Approximation properties of Lüroth expansions

Bo Tan; Qinglong Zhou

doi:10.3934/dcds.2020389

Article Contents

2021, Volume 41, Issue 6: 2873-2890. Doi: 10.3934/dcds.2020389

This issue Previous Article On stochastic porous-medium equations with critical-growth conservative multiplicative noise Next Article A symmetric Random Walk defined by the time-one map of a geodesic flow

Approximation properties of Lüroth expansions

Bo Tan^1, and
Qinglong Zhou^2, ,

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2.
School of Science, Wuhan University of Technology, Wuhan 430074, China

^* Corresponding author: Qinglong Zhou
^* Corresponding author: Qinglong Zhou

Received: April 2020

Revised: September 2020

Early access: December 18, 2020

Published online: December 18, 2020

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Abstract

For $ x\in[0,1), $ let $ [d_{1}(x),d_{2}(x),\ldots] $ be its Lüroth expansion and $ \big\{\frac{p_{n}(x)}{q_{n}(x)}, n\geq 1\big\} $ be the sequence of convergents of $ x. $ In this paper, we study the Jarník-like set of real numbers which can be well approximated by infinitely many of their convergents in the Lüroth expansion$ \colon $

$ W(\psi) = \{x\in[0,1)\colon |xq_{n}(x)-p_{n}(x)|<\psi(n) \text{ for infinitely many } n\in \mathbb{N}\}, $

where $ \psi\colon \mathbb{R}\to (0,\frac{1}{2}] $ is a positive function. We completely determine the Hausdorff dimension of $ W(\psi). $

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Mathematics Subject Classification: Primary: 11K55; Secondary: 28A80, 11J83.

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