On the ratio ergodic theorem for group actions (Q2854250)

This paper has two parts.NEWLINENEWLINEFirstly, for a general countable group \(G\) and finite subsets \(F_n\), it is shown that the ratio ergodic theorem of Hopf holds for all infinite-measure preserving \(G\)-actions along the sequence of sets \(F_n\) only if there is a finite set \(E\) such that the sequence \(EF_n\) has the Besicovitch covering property. This is a rather strong condition (Definition 2.1 in the paper); in particular, it fails for any sequence in the direct sum \(\mathbb{Z}^\infty\), and for any sequence of balls in the Heisenberg group. This necessity of the Besicovitch condition is proved by showing how, if it fails, one obtains instead the ingredients for a clever cut-and-stack construction of an infinite-measure preserving counterexample \(G\)-action. This construction is interesting partly for the great generality of the groups involved (they need not even be amenable).NEWLINENEWLINEConversely, in groups of polynomial growth, it is shown that there is some sequence of balls along which a certain weakening of the ratio ergodic theorem does always holds, in which convergence is understood as convergence in density along the sequence at almost every point of the underlying state space. This part of the paper follows the classical pattern for pointwise ergodic theorems: a proof for a dense subspace of functions (mostly spanned by coboundaries), and then a Vitali-like covering argument leading to a maximal inequality and hence the deduction of the full theorem by approximation. However, the technical innovations are again rather nontrivial, especially owing to the need to argue only in density along the indexing sequence.NEWLINENEWLINEBoth parts of the paper are nicely organized, but assume some familiarity from the reader. In particular, a reader unversed in cutting and stacking would find the construction in the first part hard to follow. There are also quite a few typos in the notation, although the reader should be able to spot most of these quite quickly.