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

TL;DR

This paper investigates the behavior of the irrationality measure function for two real numbers, establishing an optimal bound on their difference for large values of t when their sum or difference is not an integer.

Contribution

It proves an optimal bound on the difference of irrationality measure functions for two real numbers under specific conditions.

Findings

Existence of arbitrarily large t with a specific bound on the difference

The bound is shown to be optimal

Results apply when the sum or difference of the numbers is not an integer

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.