Difference of irrationality measure functions

Abstract

For an irrational number $\alpha \in \mathbb{R}$ we consider its irrationality measure function $${\psi }_{\alpha }(x)=\underset{1\le q\le x,q\in \mathbb{Z}}{\min }\Vert q\alpha \Vert .$$ It is known for all irrational numbers $\alpha$ and $\beta$ satisfying $\alpha \pm \beta \notin \mathbb{Z}$, there exist arbitrary large values of $t$ with \begin{equation*} | ψ_α(t) - ψ_β(t) | \geqslant \left( \sqrtτ - 1\right) \cdot \min( ψ_α(t), ψ_β(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 $\alpha \pm \beta \notin \mathbb{Z}$, such that at least one of them is not equivalent to $\tau$, there exist arbitrary large values of $t$ with $$|{\psi }_{\alpha }(t)-{\psi }_{\beta }(t)|⩾(\sqrt{\sqrt{2}+1}-1)\cdot \min ({\psi }_{\alpha }(t),{\psi }_{\beta }(t)).$$ Moreover, we show that the constant on the right-hand side is optimal.

15 pages, 1 figure. Comments will be appreciated