From combinatorics to ergodic theory and back again

Abstract

Multiple ergodic averages, such as the average of expressions like _f_1(Tn x) _f_2(T2n x) ... fk(Tkn x), were first studied in the ergodic theoretic proof of Szemerédi’s Theorem on arithmetic progressions. It turns out that all constraints on such averages (in a sense that we describe) have an algebraic character, arising from identities in nilpotent groups. We discuss these averages, several generalizations, and combinatorial implications of the results.