Ergodic theorems for the shift action and pointwise versions of the Abert-Weiss theorem

Let Gamma be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action Gamma & x21b7; (X, mu) and a map f is an element of L-1 (X, mu), and to compare the global average integral f d mu of f to the pointwise averages divide D divide (-1) n-ary sumation (delta is an element of D)f(delta center dot x), where x is an element of X and D is a nonempty finite subset of Gamma. The basic hope is that, when D runs over a suitably chosen infinite sequence, these pointwise averages should converge to the global value for mu-almost all x. In this paper we prove several results that refine the above basic paradigm by uniformly controlling the averages over specific sets D rather than considering their limit as divide D divide -> infinity. Our results include ergodic theorems for the Bernoulli shift action Gamma & x21b7; ([0; 1](Gamma), lambda(Gamma)) and strengthenings of the theorem of Abert and Weiss that the shift is weakly contained in every free p.m.p. action of Gamma. In particular, we establish a purely Borel version of the Abert-Weiss theorem for finitely generated groups of subexponential growth. The central role in our arguments is played by the recently introduced measurable versions of the Lovasz Local Lemma, due to the current author and to Csoka, Grabowski, Mathe, Pikhurko, and Tyros.