Permutation of values of irrationality measure functions

Abstract

For an irrational number $α\in\mathbb{R}$ we consider its irrationality measure function $$ ψ_α(t) = \min_{1\le q\le t,\, q\in\mathbb{Z}} \| qα\|. $$ Let $\boldsymbolα = (α_1, \dots, α_n)$ be $n$-tuple of pairwise independent irrational numbers. For each $t \in \mathbb{R}_{>1}$ irrationality measure functions $ψ_{α_1}, \dots, ψ_{α_n}$ can be written in an increasing order $$ψ_{α_{v_1}}(t) > ψ_{α_{v_2}}(t) > \dots > ψ_{α_{v_{n-1}}}(t) > ψ_{α_{v_n}}(t).$$ We consider the vector of functions $\boldsymbol{v}_{\boldsymbolα}(t): \mathbb{R}_{>1} \rightarrow S_n$ associated to this order and defined as $$\boldsymbol{v}_{\boldsymbolα}(t) = ( v_1, v_2, \dots, v_{n-1}, v_n ).$$ Let $\boldsymbol{k}(\boldsymbolα)$ be the number of infinitely occurring different values of $\boldsymbol{v}_{\boldsymbolα}(t)$. It is known that if $\boldsymbol{k}(\boldsymbolα)= k$ 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 $\boldsymbolα$ with $\boldsymbol{k}(\boldsymbolα) = k$. In this work we define a $k$-cyclic permutation $π$ and prove that in the extremal case $n = \frac{k(k+1)}{2}, \ \boldsymbol{k}(\boldsymbolα) = k$ the set of successive values of $\boldsymbol{v}_{\boldsymbolα}(t)$ is an orbit of $π$.