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Journal of the London Mathematical Society Advance Access originally published online on May 19, 2009
Journal of the London Mathematical Society 2009 80(1):72-84; doi:10.1112/jlms/jdp016
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© 2009 London Mathematical Society

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

Pascal Lefèvre

Univ Lille Nord de France F-59 000 LILLE U-Artois
Laboratoire de Mathématiques de Lens EA 2462
Fédération CNRS Nord-Pas-de-Calais FR 2956
F-62 300 LENS
France

Luis Rodríguez-Piazza

Faculdad de Matematica
Universidad de Sevilla
Apdo 1160
41080 Sevilla
Spain
piazza@us.es

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, {Lambda}(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.

2000 Mathematics Subject Classification 43A46 (primary), 42A20, 42A55, 43A77 (secondary).

This paper was partially supported by BFM2003-01297 and MTM2006-05622.

Received October 21, 2007; revised October 21, 2008; published online May 19, 2009.


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