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On metric theory of Diophantine approximation for complex numbers

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Chen Z.

Acta Arithmetica

2015

tom 170

nr 1

27-46

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 $∑_{n=2}^{∞}ϕ(n)ψ(n)/n = ∞$. As one of partial results, in 1978, J. D. Vaaler proved this conjecture under the additional condition $ψ(n) = 𝓞(n^{-1})$. 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 type theorem holds in this case.

Ograniczanie wyników

1 Acta Arithmetica

1 Chen Z.

1 2015

Wyniki wyszukiwania

On metric theory of Diophantine approximation for complex numbers