Ergodic-theoretic implementations of the Roth density-increment argument

Abstract

We exhibit proofs of two ergodic-theoretic results in the study of multiple recurrence using an analog of the density-increment argument of Roth and Gowers: Furstenberg's Multiple Recurrence Theorem (which implies Szemer\'edi's Theorem), and a two-dimensional special case of Furstenberg and Katznelson's multidimensional version of this theorem. The second of these requires also an analog of some recent finitary work by Shkredov. Many proofs of these multiple recurrence theorems are now known, but our main goal is to shed some further light on the heuristic correspondence principle that has grown up between the ergodic-theoretic and combinatorial aspects of multiple recurrence and Szemer\'edi's Theorem. Focusing on the density-increment strategy highlights several close points of connection between these settings.