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Exact Dimensions Of Exceptional Sets In Lãœroth Expansions

Author

Listed:
  • YAN FENG

    (School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, P. R. China)

  • BO TAN

    (School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, P. R. China)

  • QING-LONG ZHOU

    (��School of Science, Wuhan University of Technology, 430074 Wuhan, P. R. China)

Abstract

For x ∈ (0, 1], let x = [d1(x),d2(x),…,dn(x),…]L be its Lüroth expansion, and let {pn(x)/qn(x)}n≥1 be the sequence of convergents of x. Define the exceptional sets E(β) = x ∈ (0, 1]: limn→∞log dn+1(x) log qn(x) = β and U(β) = x ∈ (0, 1]: limsupn→∞log dn+1(x) log qn(x) = β. Arroyo and González Robert [Hausdorff dimension of sets of numbers with large Lüroth elements, preprint (2020), arXiv:2010.13932] presented upper and lower bound estimations for the Hausdorff dimension of E(β). In this paper, we determine the Hausdorff dimensions of the sets E(β) as well as U(β).

Suggested Citation

  • Yan Feng & Bo Tan & Qing-Long Zhou, 2021. "Exact Dimensions Of Exceptional Sets In Lãœroth Expansions," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(06), pages 1-13, September.
  • Handle: RePEc:wsi:fracta:v:29:y:2021:i:06:n:s0218348x21501425
    DOI: 10.1142/S0218348X21501425
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