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

TL;DR

This paper improves bounds on the size of sumsets of the distance set in finite fields, showing that under certain conditions, multiple sums of the distance set nearly cover the entire field.

Contribution

It provides new threshold conditions for the sumsets of the distance set in finite fields, extending previous results and improving bounds for dimensions two and higher.

Findings

For d=2, sumsets cover almost all of _q if |\u00a0| is sufficiently large.

For de 3, similar coverage occurs under different size conditions.

Results extend and refine previous bounds on additive properties of distance sets.

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 $k\Delta_{\mathbb{F}_q}(\mathcal{E}):=\Delta_{\mathbb{F}_q}(\mathcal{E})+\cdots+\Delta_{\mathbb{F}_q}(\mathcal{E}) ~(\mbox{$k$ times}).$