On the generalized Erdős–Falconer distance problems over finite fields

Abstract

In this paper we study the generalized Erdős–Falconer distance problems in the finite field setting via spectra of directed graph. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular, we give graph-theoretic proofs for various results due to Koh and Shen (2011). As a consequence, we also obtain a better bound under a weaker condition of the restricted sets.