On metric theory of Diophantine approximation for complex numbers

Zhengyu Chen. "On metric theory of Diophantine approximation for complex numbers." Acta Arithmetica 170.1 (2015): 27-46. <http://eudml.org/doc/279394>.

@article{ZhengyuChen2015,
abstract = {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.},
author = {Zhengyu Chen},
journal = {Acta Arithmetica},
keywords = {Diophantine approximation; Duffin-Schaeffer conjecture; imaginary quadratic fields},
language = {eng},
number = {1},
pages = {27-46},
title = {On metric theory of Diophantine approximation for complex numbers},
url = {http://eudml.org/doc/279394},
volume = {170},
year = {2015},
}

TY - JOUR
AU - Zhengyu Chen
TI - On metric theory of Diophantine approximation for complex numbers
JO - Acta Arithmetica
PY - 2015
VL - 170
IS - 1
SP - 27
EP - 46
AB - 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.
LA - eng
KW - Diophantine approximation; Duffin-Schaeffer conjecture; imaginary quadratic fields
UR - http://eudml.org/doc/279394
ER -