The metric, topological, and fractal properties of the sets of quasi-normal, partially anormal, and essentially anormal numbers for the factorial expansion

Abstract: This talk is devoted to the study of the fractal and topological properties of the sets of quasi-normal, partially, and essentially anormal numbers for the factorial numeral system. In particular, we prove that the set of essentially anormal numbers generated by the factorial expansion is residual. The superfractality of the set of partially anormal and essentially anormal numbers for the factorial expansion is also proved. On the other hand, in the talk, we present new properties about the calculation of the Hausdorff–Besicovitch dimension for subsets of the set of quasi-normal numbers for the factorial expansion. Possible generalizations of the obtained results and their transfer to other classes of the Cantor expansions are also discussed.

Ukrainianclassical analysis and ODEsnumber theory

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