Szemerédi's theorem, frequent hypercyclicity and multiple recurrence

Abstract

Let $T$ be a bounded linear operator acting on a complex Banach space $X$ and $(\lambda_n)_{n\in\mathsf{N}}$ a sequence of complex numbers. Our main result is that if $|\lambda_n|/|\lambda_{n+1}|\to 1$ and the sequence $(\lambda_n T^n)_{n\in\mathsf{N}}$ is frequently universal then $T$ is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemerédi's theorem in arithmetic progressions. We show that the previous assumption on the sequence $( \lambda_n)_{n\in\mathsf{N}}$ is optimal among sequences such that $|\lambda_{n}|/|\lambda_{n+1}|$ converges in $[0,\infty]$. In the case of bilateral weighted shifts and adjoints of multiplication operators we provide characterizations of topological multiple recurrence in terms of the weight sequence and the symbol of the multiplication operator respectively.