Sharp spherical extension theorem in ${\mathbb{F}}_q^{2m}$ and applications

Abstract

Let $q$ be an odd prime power and $m\ge 1$. Let $Q$ be a nondegenerate quadratic form on ${\mathbb{F}}_q^{2m}$. For every sphere $S_j=\{Q=j\}$ with $j\in {\mathbb{F}}_q^{\times }$, we prove the sharp extension estimate $R_{S_j}^{\ast }(2\to r){\lesssim }_{m}1$ for $r\ge 2(m+1)/m$, uniformly in $q$, $Q$, and $j$. As an application, we show that if $E\subseteq {\mathbb{F}}_q^{2m}$ satisfies $|E|/q^{m+1/3}\to \infty$, then almost every pin $y\in E$ determines $(1-o(1))q$ values of $Q(x-y)$. The proof combines two arithmetic Hecke operator estimates with an induction in the dimension, a centered sphere--cone estimate, and an orthogonal decomposition.

arXiv commentThis paper subsumes a previous paper of the authors arXiv:2606.03627. In this new submission, the global conjecture is solved for all even dimensions and over arbitrary finite fields, extending earlier results that were restricted to four dimensions over prime fields. The localized conjecture remains unchanged over prime fields