Misha Rudnev

Erdös distance problem in vector spaces over finite fields

We study the Erdos/Falconer distance problem in vector spaces over finite fields. Let be a finite field with elements and take , . We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in to provide estimates for minimum cardinality of the distance set in terms of the cardinality of . Bounds for Gauss and Kloosterman sums play an important role in the proof.

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.

On the number of incidences between points and planes in three dimensions

We prove an incidence theorem for points and planes in the projective space ℙ3 over any Field

On the restriction problem for discrete paraboloid in lower dimension

\mathbb{F}

Non-isotropic distance measures for lattice-generated sets

F, whose characteristic p ≠ 2. An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if p > 0.This yields a bound on the number of incidences between m points and n planes in ℙ3, with m≥n as

On an Application of Guth–Katz Theorem

O\left( {m\sqrt n + mk} \right)