  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.
