On spectra of probability measures generated by GLS-expansions

TL;DR

This paper investigates the fractal and spectral properties of distributions generated by GLS-expansions, deriving formulas for the Hausdorff--Besicovitch dimension of related Cantor-like sets and spectra of certain random variables.

Contribution

It provides new formulas for the Hausdorff--Besicovitch dimension of GLS-expansion Cantor sets and spectra of i.i.d. GLS-symbol random variables, extending understanding of their fractal structure.

Findings

Derived exact Hausdorff--Besicovitch dimension formulas for GLS-Cantor sets.

Established general formulas for spectra of i.i.d. GLS-symbol random variables.

Analyzed fractal properties of GLS-expansion related sets.

Abstract

We study properties of distributions of random variables with independent identically distributed symbols of generalized L\"{u}roth series (GLS) expansions (the family of GLS-expansions contains L\"{u}roth expansion and $Q_{\infty}$- and $G_{\infty}^2$-expansions). To this end, we explore fractal properties of the family of Cantor-like sets $C[\mathit{GLS},V]$ consisting of real numbers whose GLS-expansions contain only symbols from some countable set $V\subset N\cup\{0\}$, and derive exact formulae for the determination of the Hausdorff--Besicovitch dimension of $C[\mathit{GLS},V]$. Based on these results, we get general formulae for the Hausdorff--Besicovitch dimension of the spectra of random variables with independent identically distributed GLS-symbols for the case where all but countably many points from the unit interval belong to the basis cylinders of GLS-expansions.