Ostrogradsky-Sierpi\'nski-Pierce expansion: dynamical systems, probability theory and fractal geometry points of view

TL;DR

This paper explores the probabilistic, dynamical, and fractal properties of the Ostrogradsky-Sierpiński-Pierce expansion, revealing new phenomena and establishing foundational theories in these areas.

Contribution

It develops metric, ergodic, and dimensional theories for the expansion and analyzes the dynamical system and distribution properties of associated random variables.

Findings

Almost all real numbers have finitely many occurrences of each digit in the expansion.

No invariant, ergodic, absolutely continuous measures exist for the shift transformation.

Random variables with i.i.d. differences cannot be absolutely continuous, only discrete or singularly continuous.

Abstract

We establish several new probabilistic, dynamical, dimensional and number theoretical phenomena connected with Ostrogradsky-Sierpi\'nski-Pierce expansion. First of all, we develop metric, ergodic and dimensional theories of the Ostrogradsky-Sierpi\'nski-Pierce expansion. In particular, it is proven 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 Ostrogradsky-Sierpi\'nski-Pierce expansion. Properties of the symbolic dynamical system generated by a shift-transformation $T$ on the difference-version of the Ostrogradsky-Sierpi\'nski-Pierce expansion are also studied in details. It is shown that there are no probability measures which are invariant and ergodic (w.r.t. $T$) and absolutely continuous (w.r.t. Lebesgue measure). Thirdly, we study properties of random variables $\eta$ with independent identically distributed differences of the Ostrogradsky-Sierpi\'nski-Pierce expansion. Necessary and sufficient conditions for $\eta$ to be discrete resp. singularly continuous are found. We prove that $\eta$ can not be absolutely continuously distributed.