Sumsets of the distance set in ${\mathbb{F}}_q^d$

Abstract

Let ${\mathbb{F}}_q$ be a finite field of order $q$, where $q$ is large odd prime power. In this paper, we improve some recent results on the additive energy of the distance set, and on sumsets of the distance set due to Shparlinski (2016). More precisely, we prove that for $\mathcal{E}\subseteq {\mathbb{F}}_q^d$, if $d=2$ and $q^{1+\frac{1}{4k-1}}=o(|\mathcal{E}|)$ then we have $|k{\Delta }_{{\mathbb{F}}_q}(\mathcal{E})|=(1-o(1))q$; if $d\ge 3$ and $q^{\frac{d}{2}+\frac{1}{2k}}=o(|\mathcal{E}|)$ then we have $|k{\Delta }_{{\mathbb{F}}_q}(\mathcal{E})|=(1-o(1))q,$ where $kΔ_{\mathbb{F}_q}(\mathcal{E}):=Δ_{\mathbb{F}_q}(\mathcal{E})+\cdots+Δ_{\mathbb{F}_q}(\mathcal{E}) ~(\mbox{$k$ times}).$

V3 with some corrections

References in corpus (2)

• Some new inequalities in additive combinatorics
• Energies and structure of additive sets