Optimal lower bounds for multiple recurrence

• Published in: arXiv.org 2019-08
• Main Authors: Donoso, Sebastián, Le, Anh N, Moreira, Joel, Sun, Wenbo
• Format: Article
• Language: English
• Subjects: Colon; Ergodic processes; Integers; Linear equations; Lower bounds; Mathematical analysis; Polynomials
• ISSN: 2331-8422

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}\colon\mu(A\cap T^{-f_1(n)}A\cap T^{-f_2(n)}A\cap\ldots\cap T^{-f_k(n)}A)> \mu(A)^{k+1} - \epsilon \right\} \end{split} \end{equation*} for various families \(\{f_1,\dots,f_k\}\) of sequences \(f_i\colon \mathbb{N} \to \mathbb{N}\). For \(k \leq 3\) and \(f_{i}(n)=i f(n)\), 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.