Normal sequences of symbols produced by singular probability distribution functions with independent $Q$-symbols (qualifying seminar)

Abstract: This is a qualifying seminar organized by the Department of Dynamical Systems and Fractal Analysis for the public presentation and discussion of the Ph.D. student's research results obtained for his Ph.D. degree dissertation in specialty 111, Mathematics. The department must provide a detailed report on the scientific novelty and the theoretical and practical value of the dissertation results.

This dissertation is devoted to the generalization of the classical theory of Borel normal numbers and normal sequences of symbols to the case of numeral systems with finite alphabets, such as the $Q_s$-representation of numbers in the interval $[0, 1]$, as well as to problems of uniformly distributed sequences generated by these systems.

In this field, the results of É. Borel, H. Lebesgue, W. Sierpiński, D. Champernowne, S. Pillai, I. Piatetski-Shapiro, P. Erdős, and others are classical. Scientific interest in this topic remains high due to its deep connections with the theory of dynamical systems, fractal analysis, and fractal geometry.

The main results of this work are solutions to metric problems that are analogs of É. Borel, D. Wall, and I. Piatetski-Shapiro's results and methods (algorithms) for constructing normal (quasi-normal) symbols corresponding to the $Q_s$-representation of numbers.

In the talk, the following scientific results will be presented:

1. the necessary and sufficient conditions of the uniform and quasi-uniform distribution for sequences defined in terms of iterations of the left-shift operator for symbols of the $Q_s$-representation of numbers;
2. an analog of the Piatetski-Shapiro-type criterion for sequences of symbols generated by the left-shift operator for the $Q_s$-representation;
3. a series of properties of iterations of the left-shift operator whose indices increase nonlinearly and a full description of the structure of $Q_s$-representations corresponding to mutually inversive $Q_s$-normal sequences of symbols;
4. constructive methods for obtaining recursively computable normal (quasi-normal) sequences of symbols (normal numbers) corresponding to the $Q_s$-representation of numbers;
5. a structure of transformations that preserve a uniform distribution of sequences.

Ukrainiandynamical systemsnumber 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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