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On the Faithfulness of the Vitali Coverings for the Hausdorff Dimension Calculation and Its Applications
(National Pedagogical Dragomanov University)
One approach to the simplification of the calculation of the Hausdorff dimension
consists in some restrictions of admissible coverings. In this talk, we discuss new
results related to the faithfulness/non-faithfulness of the Vitali coverings generated
by different expansions for real numbers as well as some open questions in this
direction. Another important topic to be treated is related to transformations
preserving the Hausdorff dimension. We consider the interplay of this topic with fine
fractal analysis of singularly continuous probability measures and faithfulness of the
Vitali coverings. Finally, we discuss applications of the above results in dimensional
number theory (fractal properties of subsets of non-normal numbers).