Resumen de From combinatorics to ergodic theory and back again

Bryna Kra

• Multiple ergodic averages, such as the average of expressions like f1(T nx) f2(T 2nx) . . . fk(T knx), 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.