Quadratic distances in even dimensions over prime fields

Source abstract

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 $Δ_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $|E|\geq p^m$, $$|Δ_Q(E)|\gg \frac{p}{\log\bigl(2+p^{m+1}/|E|\bigr)},$$ with an absolute implied constant independent of $p$, $m$ and $Q$. In the planar case $m=1$, we also prove that $$|Δ_Q(E)|\gg\frac{|E|}{\log(2|E|)}, \qquad (1\leq |E|\leq p),$$ which is optimal up to a logarithmic factor.