RUN-LENGTH FUNCTION FOR REAL NUMBERS IN β-EXPANSIONS

Lixuan Zheng () and Min Wu
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Lixuan Zheng: Department of Statistics and Mathematics, Guangdong University of Finance and Economics, 510320 Guangzhou, P. R. China
Min Wu: Department of Mathematics, South China University of Technology, 510640 Guangzhou, P. R. China

FRACTALS (fractals), 2022, vol. 30, issue 03, 1-12

Abstract: Let β > 1 and x ∈ (0, 1] be two real numbers. For all x ∈ (0, 1], the run-length function with respect to x, denoted by rx(y,n), is defined as the maximal length of the prefix of the β-expansion of x amongst the first n digits of the β-expansion of y. The level set Ea,b = y ∈ (0, 1] :lim infn→∞rx(y,n) logβn = a,lim supn→∞rx(y,n) logβn = b (0 ≤ a ≤ b ≤ +∞) is investigated in our paper. We obtain the Hausdorff dimension of Ea,b which extends many known results on run-length function in β-expansions.

Keywords: Beta-Expansion; Run-Length Function; Hausdorff Dimension (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22500335

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