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 {\sl 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 $\hat{I}›(q)$-sets for all $q<\infty$, 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 ot of uniform convergence.