The generalized Erdos-Falconer distance problems in vector spaces over finite fields

TL;DR

This paper extends the Erdos-Falconer distance problem in finite fields by introducing polynomial-based generalized distances, providing formulas for distance set sizes, and generalizing previous spherical and cubic distance results.

Contribution

It develops a simple formula for estimating the size of distance sets defined by diagonal polynomials and generalizes several existing distance problems in finite fields.

Findings

Derived formulas for distance set sizes using polynomial definitions.

Generalized spherical and cubic distance problems to higher dimensions.

Extended pinned distance problem analysis in finite fields.

Abstract

In this paper we study the generalized Erdos-Falconer distance problems in the finite field setting. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular, we develop a simple formula for estimating the cardinality of distance sets determined by diagonal polynomials. As a result, we generalize the spherical distance problems due to Iosevich and Rudnev and the cubic distance problems due to Iosevich and Koh. Moreover, our results are of higher dimensional version for Vu's work on two dimension. In addition, we set up and study the generalized pinned distance problems in finite fields. We give a nice generalization of some recent work in which the pinned distance problems related to spherical distances were investigated. Discrete Fourier analysis and exponential sum estimates play an important role in our proof.