Non-recurrence sets for weakly mixing linear dynamical systems

TL;DR

This paper investigates non-recurrence sets in weakly mixing linear dynamical systems, extending recent results and providing new examples of such sets with specific growth and divisibility properties.

Contribution

It generalizes recent findings on non-recurrence sets, showing that certain growth and divisibility conditions produce non-recurrence sets in weakly mixing linear systems.

Findings

Sets with n_{k+1}/n_k o abla are non-recurrence sets.

Sets where n_k divides n_{k+1} are non-recurrence sets.

Existence of r-Bohr non-recurrence sets for weakly mixing systems.

Abstract

We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space X, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lema\'nczyk and Rosenblatt, and show in particular that sets \{n_k\} such that n_{k+1}/{n_k} tends to infinity, or such that n_{k} divides n_{k+1} for each k, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each r, of r-Bohr sets which are non-recurrence sets for some weakly mixing systems.