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.