Cantorvals: emergence, structure, open problems (part 2)

Abstract: This talk is dedicated to the study of Cantorvals—perfect sets on the real line with nonempty interior and fractal boundaries. Such sets emerge naturally in various areas of mathematics, including the study of sets of subsums of numerical series, arithmetic sums of Cantor-like sets, and attractors of iterated function systems. Despite their frequent appearance, Cantorvals remain poorly understood due to their intricate internal structure. In this talk, we establish conditions under which the set of incomplete sums of a generalized multigeometric series forms a Cantorval. We also examine the structure of such Cantorvals within a certain one-parameter family. Finally, several open problems related to the topic will be discussed.

This is joint work with Professors Mykola Pratsiovytyi, Aniceto Murillo, and Antonio Viruel.

Ukrainianclassical analysis and ODEsdynamical systemsnumber theory

Audience: researchers in the discipline

( slides )

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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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