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Vitaly Bergelson

Publications and source records attributed to Vitaly Bergelson.

At least 55 records · Page 3Linked to original sources
arXiv2018

Under- and over-independence in measure preserving systems

We introduce the notions of over- and under-independence for weakly mixing and (free) ergodic measure preserving actions and establish new results which complement and extend the theorems obtained in [BoFW] and [A]. Here is a sample of results obtained in this paper: $\cdot$ (Existence of density-1 UI and OI set) Let $(X,\mathcal{B},μ,T)$ be an invertible probability measure preserving weakly mixing system. Then for any $d\in\mathbb{N}$, any non-constant integer-valued polynomials $p_{1},p_{2},\dots,p_{d}$ such that $p_{i}-p_{j}$ are also non-constant for all $i\neq j$, (i) there is $A\in\mathcal{B}$ such that the set $$\{n\in\mathbb{N}\colonμ(A\cap T^{p_{1}(n)}A\cap\dots\cap T^{p_{d}(n)}A)<μ(A)^{d+1}\}$$ is of density 1. (ii) there is $A\in\mathcal{B}$ such that the set $$\{n\in\mathbb{N}\colonμ(A\cap T^{p_{1}(n)}A\cap\dots\cap T^{p_{d}(n)}A)>μ(A)^{d+1}\}$$ is of density 1. $\cdot$ (Existence of Cesàro OI set) Let $(X,\mathcal{B},μ,T)$ be a free, invertible, ergodic probability measure preserving system and $M\in\mathbb{N}$. %Suppose that $X$ contains an ergodic component which is aperiodic. Then there is $A\in\mathcal{B}$ such that $$\frac{1}{N}\sum_{n=M}^{N+M-1}μ(A\cap T^{n}A)>μ(A)^{2}$$ for all $N\in\mathbb{N}$. $\cdot$ (Nonexistence of Cesàro UI set) Let $(X,\mathcal{B},μ,T)$ be an invertible probability measure preserving system. For any measurable set $A$ satisfying $μ(A) \in (0,1)$, there exist infinitely many $N \in \mathbb{N}$ such that $$\frac{1}{N} \sum_{n=0}^{N-1} μ( A \cap T^{n}A) > μ(A)^2.$$

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arXiv2018

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A set $R\subset \mathbb{N}$ is called rational if it is well-approximable by finite unions of arithmetic progressions. Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form $Φ_x:=\{n\in\mathbb{N}: \frac{\boldsymbolφ(n)}{n} 0$, then the following are equivalent: (a) $R$ is divisible, i.e. $\overline{d}(R\cap u \mathbb{N})>0$ for all $u\in\mathbb{N}$. (b) $R$ is an averaging set of polynomial single recurrence. (c) $R$ is an averaging set of polynomial multiple recurrence. As an application, we show that if $R$ is rational and divisible, then for any set $E\subset\mathbb{N}$ with $\overline{d}(E)>0$ and any polynomials $p_i\in\mathbb{Q}[t]$,$i=1,\ldots,\ell$, which satisfy $p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all $i\in\{1,\ldots,\ell\}$, there exists $β>0$ such that the set $$\{n\in R:\overline{d}( E\cap (E-p_1(n))\cap\ldots\cap(E-p_\ell(n)))>β\}$$ has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences. We prove that if $\mathcal{A}$ is a finite alphabet, $η\in\mathcal{A}^\mathbb{N}$ is rationally almost periodic, $S$ denotes the left-shift on $\mathcal{A}^\mathbb{Z}$ and $$X:=\{y\in \mathcal{A}^\mathbb{Z} : \text{each finite word appearing in $y$ appears in }η\},$$ then $η$ is a generic point for an $S$-invariant probability measure $ν$ on $X$ such that $(X,ν,S)$ is ergodic and has rational discrete spectrum.

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arXiv2017

On the density of coprime tuples of the form $(n,\lfloor f_1(n)\rfloor,\ldots,\lfloor f_k(n)\rfloor)$, where $f_1,\ldots,f_k$ are functions from a Hardy field

Let $k\in\mathbb{N}$ and let $f_1,\ldots,f_k$ belong to a Hardy field. We prove that under some natural conditions on the $k$-tuple $(f_1,\ldots,f_k)$ the density of the set $$ \big\{n\in \mathbb{N}: \text{gcd}(n,\lfloor f_1(n)\rfloor,\ldots,\lfloor f_k(n)\rfloor)=1\big\} $$ exists and equals $\frac{1}{ζ(k+1)}$, where $ζ$ is the Riemann zeta function.

