Hausdorff dimension of digit frequency sets for beta-expansions and a generalization of the Hata-Yamaguchi formula

Abstract: We consider the digit frequency set of the digit 1 for beta-expansions in the case of $1<\beta\leq2$ and give an exact formula for its Hausdorff dimension via transfer operator method. As a related topic, we introduce a generalization of the Lebesgue singular function and show that it satisfies a version of the Hata-Yamaguchi formula, which yields a Takagi-like function for beta-expansions with the base $1<\beta<2$.

dynamical systemsnumber theory

Audience: researchers in the topic

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