On metric theory of Diophantine approximation for complex numbers

• Autorzy: Zhengyu Chen
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11K60: Diophantine approximation
• 11J83: Metric theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 170
• Numer: 1
• Strony: 27-46

Daty

wydano

2015

Twórcy

autor

Zhengyu Chen

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

Identyfikatory

DOI

10.4064/aa170-1-3