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Invariant means and thin sets in harmonic analysis with applications to prime numbers
Author(s) -
Lefèvre Pascal,
Rodríguez-Piazza Luis
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdp016
Subject(s) - lust , mathematics , set (abstract data type) , invariant (physics) , combinatorics , coprime integers , discrete mathematics , computer science , psychology , psychoanalysis , mathematical physics , programming language
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.