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Abstract

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.