On simultaneous Diophantine approximations. Vectors of a given Diophantine type
Let \(\psi(y)\) be a real-valued function of a real argument. A positive integer \(p\) is called a simultaneous \(\psi\)-approximation for the numbers \(\alpha_1,\dots,\alpha_s\in \mathbb R\) if \[ \max_{1\leq j\leq s}\| p\alpha_j\|\leq \psi(p)\;(\text{here }\|\alpha\|= \min_{z\in \mathbb Z}| \alpha- z|). \] The numbers \(\alpha_1,\dots, \alpha_s\) are said to possess the \(\psi\)-property if they admit infinitely many \(\psi\)-approximations, and admit no \(c\psi\)-approximation with some constant \(c\) depending on \(\alpha_1,\dots, \alpha_s\). The author proves the following theorems: Let \(\psi(y)= y^{-1/s}\omega(y)\), where \(\omega(y)\) decreases monotonously (but not necessarily strictly monotonously) for \(y\geq 1\). 1) If \[ \omega(1)\leq 2^{-(s+ 1)(s+ 2)}(s!)^{- 1/s} (\text{resp. }\omega(1)\leq 2^{-(s+ 1)(s+ 3)}(s!)^{-1/s}), \] then there is a set of numbers \(\alpha_1,\dots, \alpha_s\) (resp. a continual family of vectors \(\alpha= (\alpha_1,\dots, \alpha_s)\)) admitting infinitely many simultaneous \(\psi\)-approximations, however admitting no simultaneous \(2^{-(s+ 3)}\psi\)-approximation. 2) In any \(s\)-dimensional domain with positive volume, there is a continual family of vectors possessing the \(\psi\)-property. These theorems give a generalization of a result of \textit{J. W. S. Cassels} [Proc. Lond. Math. Soc. (3) 5, 435--448 (1955; Zbl 0065.28302)] and a refinement of that of \textit{V. Jarnik} [Math. Z. 33, 505--543 (1931; Zbl 0001.32403) and Enseign. Math. (2) 25, 171--175 (1969; Zbl 0177.07201)]. The proof of the theorems is based on ideas from Cassels' paper mentioned above, and the Minkowski-Voronoi chain is a main tool in the proof.
scientific article; zbMATH DE number 1238047
scientific article