The generalized Erdos-Falconer distance problems in vector spaces over finite fields
Journal
Journal of Number Theory
Journal Volume
132
Journal Issue
11
Pages
2455-2473
Date Issued
2012
Author(s)
Koh, D.
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 (2007) [13] and the cubic distance problems due to Iosevich and Koh (2008) [12]. Moreover, our results are higher-dimensional version of Vu's work (Vu, 2008 [24]) on two dimensions. In addition, we set up and study the generalized pinned distance problems in finite fields. We give a generalization of the work by Chapman et al. (2012) [2] who studied the pinned distance problems related to spherical distances. Discrete Fourier analysis and exponential sum estimates play an important role in our proof. © 2012 Elsevier Inc.
Type
journal article