Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

TL;DR

This paper establishes sharp extension theorems for algebraic curves in two-dimensional finite field vector spaces and applies these results to Falconer distance problems, generalizing previous work.

Contribution

It provides necessary and sufficient conditions for sharp extension estimates on algebraic curves, extending prior results and applying them to Falconer distance problems in finite fields.

Findings

Sharp $L^p-L^r$ extension estimates for algebraic curves

Necessary and sufficient conditions for extension theorems

Application to Falconer distance problems in finite fields

Abstract

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout's theorem, we obtain the sufficient and necessary conditions on general curves where sharp $L^p-L^r$ extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao and Iosevich and Koh. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.