, a+[f_1(n)],\ldots,a+[f_k(n)]\}\subset E$. We also show that if $f_1,\dots,f_k$ belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system $(X,{\mathcal B},μ,T)$ and any $A\in{\mathcal B}$, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nμ\Big(A\cap T^{-[ f_1(n) ]}A\cap\ldots\cap T^{-[f_k(n)]}A\Big)\,\geq\,μ(A)^{k+1}.$$

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arXiv2025

Weighted uniform distribution of subpolynomial functions along primes and applications

Let $u(x)$ be a subpolynomial function in a Hardy field. We establish necessary and sufficient conditions for the weighted uniform distribution of the sequences $(u(n))_{n\in\mathbb{N}}$ and $(u(p_n))_{n\in\mathbb{N}}$, where $p_n$ denotes the $n$-th prime. This extends the main result of [4] to the weighted setting and leads to new applications in uniform distribution theory, ergodic theory, and additive combinatorics.

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arXiv2025

Polynomial actions of rings of integers of global fields and quasirandomness of Paley-type graphs

The goal of this paper is to undertake an in-depth study of the phenomenon behind the Furstenberg--Sárközy theorem, which, in its modern form due to Kamae and Mendès-France, states that if $E$ is a set of integers with positive density and $P$ is an intersective polynomial, then there are distinct elements $x, y \in E$ such that $x - y = P(n)$ for some some $n$. In this paper, we identify an algebraic framework (rings of integers of global fields) for Furstenberg--Sárközy-type theorems. One of our main results establishes necessary and sufficient conditions for a polynomial to satisfy the Furstenberg--Sárközy theorem over the ring of integers of a global field, providing an extension of the result of Kamae and Mendès-France. The Furstenberg--Sárközy phenomenon goes beyond infinite rings and has interesting additional aspects in finite rings. As an example, classical exponential sum estimates can be used to show that large subsets of finite fields contain the asymptotically ``correct'' number of pairs $(x,y)$ whose difference is a square. In previous work, the class of polynomials satisfying this strong form of the Furstenberg--Sárközy theorem over finite fields was classified. In the present paper, we establish asymptotic results characterizing sequences of finite principal ideal rings that produce ``correct'' statistics in the Furstenberg--Sárközy theorem and show that these families are much more general than finite fields. As an application of our enhanced forms of the Furstenberg--Sárközy theorem over finite rings, we produce new families of examples of quasirandom graphs of algebraic origin. The production of these new examples hinges on a two-way connection between asymptotic total ergodicity -- the phenomenon responsible for enhanced versions of the Furstenberg--Sárközy theorem over finite fields and rings -- and quasirandomness.

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arXiv2025

Metric uniform distribution on analytic curves

We obtain multidimensional metric uniform distribution results involving sequences in ${\mathbb R}^k$ parametrized by analytic curves. Our theorems extend the classical theorems of Weyl and Koksma in a variety of ways. One of our main results implies that for any injective sequences $a_1,\dots,a_k:{\mathbb N}\to{\mathbb Z}$ the set $$\Big\{(x_1,\dots,x_k)\in{\mathbb R}^k:\big(a_1(n)x_1,\dots,a_k(n)x_k\big)_{n\in{\mathbb N}}\text{ is uniformly distributed in }{\mathbb T}^k\Big\}$$ has full Lebesgue measure inside any non-degenerate analytic curve $\gamma\subset{\mathbb R}^k$.

math.CA↗
arXiv2025

New Applications of Ergodic Theory to Sets of Differences

We apply the methods of ergodic theory to both simplify and significantly extend some classical results due to Stewart, Tijdeman, and Ruzsa. One of the notable features of our approach is the utilization of pointwise ergodic theory.

