On the exact rate of convergence of frequencies of digits in self-similar sets

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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 languageEnglish
Pages (from-to)85-102
Number of pages18
JournalIndagationes Mathematicae
Volume17
Issue number1
DOIs
Publication statusPublished - 27 Mar 2006

Keywords

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

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