Abstract

For an irrational number $\alpha \in ℝ$, we consider its irrationality measure function ${\psi }_{\alpha }(t)=\underset{1\le q\le t,q\in \mathbb{Z}}{\min <!--FUNCTION\; APPLICATION-->}\parallel q\alpha \parallel$. Let $\alpha =({\alpha }_1,\dots <!--FUNCTION\; APPLICATION-->,{\alpha }_n)$ be an $n$-tuple of pairwise independent irrational numbers. For each $t\in {ℝ}_{\ge 1}$, irrationality measure functions ${\psi }_{{\alpha }_1},\dots <!--FUNCTION\; APPLICATION-->,{\psi }_{{\alpha }_n}$ can be written in decreasing order: ${\psi }_{{\alpha }_{v_1}}(t)>{\psi }_{{\alpha }_{v_2}}(t)>\cdots >{\psi }_{{\alpha }_{v_{n-1}}}(t)>{\psi }_{{\alpha }_{v_n}}(t)$. We consider the vector of functions $v_{\alpha }(t):{ℝ}_{\ge 1}\to S_n$ associated to this order and defined as $v_{\alpha }(t)=(v_1,v_2,\dots <!--FUNCTION\; APPLICATION-->,v_{n-1},v_n)$. Let $k(\alpha )$ be the number of infinitely occurring different values of $v_{\alpha }(t)$. It is known that if $k(\alpha )=k$, we have $n\le \frac{1}{2}k(k+1)$. At the same time, for $k\ge 3$ and $n=\frac{1}{2}k(k+1)$, there exists an $n$-tuple $\alpha$ with $k(\alpha )=k$. In this work, we define a $k$-cyclic permutation $\pi$ and prove that in the extremal case $n=\frac{1}{2}k(k+1)$, $k(\alpha )$ equals $k$, the set of successive values of $v_{\alpha }(t)$ is an orbit of $\pi$.

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