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arXiv2025

On preservation of normality and determinism under arithmetic operations

In this paper we develop a general ergodic approach which reveals the underpinnings of the effect of arithmetic operations involving normal and deterministic numbers. This allows us to recast in new light and amplify the result of Rauzy, which states that a number $y$ is deterministic if and only if $x+y$ is normal for every normal number $x$. Our approach is based on the notions of lower and upper entropy of a point in a topological dynamical system. The ergodic approach to Rauzy theorem naturally leads to the study of various aspects of normality and determinism in the general framework of dynamics of endomorphisms of compact metric groups. In particular, we generalize Rauzy theorem to ergodic toral endomorphisms. Also, we show that the phenomena described by Rauzy do not occur when one replaces the base $2$ normality associated with the $(\frac12,\frac12)$-Bernoulli measure by the variant of normality associated with a $(p,1-p)$-Bernoulli measure, where $p\neq\frac12$. Finally, we present some rather nontrivial examples which show that Rauzy-type results are not valid when addition is replaced by multiplication.

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arXiv2025

On the Polynomial Szemerédi Theorem in Finite Commutative Rings

The polynomial Szemerédi theorem implies that, for any $δ\in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of $\{1,\ldots, N\}$ of cardinality at least $δN$ contains a nontrivial configuration of the form $\{x,x+P_1(y),\ldots, x+P_m(y)\}$. When the polynomials are assumed independent, one can expect a sharper result to hold over finite fields, special cases of which were proven recently, culminating with arXiv:1802.02200, which deals with the general case of independent polynomials. One goal of this article is to explain these theorems as the result of joint ergodicity in the presence of asymptotic total ergodicity. Guided by this concept, we establish, over general finite commutative rings, a version of the polynomial Szemerédi theorem for independent polynomials $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y_1,\ldots, y_n]$, deriving new combinatorial consequences, such as the following. Let $\mathcal R$ be a collection of finite commutative rings subject to a mild condition on their torsion. There exists $γ\in (0,1)$ such that, for every $R \in \mathcal R$, every subset $A \subset R$ of cardinality at least $|R|^{1-γ}$ contains a nontrivial configuration $\{x,x+P_1(y),\ldots, x+P_m(y)\}$ for some $(x,y) \in R \times R^n$, and, moreover, for any subsets $A_0,\ldots, A_m \subset R$ such that $|A_0|\cdots |A_m| \geq |R|^{(m+1)(1-γ)}$, there is a nontrivial configuration $(x, x+P_1(y), \ldots, x+P_m(y)) \in A_0\times \cdots \times A_m$. The fact that general rings have zero divisors is the source of many obstacles, which we overcome; for example, by studying character sums, we develop a bound on the number of roots of an integer polynomial over a general finite commutative ring, a result which is of independent interest.

math.CO↗
arXiv2024

On the interplay between notions of additive and multiplicative largeness and its combinatorial applications

Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of $\mathbb{N}$ and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give a variety of applications arising from the connection between additive and multiplicative largeness. For example, we show that given any $n, k \in \mathbb{N}$, any finite set with fewer than $n$ elements in a sufficiently large finite field can be translated so that each of its elements becomes a non-zero $k^{\text{th}}$ power. We also prove a theorem concerning Diophantine approximation along multiplicatively syndetic subsets of $\mathbb{N}$ and a theorem showing that subsets of positive upper Banach density in certain multiplicative sub-semigroups of $\mathbb{N}$ of zero density contain arbitrarily long arithmetic progressions. Along the way, we develop a new characterization of upper Banach density in a wide class of amenable semigroups and make explicit the uniformity in recurrence theorems from measure theoretic and topological dynamics. This in turn leads to strengthened forms of classical theorems of Szemerédi and van der Waerden on arithmetic progressions.

