On fractal faithfulness and fine fractal properties of random variables with independent $Q^*$-digits
Copyright policy )On fractal faithfulness and fine fractal properties of random variables with independent $Q^*$-digits
The Hausdorff measure and the Hausdorff dimension of set a \(E\) is defined using arbitrary coverings of \(E\). If one restricts to a smaller family of coverings, then one obtains variants of the Hausdorff measure and dimension. If, when considering a family \(\Phi\) of coverings, every set \(E\) has the same Hausdorff dimension as in the classical case (using arbitrary coverings), then the family \(\Phi\) is called faithful. In this paper the authors, consider coverings generated by cylinders of \(Q^*\)-expansions, a concept which unfortunately is not defined in the paper. The main result of the paper is a theorem giving sufficient conditions for the family of coverings by cylinders of \(Q^*\)-expansions to be faithful. As an application, they study fractal properties of some random variables.
- National Academy of Sciences United States
- National Pedagogical Dragomanov University Ukraine
T57-57.97, Applied mathematics. Quantitative methods, Hausdorff-Besicovitch dimension, Metric theory of other algorithms and expansions; measure and Hausdorff dimension, faithful Vitali coverings, singularly continuous probability measures, Q*-expansion, Fractals, Continuity and singularity of induced measures, \(Q^*\)-expansion, fractals, QA1-939, Hausdorff–Besicovitch dimension, Singular functions, Cantor functions, functions with other special properties, Mathematics
T57-57.97, Applied mathematics. Quantitative methods, Hausdorff-Besicovitch dimension, Metric theory of other algorithms and expansions; measure and Hausdorff dimension, faithful Vitali coverings, singularly continuous probability measures, Q*-expansion, Fractals, Continuity and singularity of induced measures, \(Q^*\)-expansion, fractals, QA1-939, Hausdorff–Besicovitch dimension, Singular functions, Cantor functions, functions with other special properties, Mathematics
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