A generalization of a theorem of Erd\H{o}s-R\'enyi to $m$-fold sums and differences

Let $m\ge 2$ be a positive integer. Given a set $E(\omega )\subseteq $\mathbb{N}$$wedefine$r_{N}^{(m)}(\omega )$ to be the number of ways to represent $N\in \mathbb{Z}$ as any combination of sums $\mathit{\text{and}}$ 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 Erd\H{o}s and R'enyi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.