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From combinatorics to dynamics and back again: 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. We start with an overview of the types of patterns that occur in any sufficiently large set, start with the classical setting of the arithmetic progressions in Szemeredi’s seminal result and then turning to the more difficult question of infinite patterns, including the recent solution of the Erdos Sumset Conjecture by Moreira, Richter, and Robertson, and its generalizations.

Translating combinatorial questions into dynamical ones: Furstenberg’s proof of Szemeredi’s Theorem introduced the Correspondence Principle, a general technique for translating a combinatorial problem into a dynamical one. While the original formulation suffices for certain patterns, including arithmetic progressions and some infinite configurations, higher order generalizations have required refinements of these tools.

Dynamical structure theorems and infinitary combinatorics: In joint work with Joel Moreira, Florian Richter, and Donald Robertson, we use ergodic methods to prove a k-fold generalization of the Erdos Sumset Conjecture. We give an overview of the dynamical structures that are used to prove this result.