Difference of irrationality measure functions
For an irrational number $α\in\mathbb{R}$ we consider its irrationality measure function $$ ψ_α(x) = \min_{1\le q\le x,\, q\in\mathbb{Z}} \| qα\|. $$ It is known for all irrational numbers $α$ and $β$ satisfying $α\pmβ\not\in\mathbb{Z}$, 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 $τ= \frac{\sqrt{5} + 1}{2}$ and this result is optimal for certain numbers equivalent to $τ$. Here we prove that for all irrational numbers $α$ and $β$, satisfying $α\pmβ\not\in\mathbb{Z}$, such that at least one of them is not equivalent to $τ$, there exist arbitrary large values of $t$ with $$ | ψ_α(t) - ψ_β(t) | \geqslant (\sqrt{\sqrt2+1}-1)\cdot \min( ψ_α(t), ψ_β(t) ). $$ Moreover, we show that the constant on the right-hand side is optimal.