Joint ergodicity along generalized linear functions

A criterion of joint ergodicity of several sequences of transformations of a probability measure space X of the form T φi(n) i is given for the case where Ti are commuting measure preserving transformations of X and φi are integer valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending of a continuous parameter, as well as a condition of joint ergodicity of sequences T φi(n) i along primes. 0. Introduction Let (X,B, μ) be a probability measure space. A measure preserving transformation T :X −→ X is said to be weakly mixing if the transformation T × T , acting on the Cartesian square X × X, is ergodic. The notion of weak mixing was introduced in [vNK] (for measure preserving flows) and has numerous equivalent forms (see, for example, [BeR] and [BeG].) The following result involving weak mixing plays a critical role in Furstenberg’s proof ([Fu]) of ergodic Szemerédi theorem and forms a natural starting point for numerous further developments (see [Be], [BeL1], [BeMc], [BeH]): Theorem 0.1. If T is an invertible weakly mixing measure preserving transformation of X, then for any k ∈ N and any A0, A1, . . . , Ak ∈ B one has