Abstract
We prove sufficient conditions for the faithfulness of the family of cylinders generated by $\widetilde{Q}$ -expansions for the Hausdorff-Besicovitch dimension calculation. We also represent the conjecture about necessary and sufficient conditions for such a family of cylinders to be faithful.
Based on these new results we study fine fractal properties of random variables with independent $\widetilde{Q}$-digits and prove exact formulae for the calculation of the Hausdorff dimension of the corresponding probability measure.
Key words: Hausdorff-Besicovitch dimension, fractals, Hausdorff dimension of probability measures, faithful Vitaly coverings, $Q^*$-expansion, $\widetilde{Q}$-expansion, singularly continuous probability measures.
Full Text
Article Information
| Title | On fractal faithfulness and fine fractal properties
of random variables with independent $\widetilde{Q}$-symbols |
| Source | Methods Funct. Anal. Topology, Vol. 32 (2026), no. 2, 181-192 |
| DOI | 10.31392/MFAT-npu26_2.2026.08 |
| Milestones | Received 21/03/2026; Revised 25/06/2026 |
| Copyright | The Author(s) 2026 (CC BY-SA) |
Authors Information
Grygoriy Torbin
Dragomanov Ukrainian State University, Pyrogova str. 9, 01601 Kyiv (Ukraine); Institute for Mathematics of NASU, Tereshchenkivs’ka str. 3, 01601 Kyiv (Ukraine)
Vladyslav Vasylenko
Dragomanov Ukrainian State University, Pyrogova str. 9, 01601 Kyiv (Ukraine)
Citation Example
Grygoriy Torbin and Vladyslav Vasylenko, On fractal faithfulness and fine fractal properties
of random variables with independent $\widetilde{Q}$-symbols, Methods Funct. Anal. Topology 32
(2026), no. 2, 181-192.
BibTex
@article {MFAT2283,
AUTHOR = {Grygoriy Torbin and Vladyslav Vasylenko},
TITLE = {On fractal faithfulness and fine fractal properties
of random variables with independent $\widetilde{Q}$-symbols},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {32},
YEAR = {2026},
NUMBER = {2},
PAGES = {181-192},
ISSN = {1029-3531},
DOI = {10.31392/MFAT-npu26_2.2026.08},
URL = {https://mfat.imath.kiev.ua/article/?id=2283},
}
