An approximation property of quadratic irrationals. (English) Zbl 1027.11047

Komatsu, Takao

Bull. Soc. Math. Fr. 130, No. 1, 35-48 (2002).

Author’s abstract: Let \(\alpha > 1\) be irrational. Several authors studied the numbers \(\ell^m(\alpha)=\inf \{|y|: y\in \Lambda_m\), \(y\neq 0\}\), where \(m\) is a positive integer and \(\Lambda_m\) denotes the set of all real numbers of the form \(y=\varepsilon_0 \alpha^n + \varepsilon_1 \alpha^{n-1} +\cdots + \varepsilon_{n-1} \alpha + \varepsilon_n\) with restricted integer coefficients \(|\varepsilon_i|\leq m\). The value of \(\ell^1(\alpha)\) was determined for many particular Pisot numbers and \(\ell^m(\alpha)\) for the golden number. In this paper the value of \(\ell^m(\alpha)\) is determined for irrational numbers \(\alpha\) satisfying \(\alpha^2=a\alpha\pm 1\) with a positive integer \(a\).

Reviewer: László Tóth (Pécs)

MSC:

11J04	Homogeneous approximation to one number
11J70	Continued fractions and generalizations

Keywords:

approximation property; quadratic irrational; continued fraction