k-Point configurations in sets of positive density of Zn

Abstract

It is shown that if $n>2k+4$ and if $A\subseteq {\mathbb{Z}}^n$ is a set of upper density $ϵ>0$, then—in a sense depending on $ϵ$—all large dilates of any given k-dimensional simplex $△=\{0,v_1,\dots ,v_k\}\subset {\mathbb{Z}}^n$ can be embedded in $A$. A simplex $△$ can be embedded in the set $A$ if $A$ contains simplex ${△}^{\prime }$, which is isometric to $△$. Moreover, the same is true if only $△\subset {\mathbb{R}}^n$ 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]