Difference sets and frequently hypercyclic weighted shifts

Abstract

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on ${\ell }^p(\mathbb{Z})$, $p\ge 1$. 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.

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