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