Ubiquity of simplices in subsets of vector spaces over finite fields

Derrick Hart, Alex Iosevich

Published 2007-03-16, updated 2007-10-11Version 2

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