On spectra of probability measures generated by GLS-expansions
Volume 3, Issue 3 (2016), pp. 213–221
Pub. online: 26 October 2016
Type: Research Article
Open Access
Open Access
Received
5 August 2016
5 August 2016
Revised
27 September 2016
27 September 2016
Accepted
27 September 2016
27 September 2016
Published
26 October 2016
26 October 2016
Abstract
We study properties of distributions of random variables with independent identically distributed symbols of generalized Lüroth series (GLS) expansions (the family of GLS-expansions contains Lüroth 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.
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