Difference sets and frequently hypercyclic weighted shifts

TL;DR

This paper characterizes frequently hypercyclic weighted shifts on ll^p(b Z), constructs counterexamples distinguishing different types of hypercyclicity, and explores the role of difference sets with positive upper density.

Contribution

It provides a characterization of frequently hypercyclic weighted shifts and constructs novel counterexamples using properties of difference sets.

Findings

Characterization of frequently hypercyclic weighted shifts on ll^p(b Z)

Existence of operators that are u- but not frequently hypercyclic

Existence of operators that are frequently but not distributionally chaotic

Abstract

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on $\ell^p(\mathbb Z)$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is $\mathcal U$-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently hypercyclic, yet not distributionally chaotic. These (surprizing) counterexamples are given by weighted shifts on $c_0$. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.