## Abstract

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).

Original language | English |
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Pages (from-to) | 465-482 |

Number of pages | 18 |

Journal | Journal of the London Mathematical Society |

Volume | 88 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2013 |

### Bibliographical note

Funding Information:Received 19 September 2012; revised 19 December 2012; published online 30 July 2013. 2010 Mathematics Subject Classification 28D15, 37A30, 37A40 (primary), 47A35 (secondary). This work was supported by ISF grant 1409/11.