Diophantine approximation and special Liouville numbers

Johannes Schleischitz

Displaying similar documents to “Diophantine approximation and special Liouville numbers”

Approximation of values of hypergeometric functions by restricted rationals

Carsten Elsner, Takao Komatsu, Iekata Shiokawa (2007)

Journal de Théorie des Nombres de Bordeaux

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We compute upper and lower bounds for the approximation of hyperbolic functions at points $1 / s$ $(s = 1, 2, \dots)$ by rationals $x / y$ , such that $x, y$ satisfy a quadratic equation. For instance, all positive integers $x, y$ with $y \equiv 0 (mod 2)$ solving the Pythagorean equation $x^{2} + y^{2} = z^{2}$ satisfy $| y sinh (1 / s) - x | ≫ \frac{log log y}{log y} .$ Conversely, for every $s = 1, 2, \dots$ there are infinitely many coprime integers $x, y$ , such that $| y sinh (1 / s) - x | ≪ \frac{log log y}{log y}$ and $x^{2} + y^{2} = z^{2}$ hold simultaneously for some integer $z$ . A generalization to the approximation of $h (e^{1 / s})$ for rational...