On some random thin sets of integers

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 $\frac{4}{3}$-Rider sets which are sets of uniform convergence and $\Lambda (q)$-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 $p > \frac{4}{3}$, the $p$-Rider sets which we had constructed in that paper are almost surely not of uniform convergence.