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

Pascal Lefèvre, Luis Rodríguez Piazza

• 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, Ë(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.