Description:

Title: $U^k(\Phi)$-uniform sets and infinite sumset patterns

Abstract: Based on the local uniformity seminorms introduced by Host and Kra in 2009, $U^k(\Phi)$\emph{-uniform} subsets of the natural numbers are \emph{pseudorandom} sets where the degree $k$ can be understood as a \emph{level of randomness}. Typically, pseudorandom sets contain a rich variety of patterns, and their study has helped to understand the structure of general subsets of the natural numbers. In this talk we introduce these sets and we relate the degree $k$ to the existence of infinite sumset patterns at prescribed vertices of $\ell$-dimensional parallelepipeds, for $k \leq \ell$. We also introduce some of the dynamical tools to prove this result, which extend the work of  Kra, Moreira, Richter and Roberson.