Rational approximations to two irrational numbers (Q2121973)

Let \(\xi\) be a real number, consider an irrationality measure function defined as follows: \[ \psi_\xi(t)=\min\limits_{1\le q \le t,q\in\mathbb Z}||q\xi||. \] The author proves the following result. Let \(\alpha\) and \(\beta\) be irrational numbers satisfying \(\alpha\pm \beta \notin \mathbb Z\). Then there exists an arbitrary large \(t\) satisfying \[ \left| \frac{1}{\psi_\alpha(t)-\psi_\beta(t)} \right| \ge \sqrt{5} \left( 1-\sqrt{\frac{\sqrt{5}-1}{2}} \right)t. \] The constant on the right-hand side is exact.