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Mykola Moroz

Publications and source records attributed to Mykola Moroz.

5 recordsLinked to original sources
arXiv2026

Faithfulness and fractal (quasi-)equivalence principles for Perron, Engel, and Pierce expansions

We establish several unifying principles that clarify the fractal properties of classical number expansions, which are generalized by the Perron expansions. In particular, we prove the fractal equivalence principle for the positive and alternating Perron expansions, the fractal quasi-equivalence principle for the classical and modified Engel expansions, and the fractal quasi-equivalence principle for the Pierce expansions in the Perron and traditional notations. These results explain several known analogies and show that the Hausdorff dimension of sets defined by one expansion often coincides with that for another. The proofs rely on faithful families of coverings. In addition to deriving a range of known theorems as direct corollaries of previous results, our approach yields new fractal properties of the Engel and Pierce expansions and provides a systematic framework for transferring Hausdorff dimension properties between different expansions.

math.NT↗
arXiv2026

Hölder exponents and fractal structure of level sets of self-affine functions associated with the $Q_s$-representation of numbers

We investigate a class of locally complicated self-affine functions defined via the $Q_s$-representation of real numbers. In particular, we compute local Hölder exponents at points with given asymptotic frequencies of digits in their $Q_s$-representation. Furthermore, we establish conditions under which these functions possess continuum level sets. Finally, for self-affine functions satisfying additional conditions, we describe the geometric structure of the set of maximum points and show that this set can be fractal.

math.CA↗
arXiv2024

A counterexample to the Karvatskyi--Pratsiovytyi conjecture concerning the achievement set of an intermediate series

We found a counterexample to the conjecture of Karvatskyi and Pratsiovytyi concerning the topological type of the achievement set of an intermediate series (Proceedings of the International Geometry Center, 2023. https://doi.org/10.15673/pigc.v16i3.2519). This conjecture is based on an analogy with the squeeze theorem from calculus. We also proposed an improved version of the conjecture, which this counterexample does not refute.

math.GM↗
arXiv2024

Representations of Real Numbers by Alternating Perron Series and Their Geometry

We consider the representation of real numbers by alternating Perron series ($P^-$-representation), which is a generalization of representations of real numbers by Ostrogradsky-Sierpiński-Pierce series (Pierce series), alternating Sylvester series (second Ostrogradsky series), alternating Lüroth series, etc. Namely, we prove the basic topological and metric properties of $P^-$-representation and find the relationship between $P$-representation and $P^-$-representation in some measure theory problems.

math.GM↗