Abstract

For an $n$-tuple $\alpha =({\alpha }_1,\dots <!--FUNCTION\; APPLICATION-->,{\alpha }_n)$ of pairwise independent numbers we consider permutations

of irrationality measure functions ${\psi }_{\alpha }(t)={\min <!--FUNCTION\; APPLICATION-->}_{1\le q\le t}\parallel q\alpha \parallel$. Let $\mathfrak{𝔨}(\alpha )$ be the number of infinitely occurring different permutations $\{{\sigma }_1,\dots <!--FUNCTION\; APPLICATION-->,{\sigma }_{\mathfrak{𝔨}(\alpha )}\}$. We prove that the length of an $n$-tuple $\alpha$ with $\mathfrak{𝔨}(\alpha )=k$ is

and this result is optimal.

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