Permutation of values of irrationality measure functions

Abstract

For an irrational number $\alpha\in\mathbb{R}$ we consider its irrationality measure function $$ \psi_\alpha(t) = \min_{1\le q\le t,\, q\in\mathbb{Z}} \| q\alpha \|. $$ Let $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_n)$ be $n$-tuple of pairwise independent irrational numbers. For each $t \in \mathbb{R}_{>1}$ irrationality measure functions $\psi_{\alpha_1}, \dots, \psi_{\alpha_n}$ can be written in an increasing order $$\psi_{\alpha_{v_1}}(t) > \psi_{\alpha_{v_2}}(t) > \dots > \psi_{\alpha_{v_{n-1}}}(t) > \psi_{\alpha_{v_n}}(t).$$ We consider the vector of functions $\boldsymbol{v}_{\boldsymbol{\alpha}}(t): \mathbb{R}_{>1} \rightarrow S_n$ associated to this order and defined as $$\boldsymbol{v}_{\boldsymbol{\alpha}}(t) = ( v_1, v_2, \dots, v_{n-1}, v_n ).$$ Let $\boldsymbol{k}(\boldsymbol{\alpha})$ be the number of infinitely occurring different values of $\boldsymbol{v}_{\boldsymbol{\alpha}}(t)$. It is known that if $\boldsymbol{k}(\boldsymbol{\alpha})= k$ we have $ n \leq \frac{k(k+1)}{2}.$ At the same time, for $k \geq 3$ and $n = \frac{k(k+1)}{2}$ there exists an $n$-tuple $\boldsymbol{\alpha}$ with $\boldsymbol{k}(\boldsymbol{\alpha}) = k$. In this work we define a $k$-cyclic permutation $\pi$ and prove that in the extremal case $n = \frac{k(k+1)}{2}, \ \boldsymbol{k}(\boldsymbol{\alpha}) = k$ the set of successive values of $\boldsymbol{v}_{\boldsymbol{\alpha}}(t)$ is an orbit of $\pi$.