Узагальнення критерію нормальності Пятецького-Шапіро для чисел з $Q_s$-цифрами / Generalization of the Piatetski-Shapiro criterion of normality for numbers with $Q_s$-digits

Abstract: У доповіді для $Q_s$-представлення, заданого стохастичним вектором $(q_0; q_1; \ldots; q_{s-1})$, та деякої послідовності стохастичних векторів $(q_{0k}; q_{1k}; \ldots; q_{(s-1)k})$, $k \in \mathbb{N}$, вводяться числа $x = \Delta^{Q_s}_{\alpha_0 \alpha_1 \ldots}$ з властивістю \[ \lim_{n\to\infty} \frac{N_n(x; (\beta_1; \beta_2; \ldots; \beta_l))}{n} = \prod_{j=1}^\infty q_{\beta_jj} \tag{1} \] для довільного блоку цифр $(\beta_1; \beta_2; \ldots; \beta_l)$, де $N_n(x; (\beta_1; \beta_2; \ldots; \beta_l))$ — кількість блоків цифр $(\beta_1; \beta_2; \ldots; \beta_l)$ серед цифр $\alpha_0$, $\alpha_1$, $\ldots$, $\alpha_n$ числа $x$. Показано, що коли існує стала $C$ така, що для довільного блоку цифр $(\beta_1; \beta_2; \ldots; \beta_l)$ \[ \limsup_{n\to\infty} \frac{N_n(x; (\beta_1; \beta_2; \ldots; \beta_l))}{n} < C \prod_{j=1}^\infty q_{\beta_jj}, \] то $x$ задовольняє умову (1). Вказано алгоритм побудови числа, що має властивість (1).

In the talk, for $Q_s$-expansion defined by stochastic vector $(q_0, q_1, \ldots, q_{s-1})$ and for some sequence of stochastic vectors $(q_{0k}, q_{1k}, \ldots, q_{(s-1)k})$, $k \in \mathbb{N}$, we introduce numbers $x = \Delta^{Q_s}_{\alpha_0 \alpha_1 \ldots}$ with the property \[ \lim_{n\to\infty} \frac{N_n(x, (\beta_1, \beta_2, \ldots, \beta_l))}{n} = \prod_{j=1}^\infty q_{\beta_jj} \tag{1} \] for any block of digits $(\beta_1, \beta_2, \ldots, \beta_l)$, where $N_n(x, (\beta_1, \beta_2, \ldots, \beta_l))$ is a number of blocks of digits $(\beta_1, \beta_2, \ldots, \beta_l)$ among digits $\alpha_0$, $\alpha_1$, $\ldots$, $\alpha_n$ of a number $x$. We show that if there exists constant $C$ such that, for any block of digits $(\beta_1, \beta_2, \ldots, \beta_l)$, \[ \limsup_{n\to\infty} \frac{N_n(x, (\beta_1, \beta_2, \ldots, \beta_l))}{n} < C \prod_{j=1}^\infty q_{\beta_jj}, \] then $x$ satisfies condition (1). Algorithm for constructing of a number with property (1) is given.

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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