On some random thin sets of integers

Daniel Li (1) , Hervé Queffélec (2) , Luis Rodriguez-Piazza (3)

1 LML - Laboratoire de Mathématiques de Lens
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 Departamento de Analisis Matematico

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 $\frac{4}{3}$-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.

Mots clés: Boucheron-Lugosi-Massart's deviation inequality; $\Lambda (q)$-sets; $p$-Rider sets; Rosenthal sets; selectors; sets of uniform convergence

Domaines: Analyse fonctionnelle [math.FA]

Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. On some random thin sets of integers. Proceedings of the American Mathematical Society, 2008, 136 (1), pp.141 - 150. ⟨hal-00376102⟩