On faithfulness and DP-transformations generated by arithmetic Cantor series expansions
Grygoriy Torbin, Yuliia Voloshyn
TL;DR
The paper addresses when Cantor-series cylinder coverings faithfully compute the Hausdorff-Besicovitch dimension and when distribution functions of random Cantor-series variables belong to the DP-class. It develops a subgeometric versus bounded-growth analysis, establishing a sufficient faithfulness criterion for Cantor-series cylinders under mixed arithmetic–geometric bounds on the basic sequence and highlighting that subgeometric faithfulness fails in general. It then surveys DP-transformations for Cantor-series expansions, proving a necessary-and-sufficient condition for the DP-class in the bounded sequence case and presenting a counterexample showing that standard DP-conditions can fail for arithmetic Cantor expansions. The results delineate the limits of faithfulness and DP-transformations in Cantor-series settings, with implications for fractal dimension calculations and the behavior of random Cantor-type measures.
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.