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

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