June 2024 Irrationality Exponents of Semi-regular Continued Fractions
Daniel DUVERNEY, Iekata SHIOKAWA
Tokyo J. Math. 47(1): 89-109 (June 2024). DOI: 10.3836/tjm/1502179394

Abstract

Let $\left[ b_{0},b_{1},b_{2},\ldots ,b_{n},\ldots \right]$ be the expansion of $\alpha \in \mathbb{R}\smallsetminus \mathbb{Q}$ in regular continued fraction, and let $p_{n}/q_{n}$ be the convergents of this continued fraction. It is known that the irrationality exponent of $\alpha$ is given by the formula $\mu \left( \alpha \right) =2+\limsup_{n\rightarrow \infty }\left( \log b_{n+1}/\log q_{n}\right)$. We prove that this formula remains valid for semi-regular continued fractions satisfying certain conditions. In particular, it remains valid for the nearest integer and singular continued fractions. We also show that it is no longer valid for the negative and farthest integer continued fractions. In application, examples of exact computation of irrationality exponents of semi-regular continued fractions are given.

Citation

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Daniel DUVERNEY. Iekata SHIOKAWA. "Irrationality Exponents of Semi-regular Continued Fractions." Tokyo J. Math. 47 (1) 89 - 109, June 2024. https://doi.org/10.3836/tjm/1502179394

Information

Published: June 2024
First available in Project Euclid: 19 August 2024

Digital Object Identifier: 10.3836/tjm/1502179394

Subjects:
Primary: 11A55
Secondary: 11J70 , 11J82

Rights: Copyright © 2024 Publication Committee for the Tokyo Journal of Mathematics

Vol.47 • No. 1 • June 2024
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