Product of Simplices and sets of positive upper density in Rd

We establish that any subset of R-d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d >= 4. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Delta(k1) and Delta(k2) are two fixed nondegenerate simplices of k(1) + 1 and k(2) + 1 points respectively, then any subset of R-d of positive upper Banach density with d >= k(1) + k(2) + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Delta(k1) x Delta(k2) A new direct proof of the fact that any subset of R-d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed nondegenerate simplex of k + 1 points provided d >= k + 1, a result originally due to Bourgain, is also presented.