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

Let $m\geq 2$ be a positive integer. Given a set $E(ω)\subseteq \mathbb{N}$ we define $r_{N}^{(m)}(ω)$ 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 $E(ω)$. In this paper, we prove the existence of a "thick" set $E(ω)$ and a positive constant $K$ such that $r_{N}^{(m)}(ω)<K$ 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.

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