On some metric and probabilistic problems given in terms of the $A_2$-continued fraction representation

Abstract: In the talk, we consider the $A_2$-continued fraction expansion with alphabet $\{\alpha_1, \alpha_2\}$, where $\alpha_2 > \alpha_1 > 0$, $\alpha_2 \alpha_1 = 0.5$. Let $\nu(\alpha_2, n, x)$ be a relative frequency of digit $\alpha_2$ among the first $n$ digits of the $A_2$-continued fraction representation of a number $x = [a_1, a_2, \ldots, a_n, \ldots]$. We show that, for $\alpha_2 > 8 - 4 \sqrt{2}$ and for some $C \in (0, 1)$, condition $\limsup_{n \to +\infty} \nu(\alpha_2, n, x) \le C$ holds for almost all (with respect to Lebesgue measure) numbers $x \in [\alpha_1, \alpha_2]$.

Let $(\Omega, S, P(\cdot))$ be a probability space and let $(\xi_n)$ be a given sequence of independent discrete random variables taking values $\alpha_1$ and $\alpha_2$ with probabilities $0.5$, respectively. For random variable $\xi = [\xi_1, \xi_2, \ldots, \xi_n, \ldots]$, we consider conditions for denominators $q_n(\xi(\omega))$ of convergents of $\xi$ that hold for almost all $\omega \in \Omega$.

Ukrainiannumber 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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