Abstract
By applying a recent result on the distributions of full and non-full words, we give a proof of the useful folklore result: 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 in order to calculate the Hausdorff dimension of frequency sets using variational formulae, one only needs to focus on the Markov measures of explicit step when the $$\beta $$ -expansion of 1 is finite. Finally, as an application, we obtain an exact formula for the Hausdorff dimension of frequency sets for an important class of $$\beta $$ ’s, which are called pseudo-golden ratios (also called multinacci numbers).
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