On spectra of probability measures generated by GLS-expansions

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 $Q_{\mathrm{\infty }}$- and $G_{\mathrm{\infty }}^2$-expansions). To this end, we explore fractal properties of the family of Cantor-like sets $C[GLS,V]$ 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 $C[GLS,V]$. 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.