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Run-Length Function For Real Numbers In β-Expansions

Author

Listed:
  • 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)

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.

Suggested Citation

  • Lixuan Zheng & Min Wu, 2022. "Run-Length Function For Real Numbers In β-Expansions," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(03), pages 1-12, May.
    • Handle: RePEc:wsi:fracta:v:30:y:2022:i:03:n:s0218348x22500335
    DOI: 10.1142/S0218348X22500335
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