Distribution of Distances in Five Dimensions and Related Problems

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