Distinct distances on regular varieties over finite fields

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 $F(\mathbf{x})\in {\mathbb{F}}_q[x_1,\dots ,x_d]$.

References in corpus (1)

• Distance sets of two subsets of vector spaces over finite fields