Difference of irrationality measure functions

Abstract

For an irrational number $\alpha \in \mathbb {R}$ we consider its irrationality measure function

$$\begin{aligned} \psi _\alpha (x) = \min _{1\le q\le x,\, q\in \mathbb {Z}} \Vert q\alpha \Vert . \end{aligned}$$

It is known that for all irrational numbers $\alpha$ and $\beta$ satisfying $\alpha \pm \beta \not \in \mathbb Z$, there exist arbitrary large values of t with

$$\begin{aligned} | \psi _\alpha (t) - \psi _\beta (t) | \geqslant \left( \sqrt{\tau } - 1\right) \cdot \min ( \psi _\alpha (t), \psi _\beta (t) ), \end{aligned}$$

where $\tau = \frac{\sqrt{5} + 1}{2}$ and this result is optimal for certain numbers $\alpha$ and $\beta$ 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

$$\begin{aligned} | \psi _\alpha (t) - \psi _\beta (t) | \geqslant (\sqrt{\sqrt{2}+1}-1)\cdot \min ( \psi _\alpha (t), \psi _\beta (t) ). \end{aligned}$$

Moreover, we show that the constant on the right-hand side is optimal.