## Abstract

We study the exact rate of convergence of frequencies of digits of "normal" points of a self-similar set. Our results have applications to metric number theory. One particular application gives the following surprising result: there is an uncountable family of pairwise disjoint and exceptionally big subsets of R-d that do not obey the law of the iterated logarithm. More precisely, we prove that there is an uncountable family of pairwise disjoint and exceptionally big sets of points x in R-d-namely, sets with full Hausdorff dimension-for which the rate of convergence of frequencies of digits in the N-adic expansion of x is either significantly faster or significantly slower than the typical rate of convergence predicted by the law of the iterated logarithm.

Original language | English |
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Pages (from-to) | 85-102 |

Number of pages | 18 |

Journal | Indagationes Mathematicae |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - 27 Mar 2006 |

## Keywords

- rate of convergence
- frequencies of digits
- Hausdorff dimension
- normal numbers
- MULTIFRACTAL DIVERGENCE POINTS
- N-ADIC EXPANSION
- HAUSDORFF DIMENSION