Invariant means and thin sets in harmonic analysis with applications to prime numbers

Abstract

We first prove a localization principle characterising Lust-Piquard sets. We obtain that the union of two Lust-Piquard sets is a Lust-Piquard set, provided that one of these two sets is closed for the Bohr topology. We also show that the closure. We first prove a localization principle characterizing Lust-Piquard sets. We obtain that the unionof two Lust-Piquard sets is a Lust-Piquard set, provided that one of these two sets is closed forthe Bohr topology. We also show that the closure of the set of prime numbers is a Lust-Piquardset, generalizing results of Lust-Piquard and Meyer, and even that the set of integers whoseexpansion uses fewer than r factors is a Lust-Piquard set. On the other hand, we use randommethods to prove that there are some sets t hat are UC,Λ(q) for every q>2andp-Sidon for everyp>1, but which are not Lust-Piquard sets. This is a consequence of the fact that a uniformly distributed set cannot be a Lust-Piquard set.for every p > 1, but which are not Lust-Piquard sets. This is a consequence of the fact that a uniformly distributed set cannot be a Lust-Piquard set.