Singular functions and $A_2$-continued fraction representations of numbers

Abstract: In this talk, we analyze the literature on the study of singular functions and on the methods used to prove their singularity, as well as on singular functions defined in terms of the $A_2$-continued fraction representation of numbers. One example of a function with complicated local behavior, which is related to the classical binary representation and $A_2$-continued fraction representation of numbers, is introduced. We prove that this function is singular.

Ukrainianclassical analysis and ODEsnumber 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