具有独立$\boldsymbol{Q^*}$-数字的随机变量的分形忠实性和精细分形性质
On fractal faithfulness and fine fractal properties of random variables with independent $\boldsymbol{Q^*}$-digits
摘要：
本文提出了一种新的方法来证明Hausdorff-Besicovitch维数计算的可靠性，该方法计算了由实数的$Q^*$-展开生成的圆柱族$\varPhi(Q^*)$的维数.所有已知的充分条件的家庭$\varPhi(Q^*)$是忠实的Hausdorff-Besicovitch维数计算使用不同的限制项目$q_{0k}$和$q_{(s-1)k}$。我们表明，这些限制是纯粹的技术性质，可以删除。在此基础上，我们研究了具有独立$Q^*$位数的随机变量的精细分形性质。
英文摘要：
We develop a new technique to prove the faithfulness of the Hausdorff--Besicovitch dimension calculation of the family $\varPhi(Q^*)$ of cylinders generated by $Q^*$-expansion of real numbers. All known sufficient conditions for the family $\varPhi(Q^*)$ 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 $Q^*$-digits.
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