Doowon Koh

Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture

We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering F q , the finite field with q elements, by A · A + ··· + A · A, where A is a subset F q of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdos-Falconer distance problem for subsets of the unit sphere in F d q and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher-dimensional vector spaces over general finite fields.

EXTENSION THEOREMS FOR SPHERES IN THE FINITE FIELD SETTING

Abstract In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields. In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, previous results by the authors in [Iosevich and Koh, Illinois J. of Mathematics: 2008]. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.

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 Bezouts theorem, we obtain the sufficient and necessary conditions on general curves where sharp

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

L^p-L^r

Distance sets of two subsets of vector spaces over finite fields

extension estimates hold. Our main result can be considered as a nice generalization of works by Mochenhaupt and Tao and Iosevich and Koh. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.

Extension and averaging operators for finite fields

We study the Erdos-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.

On the Sums of Any

We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we improve upon the results by Rainer Dietmann. In the case that one of the subsets is a product set, we obtain further improvement on the estimate.

k

In this paper we study L p L r estimates of both extension operators and averaging operators associated with the algebraic variety S = {x 2 F d : Q(x) = 0} where Q(x) is a nonde- generate quadratic form over the finite field Fq. In the case when d � 3 is odd and the surface S contains a (d 1)/2-dimensional subspace, we obtain the exponent r where the L 2 L r extension estimate is sharp. In particular, we give the complete solution to the extension problems related to specific surfaces S in three dimension. In even dimensions d � 2, we also investigates the sharp L 2 L r extension estimate. Such results are of the generalized version and extension to higher dimensions for the conical extension problems which Mochenhaupt and Tao ((10)) studied in three dimensions. The boundedness of averaging operators over the surface S is also studied. In odd dimensions d � 3 we completely solve the problems for L p L r estimates of averaging operators related to the surface S. On the other hand, in the case when d � 2 is even and S contains a d/2-dimensional subspace, using our optimal L 2 L r results for extension theorems we, except for endpoints, have the sharp L p L r estimates of the averaging operator over the surface S in even dimensions.

Points in Finite Fields

For a set

Restriction operators acting on radial functions on vector spaces over finite fields

E\subset \mathbb F_q^d