Extension theorems and a connection to the Erd?s-Falconer distance problem over finite fields

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

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 non-trivial rang...

Additive Energy and the Falconer Distance Problem in Finite Fields

We study the number of the vectors determined by two sets in d-dimensional vector spaces over finite fields. We observe that the lower bound of cardinality for the set of vectors can be given in view of an additive energy or the decay of the Fourier transform on given sets. As an application of our observation, we find sufficient conditions on sets where the Falconer distance conjecture for fin...

Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout’s theorem, we obtain the sufficient and necessary conditions on general curves where sharp L − L extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao in [16] and Iosevich and Koh in [9]. A...

The Erdös-Falconer Distance Problem on the Unit Sphere in Vector Spaces Over Finite Fields

Hart, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erdös-Falconer distance conjecture holds for subsets of the unit sphere in Fdq . In this note, we give a graph theoretic proof of this result.

Extension Theorems in Vector Spaces over Finite Fields