Hausdorff dimension of frequency sets in beta-expansions

DOI10.1007/S00209-022-03113-5arXiv1905.01481WikidataQ114231053 ScholiaQ114231053MaRDI QIDQ6318202

Y.-Q. Li

Publication date: 4 May 2019

Abstract: By applying a 2014 result on the distribution of full cylinders, we give a proof of the useful folklore: for any $$, the Hausdorff dimension of an arbitrary set in the shift space $$ is equal to the Hausdorff dimension of its natural projection in $[0,1]$. It has been used in some former papers without proof. Then we clarify that for calculating the Hausdorff dimension of frequency sets using variational formulae, one only needs to focus on the Markov measures of explicit order when the $$-expansion of $1$ is finite. Concretely, it suffices to optimize a function with finitely many variables under some restrictions. Finally, as an application, we obtain an exact formula for the Hausdorff dimension of frequency sets for an important class of $$'s, which are called pseudo-golden ratios (also called multinacci numbers).

Mathematics Subject Classification ID

Metric theory of other algorithms and expansions; measure and Hausdorff dimension (11K55) Fractals (28A80)

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