On some random thin sets of integers

Abstract

We show how different random thin sets of integers may have different behaviour. First, using a recent deviation inequality of Boucheron, Lugosi and Massart, we give a simpler proof of one of our results in Some new thin sets of integers in harmonic analysis, Journal d’Analyse Mathématique 86 (2002), 105–138, namely that there exist 4/3 -Rider sets which are sets of uniform convergence and Λ(q)-sets for all q < ∞ but which are not Rosenthal sets. In a second part, we show, using an older result of Kashin and Tzafriri, that, for p > 4/3 , the p-Rider sets which we had constructed in that paper are almost surely not of uniform convergence.