Quantitative multiple recurrence for two and three transformations

Abstract

We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that:There exists an ergodic system (X,X, mu,T (1), T (2)) with two commuting transformations such that, for every 0 < a"" < 4, there exists A a X such that There exists an ergodic system (X,X, mu,T (2), T (3)) with three commuting transformations such that, for every a"" > 0, there exists A a X such that There exists an ergodic system (X,X, mu,T (1), T (2)) with two transformations generating a 2-step nilpotent group such that, for every a"" > 0, there exists A a X such that for every > 0, there exists A. X such that mu( A n T - n 1 A n T - n 2 A) < mu( A) for every n not equal 0.