This is an announcement for the paper "On some random thin sets of integers" by Daniel Li, Herve Queffelec, Luis Rodriguez-Piazza. 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 Math\'ematique 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 ot of uniform convergence. Archive classification: math.FA Mathematics Subject Classification: 43 A 46 ; 42 A 55 ; 42 A 61 Citation: Proceedings of the American Mathematical Society 136, 1 (2008) 141 The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0904.2507 or http://arXiv.org/abs/0904.2507 or by email in unzipped form by transmitting an empty message with subject line uget 0904.2507 or in gzipped form by using subject line get 0904.2507 to: math@arXiv.org.