Simultaneous rational approximation to the successive powers of a real number
For a real number \(x\) denote by \(\lfloor x \rfloor\) the integral part of \(x\) and by \(\lceil x \rceil\) the smallest integer \(\geq x\). Let \(n\) be an integer \(\geq 1\) and \(\lambda>0\) be a real number. Set \(\kappa=\lceil n/2\rceil \). Let \(\theta\) be a real number which is either transcendental or else is algebraic of degree \(>\kappa\). Assume that for every \(\varepsilon>0\) and every sufficiently large positive real number \(X\) there exists \((x_0,\ldots,x_n)\in\mathbb{Z}^n\setminus\{0\}\) such that \(|x_0|\leq X\) and \[ \max_{1\leq j\leq n} |x_{0}\theta^{j}-x_{j}|\leq \varepsilon X^{-\lambda}. \] Then \(\lambda<1/\kappa\). This improves (when \(n\) is odd) a previous result by \textit{H. Davenport} and \textit{W. M. Schmidt} [Acta Arith. 15, 393--416 (1969; Zbl 0186.08603)] where \(\kappa\) was replaced by \(\lfloor n/2\rfloor\). The proof follows the work of Davenport and Schmidt; the improvement arises from the introduction of new lemmas involving Hankel's determinants in place of a variant of Gel'fond's transcendence criterion.
scientific article; zbMATH DE number 2075516