A generalization of a theorem of Erdős–Rényi to $m$-fold sums and differences
Volume 166 / 2014
Acta Arithmetica 166 (2014), 55-67
MSC: Primary 11K31, 11B83; Secondary 43A46.
DOI: 10.4064/aa166-1-5
Abstract
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 a combination of sums 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.
