On fractal faithfulness and fine fractal properties of random variables with independent <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="italic">Q</mi></mrow><mrow><mo>∗</mo></mrow></msup></math>-digits

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.

Keywords

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