## Abstract

Let Φ be a C^{1+γ} smooth IFS on R, where γ>0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That is, for every integer p≥2 the sequence {p^{k}x}_{k∈N} equidistributes modulo 1. We thus extend several state of the art results of Hochman and Shmerkin [29] about the prevalence of normal numbers in fractals. When Φ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless Φ has an explicit structure that is associated with some integer n≥2. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to 0 at infinity. When Φ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.

Original language | English |
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Article number | 108096 |

Journal | Advances in Mathematics |

Volume | 393 |

DOIs | |

State | Published - 24 Dec 2021 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2021 Elsevier Inc.

## Keywords

- Fourier analysis
- Local limit theorems
- Normal numbers
- Self-conformal measures

## ASJC Scopus subject areas

- Mathematics (all)