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arXiv2024

Iterated differences sets, diophantine approximations and applications

Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal R(v,ε)=\{n\in\mathbb{N}\,|\,\|v(n)\|{<ε\}}$ where $\|\cdot\|$ denotes the distance to the closest integer. We then apply the new diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-Sárközy theorem.

math.CO↗
arXiv2023

Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions

We establish two ergodic theorems which have among their corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg, a theorem of Erdős-Delange, the mean value theorem of Wirsing, and special cases of the mean value theorem of Halász. By building on the ideas behind our ergodic results, we recast Sarnak's Möbius disjointness conjecture in a new dynamical framework. This naturally leads to an extension of Sarnak's conjecture which focuses on the disjointness of additive and multiplicative semigroup actions. We substantiate this extension by providing proofs of several special cases.

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arXiv2023

Joint normality of representations of numbers: an ergodic approach

We introduce an ergodic approach to the study of {\em joint normality} of representations of numbers. For example, we show that for any integer $b \geq 2$ almost every number $x \in [0,1)$ is jointly normal with respect to the $b$-expansion and continued fraction expansion. This fact is a corollary of the following result which deals with {\em pointwise joint ergodicity}: Let $T_b:[0,1] \rightarrow [0,1]$ be the times $b$ map defined by $T_b x = bx \, \bmod \, 1 $ and let $T_G:[0,1] \rightarrow [0,1]$ be the Gauss map defined by $T_G(x) = \{\frac{1}{x}\}$ for $x \ne 0$ and $T_G (0) =0.$ (Here $\{ \cdot \}$ denotes the fractional part.) For any $f, g \in L^{\infty} (λ)$, \[ \lim_{N \rightarrow \infty} \frac{1}{N } \sum_{n=0}^{N-1} f(T_b^{n}x) \, g(T_G^n x) = \int f \, d λ\cdot \int g \, d μ_G \quad \text{for almost every } x \in [0,1], \] where $λ$ is the Lebesgue measure on $[0,1]$ and $μ_G$ is the Gauss measure on $[0,1]$ given by $μ_G (A) = \frac{1}{ \log 2} \int_A \frac{1}{1+x} dx$ for any measurable set $A \subset [0,1]$. We show that the phenomenon of the pointwise joint ergodicity takes place for a wide variety of number-theoretical maps of the interval and derive the corresponding corollaries pertaining to joint normality. We also establish the equivalence of various forms of normality and joint normality for representations of numbers, hereby providing a general framework for classical normality results.

math.DS↗
arXiv2023

Multiple recurrence and popular differences for polynomial patterns in rings of integers

We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If $K$ is a number field with ring of integers $\mathcal{O}_K$ and $E \subseteq \mathcal{O}_K$ has positive upper Banach density $d^*(E) = δ> 0$, we show, inter alia: 1. If $p(x) \in K[x]$ is an intersective $\mathcal{O}_K$-valued polynomial and $r, s \in \mathcal{O}_K$ are distinct and nonzero, then for any $\varepsilon > 0$, the set of $n \in \mathcal{O}_K$ such that \[ d^* \left( \{ x \in \mathcal{O}_K : \{x, x + rp(n), x + sp(n)\} \subseteq E \} \right) > δ^3 - \varepsilon. \] is syndetic. Moreover, if $\frac{s}{r} \in \mathbb{Q}$, then there are syndetically many $n \in \mathcal{O}_K$ such that \[ d^* \left( \{ x \in \mathcal{O}_K : \{x, x + rp(n), x + sp(n), x + (r+s)p(n)\} \subseteq E \} \right) > δ^4 - \varepsilon. \] 2. If $\{p_1, \dots, p_k\} \subseteq K[x]$ is a jointly intersective family of linearly independent $\mathcal{O}_K$-valued polynomials, then the set of $n \in \mathcal{O}_K$ such that \[ d^* \left( \{ x \in \mathcal{O}_K : \{x, x + p_1(n), \dots, x + p_k(n)\} \subseteq E \} \right)> δ^{k+1} - \varepsilon \] is syndetic. These two results generalize and extend previous work of Frantzikinakis and Kra on polynomial configurations in $\mathbb{Z}$ and build upon recent work of the authors and Best on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory, which require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalization of Weyl's equidistribution theorem.

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