On metric theory of Diophantine approximation for complex numbers

Zhengyu Chen¹

¹ Department of Mathematics Keio University Hiyoshi 3-14-1, Kohoku-ku Yokohama, Kanagawa 223-8522, Japan

Acta Arithmetica, Tome 170 (2015) no. 1, pp. 27-46

Résumé

In 1941, R. J. Duffin and A. C. Schaeffer conjectured that for the inequality $|\alpha - m/n| \psi(n)/n$ with ${\rm g.c.d.}(m,n) = 1$, there are infinitely many solutions in positive integers $m$ and $n$ for almost all $\alpha \in \mathbb{R}$ if and only if $\sum_{n=2}^{\infty}\phi(n)\psi(n)/n = \infty$. As one of partial results, in 1978, J. D. Vaaler proved this conjecture under the additional condition $\psi(n) = \mathcal O(n^{-1})$. In this paper, we discuss the metric theory of Diophantine approximation over the imaginary quadratic field $\mathbb{Q}(\sqrt{d})$ with a square-free integer $d 0$, and show that a Vaaler type theorem holds in this case.

Détail

DOI : 10.4064/aa170-1-3

Keywords: duffin schaeffer conjectured inequality alpha psi there infinitely many solutions positive integers almost alpha mathbb only sum infty phi psi infty partial results vaaler proved conjecture under additional condition psi mathcal paper discuss metric theory diophantine approximation imaginary quadratic field mathbb sqrt square free integer vaaler type theorem holds