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

Kathryn Hare; Shuntaro Yamagishi

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arXiv · 1404.2648

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

Kathryn Hare·Shuntaro Yamagishi

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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 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 Erd\H{o}s and R\'enyi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

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BibTeX RIS

Kathryn Hare, Shuntaro Yamagishi. 2014-04-09. A generalization of a theorem of Erd\H{o}s-R\'enyi to $m$-fold sums and differences. https://arxiv.org/abs/1404.2648

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