On the ratio ergodic theorem for group actions

We show that the ratio ergodic theorem of Hopf fails in general for measure-preserving actions of countable amenable groups; in fact, it already fails for the infinite-rank abelian group ⊕^∞ _n=1 Z and many groups of polynomial growth, for instance, the discrete Heisenberg group. More generally, under a technical condition, we show that if the ratio ergodic theorem holds for averages along a sequence of sets {F_n} in a group, then there is a finite set E such that {EF _n} satisfies the Besicovitch covering property. On the other hand, we prove that in groups with polynomial growth (for which the ratio ergodic theorem sometimes fails) there always exists a sequence of balls along which the ratio ergodic theorem holds if convergence is understood as almost every convergence in density (that is, omitting a sequence of density zero).