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On fractal faithfulness and fine fractal properties of random variables with independent Q∗-digits
Volume 3, Issue 2 (2016), pp. 119–131
Muslem Ibragim   Grygoriy Torbin  

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https://doi.org/10.15559/16-VMSTA55
Pub. online: 9 June 2016      Type: Research Article      Open accessOpen Access

Received
20 May 2016
Accepted
3 June 2016
Published
9 June 2016

Abstract

We develop a new technique to prove the faithfulness of the Hausdorff–Besicovitch dimension calculation of the family $\varPhi ({Q}^{\ast })$ of cylinders generated by ${Q}^{\ast }$-expansion of real numbers. All known sufficient conditions for the family $\varPhi ({Q}^{\ast })$ to be faithful for the Hausdorff–Besicovitch dimension calculation use different restrictions on entries $q_{0k}$ and $q_{(s-1)k}$. We show that these restrictions are of purely technical nature and can be removed. Based on these new results, we study fine fractal properties of random variables with independent ${Q}^{\ast }$-digits.

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Keywords
Hausdorff–Besicovitch dimension fractals faithful Vitali coverings Q*-expansion singularly continuous probability measures

MSC2010
11K55 26A30 28A80 60G30

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