A generalization of a theorem of Erdős-Rényi to $m$-fold sums and differences

Abstract

Let $m\ge 2$ be a positive integer. Given a set $E(\omega )\subseteq \mathbb{N}$ we define $r_N^{(m)}(\omega )$ to be the number of ways to represent $N\in \mathbb{Z}$ as any combination of sums $\text{ and }$ differences of $m$ distinct elements of $E(\omega )$. In this paper, we prove the existence of a "thick" set $E(\omega )$ and a positive constant $K$ such that $r_N^{(m)}(\omega )<K$ for all $N\in \mathbb{Z}$. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.