On a class of random sets of positive integers

TL;DR

This paper investigates a class of randomly generated positive integer sets, establishing conditions under which they are lacunary, have unbounded gaps, and contain infinitely many arithmetic progressions.

Contribution

It provides new probabilistic conditions for the structural properties of random subsets of positive integers induced by Bernoulli variables.

Findings

Sets are almost surely lacunary under certain conditions.

Sets have unbounded gaps with high probability.

Sets contain infinitely many arithmetic progressions under specified criteria.

Abstract

In this note, we study a class of random subsets of positive integers induced by Bernoulli random variables. We obtain sufficient conditions such that the random set is almost surely lacunary, does not have bounded gaps and contains infinitely many arithmetic progressions, respectively.