Quadratic distances in even dimensions over prime fields

Abstract

Let $p$ be an odd prime, let $m\ge 1$ be an integer, and let $Q$ be a nondegenerate quadratic form on ${\mathbb{F}}_p^{2m}$. 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 $|E|>p^m$, \[ |Δ_Q(E)|\gg \min\left\{\frac{p}{L_E},\, \frac{|E|}{p^{m-1}L_E^2}\right\}, \qquad L_E=\log\left(2+\frac{p^{m+1}}{|E|}\right), \] with an absolute implied constant independent of $p$, $m$ and $Q$. This confirms the Erdős--Falconer distance conjecture up to a logarithmic factor. In the anisotropic planar case $m=1$, for small sets, we also prove that \[ |Δ_Q(E)|\gg\frac{|E|}{\log(2|E|)}, \qquad (1\leq |E|\leq p), \] which is also optimal up to a logarithmic factor.

arXiv comment50 pages