EXACT DIMENSIONS OF EXCEPTIONAL SETS IN LÃœROTH EXPANSIONS

Yan Feng (), Bo Tan and Qing-Long Zhou ()
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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

FRACTALS (fractals), 2021, vol. 29, issue 06, 1-13

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).

Keywords: Lüroth Expansion; Diophantine Approximation; Jarník-like Set; Hausdorff Dimension (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21501425

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