Numeral systems with natural base and redundant alphabet and their applications in number theory, probability theory, and geometry of numerical series

Abstract: This talk is devoted to the function defined by equality \[ f(x = \Delta^{r+1}_{\alpha_1\ldots\alpha_n\ldots}) = \Delta^{r_s}_{\alpha_1\ldots\alpha_n\ldots}, \] where $s$ and $r$ are fixed natural numbers such that $2 \leq s \leq r$, \[ \Delta^{r+1}_{\alpha_1\ldots\alpha_n\ldots} = \sum_{n=1}^\infty \alpha_n (r+1)^{-n}, \quad \Delta^{r_s}_{\alpha_1\ldots\alpha_n\ldots} = \sum_{n=1}^\infty \alpha_n s^{-n}, \quad \alpha_n \in A = \{ 0, 1, \ldots, r \}. \]

We study structural, variational, topological, metric, and fractal properties of the function, in particular its level sets.

Since the values of the function are given in the system of encoding of numbers with base $s$ and redundant alphabet $A$, in the talk, we discuss such systems and their applications in the theory of singular probability distributions in detail.

We also clarify a connection between numeral systems with redundant alphabet and the geometry of numerical series (i.e., topological and metric analysis of their subsums).

Ukrainianclassical analysis and ODEsfunctional analysisnumber 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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