## Abstract

We prove a ratio ergodic theorem for non-singular free ℤ^{d} and ℝ^{d} actions, along balls in an arbitrary norm. Using a Chacon-Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in ℝ^{d}. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group actions, the Besicovitch covering property not only implies the maximal inequality, but is equivalent to it, implying that further generalization may require new methods.

Original language | American English |
---|---|

Pages (from-to) | 365-383 |

Number of pages | 19 |

Journal | Journal of the European Mathematical Society |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - 2010 |

Externally published | Yes |

## Keywords

- Commuting transformations
- Ergodic theorem
- Group actions
- Maximal inequality
- Measure preserving transformations
- Non-singular actions