Return times theorem for amenable groups (Q480793)

Bourgain's return times theorem says the following: Let \((X,T)\) be a measure-preserving system and \(f\) an element of \(L^\infty (X)\). Then, for a.e. \(x\) in \(X\), the sequence \(c_n=f(T^nx)\) is a \textit{good sequence of weights} for the pointwise ergodic theorem. This means that for every measure-preserving system \((Y,S)\) and \(g\) in \(L^\infty (Y)\), for a.e. \(y\) in \(Y\) the averages \[ \frac{1}{N}\sum_{n=1}^N f(T^nx)g(S^ny) \] converge as \(N\to\infty\). In the paper under review, the author uses the Lindenstrauss random covering lemma to extend this result to not necessarily discrete locally compact second countable amenable groups. At the same time, he proves a generalized version of the Bourgain-Furstenberg-Katznelson-Ornstein orthogonality criterion. This criterion gives a sufficient condition for the values of a function along an orbit of a measure-preserving transformation to be a good sequence of weights for the convergence to zero in the pointwise ergodic theorem.