Permutation of values of irrationality measure functions

TL;DR

This paper studies the ordering of irrationality measure functions for multiple irrational numbers, characterizing the structure of their value permutations and extremal cases using cyclic permutations.

Contribution

It introduces the concept of $k$-cyclic permutations and proves their role in the extremal behavior of the ordering of irrationality measure functions.

Findings

The number of distinct orderings is bounded by $n \,\leq\, \frac{k(k+1)}{2}$.

In extremal cases, the sequence of orderings forms an orbit of a $k$-cyclic permutation.

The set of successive orderings corresponds to the orbit of a cyclic permutation in extremal scenarios.

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 $\pi$.