M. V. Pratsiovytyi

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Articles: 2

Level sets of asymptotic mean of digits function for $4$-adic representation of real number

M. V. Pratsiovytyi, S. O. Klymchuk, O. P. Makarchuk

↓ Abstract   |   Article (.pdf)

MFAT 22 (2016), no. 2, 184-196


We study topological, metric and fractal properties of the level sets $$S_{\theta}=\{x:r(x)=\theta\}$$ of the function $r$ of asymptotic mean of digits of a number $x\in[0;1]$ in its $4$-adic representation, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}\alpha_i(x)$$ if the asymptotic frequency $\nu_j(x)$ of at least one digit does not exist, were $$ \nu_j(x)=\lim_{n\to\infty}n^{-1}\#\{k: \alpha_k(x)=j, k\leqslant n\}, \:\: j=0,1,2,3. $$

On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols

S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, G. Torbin

↓ Abstract   |   Article (.pdf)

MFAT 17 (2011), no. 2, 97-111


We introduce a new fine classification of singularly continuous probability measures on $R^1$ on the basis of spectral properties of such measures (topological and metric properties of the spectrum of the measure as well as local behavior of the measure on subsets of the spectrum). The theorem on the structural representation of any one-dimensional singularly continuous probability measure in the form of a convex combination of three singularly continuous probability measures of pure spectral type is proved.

We introduce into consideration and study a $\widetilde{Q}$-representation of real numbers and a family of probability measures with independent $\widetilde{Q}$-symbols. Topological, metric and fractal properties of the above mentioned probability distributions are studied in details. We also show how the methods of $\widetilde{P}-\widetilde{Q}$-measures can be effectively applied to study properties of generalized infinite Bernoulli convolutions.

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