On metric, dynamical, probabilistic and fractal phenomena connected with the second Ostrogradsky expansion

TL;DR

This paper explores the second Ostrogradsky expansion from multiple mathematical perspectives, revealing unique probabilistic, dynamical, and fractal properties, including singularity and measure invariance issues, and contrasting it with other expansions.

Contribution

It establishes new phenomena related to the second Ostrogradsky expansion, including singularity of the associated measure and development of metric and dimensional theories using probabilistic methods.

Findings

Proves the singularity of the random second Ostrogradsky expansion.

Shows no invariant and ergodic probability measures are absolutely continuous w.r.t. Lebesgue measure.

Demonstrates that digits in the expansion appear finitely often for almost all reals and the set of bounded digit reals has zero Hausdorff dimension.

Abstract

We consider the second Ostrogradsky expansion from the number theory, probability theory, dynamical systems and fractal geometry points of view, and establish several new phenomena connected with this expansion. First of all we prove the singularity of the random second Ostrogradsky expansion. Secondly we study properties of the symbolic dynamical system generated by the natural one-sided shift-transformation $T$ on the second Ostrogradsky expansion. It is shown, in particular, that there are no probability measures which are simultaneously invariant and ergodic (w.r.t. $T$) and absolutely continuous (w.r.t. Lebesgue measure). So, the classical ergodic approach to the development of the metric theory is not applicable for the second Ostrogradsky expansion. We develop instead the metric and dimensional theories for this expansion using probabilistic methods. We show, in particular, that for Lebesgue almost all real numbers any digit $i$ from the alphabet $A= \mathbb{N} $ appears only finitely many times in the difference-version of the second Ostrogradsky expansion, and the set of all reals with bounded digits of this expansion is of zero Hausdorff dimension. Finally, we compare metric, probabilistic and dimensional theories for the second Ostrogradsky expansions with the corresponding theories for the Ostrogradsky-Pierce expansion and for the continued fractions expansion.