Higher Order Sumsets in Sets of Positive Density

In this talk we present a natural generalization of a sumset conjecture of Erdos to higher orders, asserting that every subset of the integers with positive density contains a sumset $B_1+\ldots +B_k$ where $B_1, \ldots , B_k$ are infinite. Our method relies on a newfound connection between k-fold sumsets in the integers and return times of orbits in k-fold joinings of measure preserving systems arising from the Host-Kra structure theory. This talk is based on joint work with Kra, Moreira, and Robertson.