Abstract
Let $\xi=\sqrt{v/u}\tanh(uv)^{-1/2}$, where $u$ and $v$ are positive integers, and let $\eta=|h(\xi)|$, where $h(t)$ is a non-constant rational function with algebraic coefficients. We compute upper and lower bounds for the approximation of certain values $\eta$ of hyperbolic functions by rationals $x/y$ such that $x$ and $y$ satisfy Diophantine equations. We show that there are infinitely many coprime integers $x$ and $y$ such that $|y\eta-x|\ll\log\log y/\log y$ and a Diophantine equation holds simultaneously relating $x$ and $y$ and some integer $z$. Conversely, all positive integers $x$ and $y$ with $y\ge c_0$ solving the Diophantine equation satisfy $|y\eta-x|\gg\log\log y/\log y$.
Citation
Tuangrat CHAICHANA. Takao KOMATSU. Vichian LAOHAKOSOL. "On Convergents of Certain Values of Hyperbolic Functions Formed from Diophantine Equations." Tokyo J. Math. 36 (1) 239 - 251, June 2013. https://doi.org/10.3836/tjm/1374497522
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