Simultaneous approximation of algebraic numbers (Q1316849)

Let 1, \(\theta_ 1, \dots, \theta_ s\) \((s \geq 2)\) be a basis of a purely algebraic field \(\mathbb{K}\) of degree \(s+1\). The author proves the following theorem: assume that a natural number simultaneously approximates the number \(\theta_ 1, \dots, \theta_ s\), \[ \| q \theta_ i \| = \min_{a \in \mathbb{Z}} | q \theta_ i - a |<cq^{-1/s}, \quad i = 1, \dots,s, \] where \(c>0\) is some constant. Then \[ \| q\theta_ i\|>Cq^{-1/q}\ln^{-\beta}q,\quad i=1,\dots,s, \] with constant \(C\) and \(\beta>0\) depending on \(\theta_ 1, \dots, \theta_ s\) and \(c\). From this theorem the author obtains some sharp results of \textit{L. G. Peck} [Bull. Am. Math. Soc., 67, 197-201 (1961; Zbl 0098.263)] and \textit{B. F. Skubenko} [Zap. Nauchn. Semin. LOMI 134, 226-231 (1984; Zbl 0535.10038)]. The proof of this theorem is based on ideas of Skubenko and applies the lower bounds of linear forms of logarithms by A. Baker.