Über die Funktionen des Irrationalitätsmaßes für zwei irrationale Zahlen

Abstract

For real \(\xi \) we consider the irrationality measure function \(\psi _\xi (t) = \min _{1\leqslant q \leqslant t, \, q\in \mathbb {Z}} ||q\xi ||\). We prove that in the case \(\alpha \pm \beta \not \in \mathbb {Z}\) there exist arbitrary large values of t with \(|\psi _\alpha (t) -\psi _\beta (t)| \geqslant \left( \sqrt{\frac{\sqrt{5}+1}{2}}-1\right) \min (\psi _\alpha (t), \psi _\beta (t))\). This result is optimal.