Distinct distances on regular varieties over finite fields

TL;DR

This paper extends results on distance sets in finite fields, showing that large subsets of regular varieties determine almost all distances for certain polynomial families.

Contribution

It generalizes previous work by establishing conditions under which subsets of regular varieties cover all nonzero values of specific polynomial distance functions.

Findings

Large subsets in regular varieties determine all nonzero polynomial distances.

Conditions on subset size ensure full coverage of distance values.

Results extend known theorems to broader polynomial families.

Abstract

In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset $\mathcal{E}$ in a regular variety satisfies $|\mathcal{E}|\gg q^{\frac{d-1}{2}+\frac{1}{k-1}}$, then $\Delta_{k, F}(\mathcal{E})\supseteq \mathbb{F}_q\setminus \{0\}$ for some certain families of polynomials $F(\mathbf{x})\in \mathbb{F}_q[x_1, \ldots, x_d]$.