Result: Interactions between Ergodic Theory and Combinatorial Number Theory

The seminal work of Furstenberg on his ergodic proof of Szemerédi’s Theorem gave rise to a very rich connection between Ergodic Theory and Combinatorial Number Theory (Additive Combinatorics). The former is concerned with dynamics on probability spaces, while the latter is concerned with Ramsey theoretic questions about the integers, as well as other groups. This thesis further develops this symbiosis by establishing various combinatorial results via ergodic techniques, and vice versa. Let us now briefly list some examples of such. A shorter ergodic proof of the following theorem of Magyar is given: If B Zd, where d 5, has upper Banach density at least > 0, then the set of all squared distances in B, i.e., the set fkb1 􀀀 b2k2 j b1; b2 2 Bg, contains qZ>R for some integer q = q( ) > 0 and R = R(B). Our technique also gives rise to results on the abundance of many other higher order Euclidean configurations in such sets. Next, we turn to establishing analogues of this result of Magyar, where k k2 is replaced with other quadratic forms and various other algebraic functions. Such results were initially obtained by Björklund and Fish, but their techniques involved some deep measure rigidity results of Benoist-Quint. We are able to recover many of their results and prove some completely new ones (not obtainable by their techniques) in a much more self-contained way by avoiding these deep results of Benoist-Quint and using only classical tools from Ergodic Theory. Finally, we extend some recent ergodic analogues of the classical Plünnecke inequalities for sumsets obtained by Björklund-Fish and establish some estimates of the Banach density of product sets in amenable non-abelain groups. We have aimed to make this thesis accesible to readers outside of Ergodic Theory who may be primarily interested in the arithmetic and combinatorial applications.