Distribution of distances in five dimensions and related problems

TL;DR

This paper investigates the Erdős-Falconer distance problem in five dimensions over finite fields, establishing new bounds and conditions for the distance set to cover the entire field, especially for Cartesian product sets.

Contribution

It provides new bounds for the size of sets needed to ensure the distance set covers the entire field in five dimensions, with improved results for sets with small difference sets.

Findings

For sets A with size greater than p^{13/22}, the distance set covers all of _p.

Stronger results are obtained when |A-A|\u223c|A|, including explicit sumset conditions.

New bounds on the size of A^2+A^2 in relation to |A|, K, and p are established.

Abstract

In this paper, we study the Erd\H{o}s-Falconer distance problem in five dimensions for sets of Cartesian product structures. More precisely, we show that for $A\subset \mathbb{F}_p$ with $|A|\gg p^{\frac{13}{22}}$, then $\Delta(A^5)=\mathbb{F}_p$. When $|A-A|\sim |A|$, we obtain stronger statements as follows: If $|A|\gg p^{\frac{13}{22}}$, then $(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$ If $|A|\gg p^{\frac{4}{7}}$, then $(A-A)^2+(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$ We also prove that if $p^{4/7}\ll |A-A|=K|A|\le p^{5/8}$, then \[|A^2+A^2|\gg \min \left\lbrace \frac{p}{K^4}, \frac{|A|^{8/3}}{K^{7/3}p^{2/3}}\right\rbrace.\] As a consequence, $|A^2+A^2|\gg p$ when $|A|\gg p^{5/8}$ and $K\sim 1$, where $A^2=\{x^2\colon x\in A\}$.