Approximation to real numbers by cubic algebraic integers. II
Annals of mathematics, Tome 158 (2003) no. 3, pp. 1081-1087
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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 $(3+\sqrt{5})/2\simeq 2.618$.
@article{10_4007_annals_2003_158_1081,
author = {Damien Roy},
title = {Approximation to real numbers by cubic algebraic integers. {II}},
journal = {Annals of mathematics},
pages = {1081--1087},
publisher = {mathdoc},
volume = {158},
number = {3},
year = {2003},
doi = {10.4007/annals.2003.158.1081},
mrnumber = {2031862},
zbl = {1044.11061},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.158.1081/}
}
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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
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