Ergodic-theoretic implementations of the Roth density-increment argument

We exhibit proofs of Furstenberg’s Multiple Recurrence Theorem and of a special case of Furstenberg and Katznelson’s multidimensional version of this theorem, using an analog of the density-increment argument of Roth and Gowers. The second of these results requires also an analog of some recent finitary work by Shkredov.

Many proofs of these multiple recurrence theorems are already known. However, the approach of this paper sheds some further light on the well-known heuristic correspondence between the ergodic-theoretic and combinatorial aspects of multiple recurrence and Szemeredi’s Theorem. Focusing on the density- increment strategy highlights several close points of connection between these settings.