Rational approximations to two irrational numbers

For real $ξ$ we consider the irrationality measure function $ψ_ξ(t) = \min_{1\leqslant q \leqslant t, q\in\mathbb{Z}} || qξ||$, where $||\cdot||$ - distance to the nearest integer. We prove that in the case $α\pmβ\notin\mathbb{Z}$ there exist arbitrary large values of $t$ with $\Bigl | \frac{1}{ψ_α(t)} - \frac{1}{ψ_β(t)} \Bigl | \geqslant \sqrt5\left(1-\sqrt{\frac{\sqrt5-1}{2}}\right)t.$ The constant on the right-hand side is optimal.

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