Singular distributions of random variables with independent digits of representation in numeral system with natural base and redundant alphabet

  • M. V. Pratsiovytyi Institute of Mathematics of NASU Dragomanov Ukrainian State University Kyiv, Ukraine
  • S. P. Ratushniak Institute of Mathematics of NASU Dragomanov Ukrainian State University, Kyiv, Ukraine
Keywords: singular distribution, absolutely continuous distribution, infinite Bernoulli convolution governed by a series, numeral system with base 3 and a redundant digit, set of incomplete sums of series, self-similar set, fractal dimension

Abstract

Given natural parameters s and r, where 2sr, we consider the distribution of a~ran\-dom variable ξ=k=1skξkΔrsξ1ξ2...ξk...,
where (ξk) is a sequence of independent random variables taking values in {0,1,...,r} with probabilities p0, p1, , pr, respectively, and all pi<1.

In the case s=3=r, necessary and sufficient conditions for the singularity and absolute continuity of the distribution of ξ are established. It is shown that the distribution of ξ is absolutely continuous if and only if p1=13=p2. In all other cases, the distribution is singular (i.e., supported on a set of zero Lebesgue measure).
For p0p1p2p3=0, the fractal Hausdorff–Besicovitch dimension of the spectrum (i.e., the minimal closed support) of the distribution of ξ and of the essential support of its density is explicitly determined under the condition pipi+1pi+20 for i=0,1.

The work also discusses the connection between the distribution of ξ and infinite Bernoulli convolutions governed by the corresponding series as well as representations of numbers in the base-3 numeral system with one redundant digit. Several open problems are formulated.

For the numeral system with the base 3 and the alphabet A={0,1,2,3}, the problem of determining the number of representations of a number is completely solved. It is proven that almost all numbers (with respect to the Lebesgue measure) in the interval [0;32] have a~continuum of distinct representations, while those with a unique representation form a fractal set of Hausdorff–Besicovitch dimension log32.

Author Biographies

M. V. Pratsiovytyi, Institute of Mathematics of NASU Dragomanov Ukrainian State University Kyiv, Ukraine

Institute of Mathematics of NASU Dragomanov Ukrainian State University Kyiv, Ukraine

S. P. Ratushniak, Institute of Mathematics of NASU Dragomanov Ukrainian State University, Kyiv, Ukraine

Institute of Mathematics of NASU
Dragomanov Ukrainian State University, Kyiv, Ukraine

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Published
2025-06-24
How to Cite
Pratsiovytyi, M. V., & Ratushniak, S. P. (2025). Singular distributions of random variables with independent digits of representation in numeral system with natural base and redundant alphabet. Matematychni Studii, 63(2), 199-209. https://doi.org/10.30970/ms.63.2.199-209
Section
Articles