## Abstract

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand’s projection theorem for planar self-similar sets.

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

Pages (from-to) | 5125-5151 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2016 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2015 American Mathematical Society.

## Keywords

- Absolute continuity
- Convolutions
- Hausdorff dimension
- Self-similar measures