\"Uber die Funktionen des Irrationalit\"atsma\ss es f\"{u}r zwei irrationalen Zahlen

Abstract

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