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We study properties of distributions of random variables with independentidentically 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 fractalproperties of the family of Cantor-like sets $C[\mathit{GLS},V]$ consisting ofreal numbers whose GLS-expansions contain only symbols from some countable set$V\subset N\cup\{0\}$, and derive exact formulae for the determination of theHausdorff--Besicovitch dimension of $C[\mathit{GLS},V]$. Based on theseresults, we get general formulae for the Hausdorff--Besicovitch dimension ofthe spectra of random variables with independent identically distributedGLS-symbols for the case where all but countably many points from the unitinterval belong to the basis cylinders of GLS-expansions.

On spectra of probability measures generated by GLS-expansions