Progressions with a pseudorandom step

Combinatorics Seminar
Friday, October 25, 2013 - 15:00
1 hour (actually 50 minutes)
Skiles 005
University of Hamburg
An open problem of interest in combinatorial number theory is that of providing a non-ergodic proof to the so called polynomial Szemeredi theorem. So far, the landmark result in this venue is that of Green who considered the emergence of 3-term arithmetic progressions whose gap is a sum of two squares (not both zero) in dense sets of integers. In view of this we consider the following problem. Given two dense subsets A and S of a finite abelian group G, what is the weakest "pseudorandomness assumption" once put on S implies that A contains a 3-term arithmetic progressions whose gap is in S?  We answer this question for G=Z_n and G = F_p^n. To quantify pseudorandomness we use Gowers norms.