On metric theory of Diophantine approximation for complex numbers
Zhengyu Chen
Acta Arithmetica, Tome 168 (2015), p. 27-46 / Harvested from The Polish Digital Mathematics Library
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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 ∑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)=(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.

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:279394 
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     title = {On metric theory of Diophantine approximation for complex numbers},
     journal = {Acta Arithmetica},
     volume = {168},
     year = {2015},
     pages = {27-46},
     zbl = {1337.11046},
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Zhengyu Chen. On metric theory of Diophantine approximation for complex numbers. Acta Arithmetica, Tome 168 (2015) pp. 27-46. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-1-3/