On some metric and probabilistic results for the $A_2$-continued fraction representation of real numbers

Abstract: In this talk, we consider the $A_2$-continued fraction representation with alphabet $\{\alpha_1, \alpha_2\}$ ($0 < \alpha_1 < \alpha_2$, $\alpha_1 \alpha_2 = \frac{1}{2}$) for numbers of interval $[\alpha_1, \alpha_2]$. We obtain metric results for this representation that are analogous to the results of A. Khinchin and P. Lévy for the classical continued fraction representation. The Lebesgue structure of the random variable with independent digits of its $A_2$-continued fraction representation is studied.

Ukrainiannumber theoryprobability

Audience: researchers in the discipline

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