On approximations for \(n\) real numbers (Q6097054)

The authors study the irrationality function of a real number \(\alpha\), introduced by \[\psi_{\alpha}(t)=\min_{q\in\mathbb{Z}_{+},q\leq t}\,\|q\alpha\|,\text{ with }\|x\|=\min_{a\in\mathbb{Z}}\,|x-a|.\] In a paper by \textit{I. D. Kan} and \textit{N. G. Moshchevitin} [Unif. Distrib. Theory 5, No. 2, 79--86 (2010; Zbl 1249.11064)] it was shown that the difference \(\psi_{\alpha}(t)-\psi_{\beta}(t)\) for two irrational numbers changes sign infinitely often as \(t\rightarrow\infty\). In this paper the authors give a generalization of this theorem to the case of \(n\geq 2\) real numbers. An important quantity here is the so called \(k\)-index. Let \({\mathbf{a}}=(\alpha_1,\ldots,\alpha_n)\) be an \(n\)-tuple pairwise incommensurable numbers (i.e., \(\psi_{\alpha_i}(t)\not=\psi_{\alpha_j}(t)\) for \(i\not= j\) and all sufficiently large \(t\)). Then there exists for all sufficiently large \(t\) a permutation \[\vec{\sigma}(t):\{1,2,3,\ldots,n\}\rightarrow \{\sigma_1,\sigma_2,\sigma_3,\ldots,\sigma_n\} \] with \[\psi_{\alpha_{\sigma_1}}(t) >\psi_{\alpha_{\sigma_2}}(t)>\psi_{\alpha_{\sigma_3}}(t)>\ldots>\psi_{\alpha_{\sigma_n}}(t).\] The \(k\)-index \(\mathbf{k(a)}=\mathbf{k}(\alpha_1,\ldots,\alpha_n)\) is now defined by \[{\mathbf{k(a)}}=\max\{k:\text{ exists } k\text{ permutations with }\sigma_j\ (1\leq j\leq k\text{ with }\text{ for all } j\text{ for all } t_0>0\] \[ \text{ exists } t>t_0\text{ with }\sigma(t)=\sigma_j\}.\] The main result of the paper is now: Theorem 1. Let \({\mathbf{a}}=(\alpha_1,\ldots,\alpha_n)\) be an \(n\)-tuple pairwise incommensurable numbers, then \[\mathbf{k(a)}\geq \sqrt{\frac{n}{2}}.\] Moreover, the following theorem shows that this result leads to the optimal number for the inequality: Theorem 2. Let \(k\geq 3\) and \(n=\frac{(k(k+1)}{2}\), then there exists an \(n\)-tuple \(\mathbf{a}\) pairwise incommensurable numbers with \[\mathbf{k(a)}=k.\] The layout of the paper is as follows: \S1 The \(k\)-index. Contains the main results. \S2 Continued fractions. \S3 The main Lemma (\(3\) pages). \S4 Proof of Theorem 1 (\(2\frac{1}{2}\) pages). \S5 Four numbers (\(1\) page). \S6 Proof of Theorem 2 (\(3\) pages). References (\(11\) items). A nicely written paper.