Central sets and a non-commutative Roth theorem

Abstract

In 1952 K. Roth showed that any subset of N having positive upper density contains arithmetic progressions of length 3, a result E. Szemer´edi extended to progressions of arbitrary length in 1975. An approach involving multiple recurrence for measure preserving systems developed by H. Furstenberg led to extensions to linear configurations in more general groups while demonstrating families of such configurations to be large in various senses. Using ultrafilter techniques for doing ergodic theory without averaging, we prove a version of Roth's theorem, in its ergodic-theoretic formulation, for general countable groups G. A combinatorial consequence, valid in amenable groups, is the presence of many configurations of the form {(a, b), (ag, b), (ag, bg)} in positive density subsets of G × G. Here "many" implies in particular "central* in the variable g".