Chun-Yen Shen

Quadratic distances in even dimensions over prime fields

Let $p$ be an odd prime, let $m\geq1$ be an integer, and let $Q$ be a nondegenerate quadratic form on $\mathbb{F}_p^{2m}$ with Witt index $m-1$. For a nonempty set $E\subseteq\mathbb{F}_p^{2m}$, write $\Delta_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $|E|\geq p^m$, $$ |\Delta_Q(E)|\gg \frac{p}{\log\bigl(2+p^{m...