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

Kathryn E. Hare¹ ; Shuntaro Yamagishi¹

¹ Department of Pure Mathematics University of Waterloo Waterloo, ON, Canada N2L 3G1

Acta Arithmetica, Tome 166 (2014) no. 1, pp. 55-67

Résumé

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.

Détail

DOI : 10.4064/aa166-1-5

Keywords: geq positive integer given set omega subseteq mathbb define omega number ways represent mathbb combination sums differences distinct elements omega paper prove existence thick set omega positive constant omega mathbb generalization known theorem erd nyi apply results harmonic analysis where prove existence certain thin sets