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In 1941, R. J. Duffin and A. C. Schaeffer conjectured that for the inequality |α - m/n| < ψ(n)/n with g.c.d.(m,n) = 1, there are infinitely many solutions in positive integers m and n for almost all α ∈ ℝ if and only if ${\sum }_{n=2}^{\infty }\varphi \left(n\right)\psi \left(n\right)/n=\infty$. As one of partial results, in 1978, J. D. Vaaler proved this conjecture under the additional condition $\psi \left(n\right)=\left(n^{-1}\right)$. In this paper, we discuss the metric theory of Diophantine approximation over the imaginary quadratic field ℚ(√d) with a square-free integer d < 0, and show that a Vaaler...