K-POINT CONFIGURATIONS IN SETS OF POSITIVE DENSITY OF Zn

It is shown that if n > 2k + 4 and if A ⊆ Z is a set of upper density ε > 0, then in a sense depending on ε all large dilates of any given k-dimensional simplex 4 = {0, v1, . . . , vk} ⊂ Z can be embedded in A. A simplex 4 can be embedded in the set A, if A contains simplex 4′ which is isometric to 4. Moreover, the same is true if only 4 ⊂ R is assumed, and 4 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.