Difference of irrationality measure functions

Abstract

For an irrational number $\alpha\in\mathbb{R}$ we consider its irrationality measure function $$ \psi_\alpha(x) = \min_{1\le q\le x,\, q\in\mathbb{Z}} \| q\alpha \|. $$ It is known for all irrational numbers $\alpha$ and $\beta$ satisfying $\alpha\pm\beta\not\in\mathbb{Z}$, there exist arbitrary large values of $t$ with \begin{equation*} | \psi_\alpha(t) - \psi_\beta(t) | \geqslant \left( \sqrt{\tau} - 1\right) \cdot \min( \psi_\alpha(t), \psi_\beta(t) ), \end{equation*} where $\tau = \frac{\sqrt{5} + 1}{2}$ and this result is optimal for certain numbers equivalent to $\tau$. Here we prove that for all irrational numbers $\alpha$ and $\beta$, satisfying $\alpha\pm\beta\not\in\mathbb{Z}$, such that at least one of them is not equivalent to $\tau$, there exist arbitrary large values of $t$ with $$ | \psi_\alpha(t) - \psi_\beta(t) | \geqslant (\sqrt{\sqrt2+1}-1)\cdot \min( \psi_\alpha(t), \psi_\beta(t) ). $$ Moreover, we show that the constant on the right-hand side is optimal.