164 research outputs found
Difference sets and frequently hypercyclic weighted shifts
We solve several problems on frequently hypercyclic operators. Firstly, we
characterize frequently hypercyclic weighted shifts on ,
. 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 -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 . The construction of these shifts lies on the
construction of sets of positive integers whose difference sets have very
specific properties
Difference Sets and Positive Exponential Sums I. General Properties
We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set A the maximal size of a set whose difference set avoids A will be related to positive exponential sums using frequencies from A. © 2013 Springer Science+Business Media New York
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