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arXiv2016

Multiplicative richness of additively large sets in $\mathbb{Z}^d$

In their proof of the IP Szemerédi theorem, a far reaching extension of the classic theorem of Szemerédi on arithmetic progressions, Furstenberg and Katznelson introduced an important class of additively large sets called $\text{IP}_{\text{r}}^*$ sets which underlies recurrence aspects in dynamics and is instrumental to enhanced formulations of combinatorial results. The authors recently showed that additive $\text{IP}_{\text{r}}^*$ subsets of $\mathbb{Z}^d$ are multiplicatively rich with respect to every multiplication on $\mathbb{Z}^d$ without zero divisors (e.g. multiplications induced by degree $d$ number fields). In this paper, we explain the relationships between classes of multiplicative largeness with respect to different multiplications on $\mathbb{Z}^d$. We show, for example, that in contrast to the case for $\mathbb{Z}$, there are infinitely many different notions of multiplicative piecewise syndeticity for subsets of $\mathbb{Z}^d$ when $d \geq 2$. This is accomplished by using the associated algebra representations to prove the existence of sets which are large with respect to some multiplications while small with respect to others. In the process, we give necessary and sufficient conditions for a linear transformation to preserve a class of multiplicatively large sets. One consequence of our results is that additive $\text{IP}_{\text{r}}^*$ sets are multiplicatively rich in infinitely many genuinely different ways. We conclude by cataloging a number of sources of additive $\text{IP}_{\text{r}}^*$ sets from combinatorics and dynamics.

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arXiv2016

New examples of complete sets, with connections to a Diophantine theorem of Furstenberg

A set $A\subseteq\mathbb N$ is called $complete$ if every sufficiently large integer can be written as the sum of distinct elements of $A$. In this paper we present a new method for proving the completeness of a set, improving results of Cassels ('60), Zannier ('92), Burr, Erdős, Graham, and Li ('96), and Hegyvári ('00). We also introduce the somewhat philosophically related notion of a $dispersing$ set and refine a theorem of Furstenberg ('67).

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arXiv2016

Triangles in cartesian squares of quasirandom groups

We prove that triangular configurations are plentiful in large subsets of cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.

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arXiv2016

New polynomial and multidimensional extensions of classical partition results

In the 1970s Deuber introduced the notion of $(m,p,c)$-sets in $\mathbb{N}$ and showed that these sets are partition regular and contain all linear partition regular configurations in $\mathbb{N}$. In this paper we obtain enhancements and extensions of classical results on $(m,p,c)$-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in $\mathbb{Z}^d$. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups. We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of $(m,p,c)$-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over $\mathbb{N}$ to commutative semigroups.

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arXiv2015

Measure preserving actions of affine semigroups and {x+y,xy} patterns

Ergodic and combinatorial results obtained in [10] involved measure preserving actions of the affine group ${\mathcal A}_K$ of a countable field $K$. In this paper we develop a new approach based on ultrafilter limits which allows one to refine and extend the results obtained in [10] to a more general situation involving the measure preserving actions of the non-amenable affine semigroups of a large class of integral domains. (The results in [10] heavily depend on the amenability of the affine group of a field). Among other things, we obtain, as a corollary of an ultrafilter ergodic theorem, the following result: Let $K$ be a number field and let ${\mathcal O}_K$ be the ring of integers of $K$. For any finite partition $K=C_1\cup\cdots\cup C_r$ there exists $i\in\{1,\dots,r\}$ and many $x\in K$ and $y\in{\mathcal O}_K$ such that $\{x+y,xy\}\subset C_i$.

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arXiv2015

Ergodic Theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns

We establish a "diagonal" ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg's correspondence principle, prove that any "large" set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular we obtain an alternative proof for a result obtained by Cilleruelo [11], showing that for any finite field $F$ and any subsets $E_1,E_2\subset F$ with $|E_1||E_2|>6|F|$, there exist $u,v\in F$ such that $u+v\in E_1$ and $uv\in E_2$.

