Parabolic distance in $\mathbb F_q^2$: a sharp exponent and new results

TL;DR

This paper investigates the parabolic distance problem in finite fields, establishing size thresholds for sets to determine many distances, using Fourier analysis and additive combinatorics to derive sharp results.

Contribution

It introduces new bounds for the parabolic distance problem in finite fields and combines Fourier analytic and additive combinatorics techniques for sharper results.

Findings

Established size thresholds for sets to determine many parabolic distances

Provided sharpness examples demonstrating the bounds' optimality

Extended understanding of the parabolic distance functional in finite fields

Abstract

We study the parabolic variant of the Erd\H os--Falconer distance problem in finite fields. That is, if $q$ is odd, we seek size thresholds beyond which any subset $E\subset \mathbb F_q^2$ will determine many distinct parabolic distances. This problem has a rich history because the parabolic distance functional shares many properties with the standard distance functional, but exhibits many distinct behaviors. Here we begin with rather standard Fourier analytic arguments, but diverge into additive combinatorics to handle the central obstructions. We provide a suite of positive results and corresponding sharpness examples.