  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 Math\'ematique 86
(2002), 105--138}, namely that there exist 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 $p > {4/3}$, the $p$-Rider sets which we had constructed
in that paper are almost surely ot of uniform convergence.
