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

We first prove a localization principle characterizing 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 of the set of prime numbers is a Lust‐Piquard set, generalizing results of Lust‐Piquard and Meyer, and even that the set of integers whose expansion uses fewer than r factors is a Lust‐Piquard set. On the other hand, we use random methods to prove that there are some sets that are UC , Λ( q ) for every q >2 and p ‐Sidon 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.