Geodesic
Approximation to real numbers by cubic algebraic integers. II
Damien Roy1
1 Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5
Annals of mathematics, Tome 158 (2003) no. 3, pp. 1081-1087.

Voir la notice de l'article provenant de la source Annals of Mathematics website

It has been conjectured for some time that, for any integer $n\ge 2$, any real number $\varepsilon >0$ and any transcendental real number $\xi$, there would exist infinitely many algebraic integers $\alpha$ of degree at most $n$ with the property that $|\xi-\alpha|\le H(\alpha)^{-n+\varepsilon}$, where $H(\alpha)$ denotes the height of $\alpha$. Although this is true for $n=2$, we show here that, for $n=3$, the optimal exponent of approximation is not $3$ but $(3+\sqrt{5})/2\simeq 2.618$.
DOI : 10.4007/annals.2003.158.1081

Damien Roy 1

1 Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 
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Damien Roy. Approximation to real numbers by cubic algebraic integers. II. Annals of mathematics, Tome 158 (2003) no. 3, pp. 1081-1087. doi : 10.4007/annals.2003.158.1081. http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.158.1081/

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