K-Point configurations in sets of positive density of Zn

Abstract

It is shown that if n>2k+4 and if A⊆Zn is a set of upper density ϵ>0, then—in a sense depending on ϵ—all large dilates of any given k-dimensional simplex △={0,v1,…,vk}⊂Zn can be embedded in A. A simplex △ can be embedded in the set A if A contains simplex △′, which is isometric to △. Moreover, the same is true if only △⊂Rn is assumed, and △ satisfies some immediate necessary conditions. The proof uses techniques of harmonic analysis developed for the continuous case, as well as a variant of the circle method due to Siegel [S]