Akos Magyar

Geometric Ramsey theory on the integer lattice

Let S be a subset of the integer lattice Z^n of positive upper density and let Q be a finite point configuration. We will study the question whether all sufficiently large dilates of Q have an isometric copy in S. We will show the answer is affirmative if Q is a d-dimensional cube in dimensions n ≥ 5d, in an appropriate sense, and will also indicate how our approach can be modified to handle other configurations such as direct products of non-degenerate simplices.

The proof combines methods of number theory - such as the circle method - with those of additive combinatorics, in particular the use of box norms and notions of weak regularity, adapted to our settings.