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 We also show that it is no longer valid for the negative and farthest integer continued fractions.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
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
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