Optimal lower bounds for multiple recurrence

Abstract

Let $(X,\mathcal{B},\mu ,T)$ be an ergodic measure preserving system, $A\in \mathcal{B}$ and $\epsilon >0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \left\{ n\in\mathbb{N}\colonμ(A\cap T^{-f_1(n)}A\cap T^{-f_2(n)}A\cap\ldots\cap T^{-f_k(n)}A)> μ(A)^{k+1} - ε\right\} \end{split} \end{equation*} for various families $\{f_1,\dots ,f_k\}$ of sequences $f_i:\mathbb{N}\to \mathbb{N}$. For $k\le 3$ 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 $k\ge 4$, 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.

29 pages