Representation of real numbers by alternating Perron series and its metric equivalence to representation of numbers by positive Perron series

Abstract: In this talk, we consider the representation of real numbers by alternating Perron series (an analog of positive Perron series), which generalizes expansions of numbers in alternating Lüroth series, Ostrogradsky–Sierpiński–Pierce series, second Ostrogradsky series, etc. The main result is a metric equivalence of representations of numbers by positive and alternating Perron series. This equivalence explains analogies in the individual theories of representations of numbers by various positive and alternating series on a basic level. The series are the following: positive and alternating Lüroth series, modified Engel series and Ostrogradsky–Sierpiński–Pierce series, Sylvester series and second Ostrogradsky series. We also analyze the known “identical” results, which are explained by the discovered metric equivalence.

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