On spectra of probability measures generated by GLS-expansions

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.