The Erdös–Falconer Distance Problem, Exponential Sums, and Fourier Analytic Approach to Incidence Theorems in Vector Spaces over Finite Fields

Abstract

We study the Erdös–Falconer distance problem in vector spaces over finite fields with respect to the cubic metric. Estimates for discrete Airy sums and Adolphson–Sperber estimates for exponential sums in terms of Newton polyhedra play a crucial role. Similar techniques are used to study the incidence problem between points and cubic and quadratic curves. As a result we obtain a nontrivial range of exponents that appear to be difficult to attain using combinatorial methods.