Infinite Bernoulli convolutions governed by series with Cantorval sets of subsums and their fractal properties

Abstract: In this talk, we discuss infinite Bernoulli convolutions governed by positive multigeometric series as well as distributions of random variables such that digits of their representation in the system with an even natural base $s$ and two redundant digits are independent identically distributed random variables.

The main objects of study are the random variables \[ \xi = \sum_{n=1}^\infty \frac{\xi_n}{s^n}, \] where $(\xi_n)$ is a sequence of independent random variables taking the values $0$, $1$, …, $s$, $s+1$ with probabilities $p_0$, $p_1$, …, $p_s$, $p_{s+1}$, respectively ($3 < s \in \mathbb{N}$) and \[ \eta = \sum_{n=1}^\infty \left[ \frac{3\eta_{(n-1)(m+1)+1}}{s^n} + \sum_{j=1}^m \frac{2\eta_{(n-1)(m+1)+1+j}}{s^n} \right], \] where $(\eta_n)$ is a sequence of independent identically distributed random variables taking the values $0$ and $1$ with probabilities $q_0 > 0$ and $q_1 = 1 - q_0 > 0$, respectively. We study the conditions of absolute continuity and singularity of the distributions of these random variables and topological, metric, and fractal properties of their essential supports. The cases when the spectrum of distribution is a Cantorval are considered in detail.

When $s = 4$, we find the necessary and sufficient conditions for the distributions of the random variables $\xi$ and $\eta$ to be singular and absolutely continuous, including the case when they are supported on the Guthrie–Nymann Cantorval. For an arbitrary even number $s > 4$, we find the necessary and sufficient conditions for the random variable $\xi$ to be decomposed into a sum of two independent random variables such that one of them has a uniform distribution on the unit interval, which is equivalent to its absolute continuity. Using the method of characteristic functions, we obtain sufficient conditions of singularity and absolute continuity. For Cantorvals that are the spectra of distributions, their structure and fractal properties of their boundary are studied.

Ukrainianclassical analysis and ODEsfunctional analysisnumber theoryprobability

Audience: researchers in the discipline

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Семінар з фрактального аналізу / Fractal analysis seminar

Series comments: Weekly research seminar on fractal analysis (online)

Topics:

• theory of fractals (fractal geometry and fractal analysis)
• Hausdorff–Besicovitch dimension, techniques and methods for its calculation and estimation
• functions and transformations preserving fractal (Hausdorff–Besicovitch, entropic, box-counting, packing, etc.) dimension
• sets of metric spaces that are essential for functions, sets, and dynamical systems
• self-similar, self-affine properties of mathematical objects
• systems of encoding for real numbers (numeral systems) and their applications
• metric number theory and metric theory of representations of numbers
• probabilistic number theory and probabilistic theory of representations of numbers
• measures supported on fractals, particularly singular measures and probability distributions
• nowhere monotonic and nowhere differentiable functions, functions with fractal properties
• theory of groups determined by invariants of transformations preserving tails of representation of numbers, digit frequencies, etc.

The talks are mostly in Ukrainian but English is also acceptable

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