A generalization of Erd\H{o}s-R\'enyi to $m$-fold sums and differences
Kathryn E. Hare, Shuntaro Yamagishi · arXiv (Cornell University) · 2014
Let $m\geq 2$ 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 $\textit{ and }$ 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 Renyi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.