A spherical extension theorem and applications in positive characteristic

TL;DR

This paper proves a new extension theorem for spheres in finite fields, improving previous results, and introduces a novel incidence bound with applications to distance problems in positive characteristic.

Contribution

It presents a new extension theorem for spheres in finite fields and derives a point-hyperplane incidence bound using a cone restriction theorem.

Findings

Improved extension theorem for spheres of square radii in finite fields.

New point-hyperplane incidence bound derived via cone restriction theorem.

Applications to distance problems in positive characteristic.

Abstract

In this paper, we prove an extension theorem for spheres of square radii in $\mathbb{F}_q^d$, which improves a result obtained by Iosevich and Koh (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restriction theorem due to the authors and Lee (2022). Applications on the distance problems will also be discussed.