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Optimal lower bounds for multiple recurrence

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Ergodic theory

Published online by Cambridge University Press:  07 October 2019

SEBASTIÁN DONOSO Open the ORCID record for SEBASTIÁN DONOSO [Opens in a new window] ,
ANH NGOC LE Open the ORCID record for ANH NGOC LE [Opens in a new window] ,
JOEL MOREIRA Open the ORCID record for JOEL MOREIRA [Opens in a new window]  and
WENBO SUN Open the ORCID record for WENBO SUN [Opens in a new window]
SEBASTIÁN DONOSO
Affiliation:
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgings, Av. Libertador Bernardo O’Higgins 611, Rancagua, Chile email [email protected]
ANH NGOC LE
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL60208-2730, USA email [email protected], [email protected]
JOEL MOREIRA
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL60208-2730, USA email [email protected], [email protected]
WENBO SUN
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH43210-1174, USA email [email protected]
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Abstract

Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$. We study the largeness of sets of the form

$$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$
for various families $\{f_{1},\ldots ,f_{k}\}$ of sequences $f_{i}:\mathbb{N}\rightarrow \mathbb{N}$. For $k\leq 3$ and $f_{i}(n)=if(n)$, we show that $S$ has positive density if $f(n)=q(p_{n})$, where $q\in \mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1)=0$ and $p_{n}$ denotes the $n$th prime; or when $f$ is a certain Hardy field sequence. If $T^{q}$ is ergodic for some $q\in \mathbb{N}$, then, for all $r\in \mathbb{Z}$, $S$ is syndetic if $f(n)=qn+r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and, for $k=3$, we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.

Keywords

optimal multiple recurrencesystems of linear equationsnilsystems

MSC classification

Primary: 37A45: Relations with number theory and harmonic analysis
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators 11K36: Well-distributed sequences and other variations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021 , pp. 379 - 407
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Copyright
© Cambridge University Press, 2019

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