An approximation property of quadratic irrationals
Bulletin de la Société Mathématique de France, Volume 130 (2002) no. 1, pp. 35-48.

Let $\alpha >1$ be irrational. Several authors studied the numbers

 ${\ell }^{m}\left(\alpha \right)=inf\left\{\phantom{\rule{0.166667em}{0ex}}|y|:y\in {\Lambda }_{m},\phantom{\rule{0.166667em}{0ex}}y\ne 0\right\},$
where $m$ is a positive integer and ${\Lambda }_{m}$ denotes the set of all real numbers of the form $y={ϵ}_{0}{\alpha }^{n}+{ϵ}_{1}{\alpha }^{n-1}+\cdots +{ϵ}_{n-1}\alpha +{ϵ}_{n}$ with restricted integer coefficients $|{ϵ}_{i}|\le m$. The value of ${\ell }^{1}\left(\alpha \right)$ was determined for many particular Pisot numbers and ${\ell }^{m}\left(\alpha \right)$ for the golden number. In this paper the value of ${\ell }^{m}\left(\alpha \right)$ is determined for irrational numbers $\alpha$, satisfying ${\alpha }^{2}=a\alpha ±1$ with a positive integer $a$.

Soit $\alpha >1$ un irrationnel. Plusieurs auteurs ont étudié les nombres

 ${\ell }^{m}\left(\alpha \right)=inf\left\{\phantom{\rule{0.166667em}{0ex}}|y|:y\in {\Lambda }_{m},\phantom{\rule{0.166667em}{0ex}}y\ne 0\right\},$
$m$ est un entier positif et ${\Lambda }_{m}$ est l’ensemble de tous les réels de la forme $y={ϵ}_{0}{\alpha }^{n}+{ϵ}_{1}{\alpha }^{n-1}+\cdots +{ϵ}_{n-1}\alpha +{ϵ}_{n}$ avec des $|{ϵ}_{i}|\le m$ entiers. La valeur de ${\ell }^{1}\left(\alpha \right)$ a été précisée pour beaucoup de nombres de Pisot et ${\ell }^{m}\left(\alpha \right)$ pour le nombre d’or. Dans cet article, on détermine ${\ell }^{m}\left(\alpha \right)$ lorsque $\alpha$ est un irrationnel qui satisfait ${\alpha }^{2}=a\alpha ±1$ avec $a$ entier positif.

DOI: 10.24033/bsmf.2411
Classification: 11A63,  11J04,  11J70
Keywords: approximation property, quadratic irrationals, continued fractions
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Komatsu, Takao. An approximation property of quadratic irrationals. Bulletin de la Société Mathématique de France, Volume 130 (2002) no. 1, pp. 35-48. doi : 10.24033/bsmf.2411. http://archive.numdam.org/articles/10.24033/bsmf.2411/

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