Zero-sum subsets in vector spaces over finite fields

Ubiquity of simplices in subsets of vector spaces over finite fields

We prove that a sufficiently large subset of the d-dimensional vector space over a finite field with q elements, Fq , contains a copy of every k-simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.

On Directions Determined by Subsets of Vector Spaces over Finite Fields

We prove that if a subset of a d-dimensional vector space over a finite field with q elements has more than qd−1 elements, then it determines all the possible directions. We obtain a complete characterization if the size of the set is ≥ qd−1. If a set has more than qk elements, it determines a k-dimensional set of directions. We prove stronger results for sets that are sufficiently random. This...

Extension and averaging operators in vector spaces over finite fields

We study the mapping properties of both extension operators and averaging operators associated with the algebraic variety S = {x ∈ Fq : Q(x) = 0} where Q(x) is a nondegenerate quadratic form over the finite field Fq. Carbery, Stones, and Wright [3] established a method to deduce L − L estimates for averaging operators over a general variety. As a core result in this paper, we give the sharp est...

Extension Theorems in Vector Spaces over Finite Fields

Orthogonal Systems in Vector Spaces over Finite Fields

We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors.