On fractal faithfulness and fine fractal properties of random variables with independent -digits
Volume 3, Issue 2 (2016), pp. 119–131
Pub. online: 9 June 2016
Type: Research Article
Open Access
Open Access
Received
20 May 2016
20 May 2016
Accepted
3 June 2016
3 June 2016
Published
9 June 2016
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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