Distribution of distances in five dimensions and related problems

Abstract

In this paper, we study the Erdős-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 $A^2=\{x^2:x\in A\}$.

13 pages