The Erdos sumset conjecture and its generalizations

A striking example of the interactions between additive combinatorics and ergodic theory is  Szemeredi’s Theorem that a set of integers with positive upper density contains arbitrarily long arithmetic progressions.  Soon thereafter, Furstenberg used Ergodic Theory to gave a new proof of this result, leading to the development of combinatorial ergodic theory.  These tools have led to uncovering new patterns that must occur in sufficiently large sets of integers and an understanding of what types of structures control these behaviors.  Only recently have we been able to extend these methods to infinite patterns, and in recent work we show that any set of integers with positive upper density contains a k-fold sumset. This is joint work with Joel Moreira, Florian Richter, and Donald Robertson.