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