Geometric patterns in dense subsets of \Z ^n

Abstract: A basic problem of arithmetic Ramsey theory is to show the existence of structured subsets in large but otherwise arbitrary sets of integers or integer points. One of the most fundamental results in this area, proved originally by Furstenberg and Katznelson via ergodic theory, states that if A\subseteq \Z^n is a set of positive upper density then A contains a translated and dilated copy of any given finite set F of lattice points. Though there is a more recent combinatorial proof, current approaches give no or very weak quantitative bounds. In the talk we will discuss some quantitative results concerning an analogues problem, made more approachable to Fourier analytic methods by allowing rotations. Namely, we will show the existence of translated, dilated and rotated copies of certain finite sets F in sets of positive density A\subseteq \Z^n.

Refreshments will be served at 3:45 p.m. (MATX 1115, Math Lounge).