On some random thin sets of integers

TL;DR

This paper investigates the behavior of various random thin sets of integers, demonstrating their differing properties in harmonic analysis, including convergence and set classifications, through new proofs and probabilistic methods.

Contribution

It provides new probabilistic proofs of properties of thin sets of integers and explores their convergence behaviors using recent deviation inequalities.

Findings

Existence of 4/3-Rider sets with specific convergence properties

Construction of p-Rider sets that are almost surely not of uniform convergence for p > 4/3

Application of recent deviation inequalities to harmonic analysis problems

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 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.