Permutation of values of irrationality measure functions

Abstract

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

22 pages, 10 figures

References in corpus (1)

• On irrationality measure functions for several real numbers