## Abstract

Given ergodic p-invariant measures {μ_{i}} on the 1-torus double-struck T sign = double-struck R sign/double-struck Z sign, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution μ_{1} * ⋯ * μ_{n} converges to log p. We also prove a variant of this result for joinings of full entropy on double-struck T sign^{double-struck N sign}. In conjunction with a method of Host, this yields the following. Denote σ_{q}(cursive Greek chi) = qcursive Greek chi (mod 1). Then for every p-invariant ergodic μ with positive entropy, 1/N Σ^{N-1}_{n=0} σ_{cn}μ converges weak* to Lebesgue measure as N → ∞, under a certain mild combinatorial condition on {c_{k}}. (For instance, the condition is satisfied if p = 10 and c_{k} = 2^{k} + 6^{k} or c_{k} = 2^{2k}.) This extends a result of Johnson and Rudolph, who considered the sequence c_{k} = q^{k} when p and q are multiplicatively independent. We also obtain the following corollary concerning Hausdorff dimension of sum sets: For any sequence {S_{i}} of p-invariant closed subsets of double-struck T sign, if Σ dim_{H}(S_{i})/| log dim_{H} (S_{i})| = ∞, then dim_{H}(S_{1} + ⋯ + S_{n}) → 1.

Original language | American English |
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Pages (from-to) | 871-904 |

Number of pages | 34 |

Journal | Annals of Mathematics |

Volume | 149 |

Issue number | 3 |

DOIs | |

State | Published - May 1999 |

## Keywords

- Convolution
- Entropy
- Hausdorff dimension
- Uniform distribution