Про множину нормальних чисел, побудовану в термінах $Q_s^*$-представлення дійсних чисел / On the set of normal numbers constructed in terms of $Q_s^*$-representation of real numbers

Abstract: У доповіді вводиться поняття $Q_s^*$-нормального числа, що є аналогом відповідного поняття для класичного $s$-кового представлення, за умови, що послідовність стохастичних векторів $(q_{0n}; q_{1n}; \ldots; q_{(s-1)n})$, які відповідають $Q_s^*$-представленню, збігається до стохастичного вектора $(q_0; q_1; \ldots; q_{s-1})$ зі строго додатними координатами. Показано, що за умови \[ \sum_{n=1}^\infty \left( (q_{0n} - q_0)^2 + (q_{1n} - q_1)^2 + \ldots + (q_{(s-1)n} - q_{s-1})^2 \right) < +\infty \] майже всі числа є $Q_s^*$-нормальними.

In the talk, we introduce the notion of a $Q_s^*$-normal number, which is an analogue of the corresponding notion for classic $s$-adic representation, with condition that the sequence of stochastic vectors $(q_{0n}, q_{1n}, \ldots, q_{(s-1)n})$ corresponding to $Q_s^*$-representation tends to the stochastic vector $(q_0, q_1, \ldots, q_{s-1})$ with strictly positive coordinates. We show that almost all numbers are $Q_s^*$-normal if \[ \sum_{n=1}^\infty \left( (q_{0n} - q_0)^2 + (q_{1n} - q_1)^2 + \ldots + (q_{(s-1)n} - q_{s-1})^2 \right) < +\infty. \]

Ukrainiannumber theory

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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