  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.