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arXiv2015

Uniform distribution of subpolynomial functions along primes and applications

Let $H$ be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let $f \in H$ be a subpolynomial function. Let $\mathcal{P} = \{2, 3, 5, 7, \dots \}$ be the (naturally ordered) set of primes. We show that $(f(n))_{n \in \mathbb{N}}$ is uniformly distributed mod 1 if and only if $(f(p))_{p \in \mathcal{P}}$ is uniformly distributed mod 1. This result is then utilized to derive various ergodic and combinatorial statements which significantly generalize the results obtained in [BKMST].

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arXiv2014

Polynomial multiple recurrence over rings of integers

We generalize the polynomial Szemerédi theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of $\mathbb{Z}^m$ and strengthens and extends recent results of Bergelson, Leibman and Lesigne on polynomials over the integers.

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arXiv2014

Joint ergodicity along generalized linear functions

A criterion of joint ergodicity of several sequences of transformations of a probability measure space $X$ of the form $T_{i}^{ϕ_{i}(n)}$ is given for the case where $T_{i}$ are commuting measure preserving transformations of $X$ and $ϕ_{i}$ are integer valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending of a continuous parameter, as well as a condition of joint ergodicity of sequences $T_{i}^{ϕ_{i}(n)}$ along primes.

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arXiv2014

Simultaneous dense and nondense orbits for commuting maps

We show that, for two commuting automorphisms of the torus and for two elements of the Cartan action on compact higher rank homogeneous spaces, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, we show that the set of points that have dense orbit under one map and nondense orbit under the second has full Hausdorff dimension.

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arXiv2014

Finite Products Sets and Minimally Almost Periodic Groups

We construct, in locally compact, second countable, amenable groups, sets with large density that fail to have certain combinatorial properties. For the property of being a shift of a set of measurable recurrence we show that this is possible when the group does not have cocompact von Neumann kernel. For the stronger property of being piecewise-syndetic we show that this is always possible. For minimally almost periodic, locally compact, second countable, amenable groups, we prove that any dilation of a positive density set by an open neighborhood of the identity contains a set of measurable recurrence, and that the same result holds, up to a shift, when the von Neumann kernel is cocompact. This leads to a trichotomy for locally compact, second countable, amenable groups based on combinatorial properties of large sets. We also prove, using a two-sided Furstenberg correspondence principle, that any two-sided dilation of a positive density set contains a two-sided finite products set.

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arXiv2014

Polynomial actions of unitary operators and idempotent ultrafilters

Let $p$ be an idempotent ultrafilter over $\mathbb{N}$. For a positive integer $N$, let ${\cal P}_{\leq N}$ denote the additive group of polynomials $P\in\mathbb{Z}[x]$ with ${\rm deg}\, P\leq N$ and $P(0)=0$. Given a unitary operator $U$ on a Hilbert space ${\cal H}$, we prove, for each $N\geq1$, the existence of a unique decomposition ${\cal H}=\bigoplus_{r\geq 1}{\cal H}^{(N)}_r$ into closed, $U$-invariant subspaces such that (a) for any polynomial $P\in{\cal P}_{\leq N}$, we have $$ p\, \text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_r^{(N)}}\right)^{P(n)}=0_{{\cal H}_r^{(N)}}\;\mbox{or}\; Id_{{\cal H}_r^{(N)}},\; \mbox{for each}\; r\geq1 ; $$ (b) for each $r\neq s$ there exists $Q\in{\cal P}_{\leq N}$ such that $$ p\,\text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_r^{(N)}}\right)^{Q(n)}\neq p\,\text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_s^{(N)}}\right)^{Q(n)}. $$ In connection with this result we introduce the notion of rigidity group. Namely, a subgroup $G\subset {\cal P}_{\leq N}$ is called an $N$-rigidity group if there exist an idempotent ultrafilter $p$ over $\mathbb{N}$ and a unitary operator $U$ on a Hilbert space $\cal H$ such that $$\label{ab1} G=\{P\in{\cal P}_{\leq N}:\: p\,\text{-}\!\lim_{n\in\mathbb{N}} U ^{P(n)}=Id\}$$ and $p\,\text{-}\!\lim_{n\in\mathbb{N}} U ^{Q(n)}=0\;\;\mbox{for each}\;\;Q\in{\cal P}_{\leq N}\setminus G.$ The main result of the paper states that a subgroup $G\subset {\cal P}_{\leq N}$ satisfying $\max\{{\rm deg}\, P:\:P\in G\}=N$ is an $N$-rigidity group if and only if $G$ has finite index in ${\cal P}_{\leq N}$.

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