Szemeredi's theorem, frequent hypercyclicity and multiple recurrence

TL;DR

This paper links Szemerédi's theorem with frequent hypercyclicity and multiple recurrence in linear operators, establishing new conditions under which operators exhibit topological multiple recurrence.

Contribution

It introduces a novel connection between Szemerédi's theorem and operator recurrence, providing optimal conditions on sequences for multiple recurrence in linear operators.

Findings

If |_n|/|_{n+1}| o 1 and (_n T^n) is frequently universal, then T is topologically multiply recurrent.

The assumption on the sequence (_n) is optimal among sequences with |_n|/|_{n+1}| converging in [0,+].

Characterizations of topological multiple recurrence are provided for bilateral weighted shifts and multiplication operators in terms of their weights and symbols.

Abstract

Let T be a bounded linear operator acting on a complex Banach space X and (\lambda_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) is frequently universal then T is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemer\'edi's theorem in arithmetic progressions. We show that the previous assumption on the sequence (\lambda_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.