Abstract
The paper is devoted to the study of conditions for the Hausdorff-Besicovitch faithfulness of the family of cylinders generated by Cantor series expansions.
We show that there exist subgeometric Cantor series expansions for which the corresponding families of cylinders are not faithful for the Hausdorff-Besicovitch dimension on the unit interval. On the other hand we found a rather
wide subfamily of subgeometric Cantor series expansions generating faithful families of cylinders.
We also study conditions for the Hausdorff-Besicovitch dimension preservation on [0;1] by probability distribution functions of random variables with independent symbols of arithmetic Cantor series expansions.
Key words: Fractals, Hausdorff-Besicovitch dimension, DP-transformation, locally fine covering families, faithful covering families, Cantor series expansion, random variable with independent symbols of Cantor series expansion, singular probability measures.
Full Text
Article Information
| Title | On faithfulness, DP-transformations and Cantor series expansions |
| Source | Methods Funct. Anal. Topology, Vol. 31 (2025), no. 4, 360-370 |
| DOI | 10.31392/MFAT-npu26_4.2025.06 |
| Milestones | Received 03/12/2025; Revised 15/12/2025 |
| Copyright | The Author(s) 2025 (CC BY-SA) |
Authors Information
Grygoriy Torbin
Dragomanov Ukrainian State University, Pyrogova str. 9, 01601 Kyiv
Yuliia Voloshyn
Dragomanov Ukrainian State University, Pyrogova str. 9, 01601 Kyiv (Ukraine); Institute for Mathematics of NASU, Tereshchenkivs’ka str. 3, 01601 Kyiv (Ukraine)
Citation Example
Grygoriy Torbin and Yuliia Voloshyn, On faithfulness, DP-transformations and Cantor series expansions, Methods Funct. Anal. Topology 31
(2025), no. 4, 360-370.
BibTex
@article {MFAT2144,
AUTHOR = {Grygoriy Torbin and Yuliia Voloshyn},
TITLE = {On faithfulness, DP-transformations and Cantor series expansions},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {31},
YEAR = {2025},
NUMBER = {4},
PAGES = {360-370},
ISSN = {1029-3531},
DOI = {10.31392/MFAT-npu26_4.2025.06},
URL = {https://mfat.imath.kiev.ua/article/?id=2144},
}
