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Approximation to real numbers by cubic algebraic integers. II. (English) Zbl 1044.11061
The study of approximation of real numbers by algebraic numbers of bounded degree $$n$$ began in 1961 with a paper of E. Wirsing. In 1967, H. Davenport and W. Schmidt studied the case $$n=2$$ and in 1969 the approximation by algebraic integers of bounded degree and obtained an optimal result for $$n=2$$ and a certain result of ‘good’ approximation for $$n=3$$: they proved that there exists a positive constant $$c$$ such that the inequality $| \xi-\alpha| \leq c H(\alpha)^{-\gamma^2},$ where $$H(\alpha)$$ is the naive height of $$\alpha$$ and $$\gamma=(1+\sqrt 5)/2$$, so that $$\gamma^2=(3+\sqrt 5)/2=2{.}618\ldots$$, has infinitely many solutions in algebraic integer $$\alpha$$ of degree at most $$3$$ over $$\mathbb Q$$.
In this paper, the author studies the case $$n=3$$, the first case for which the optimal exponent of approximation was not known; he shows that the exponent of approximation $$\gamma^2$$ appearing in the result of Davenport and Schmidt for $$n=3$$ is optimal. More precisely he proves the
Theorem. There exists a real number $$\xi$$ which is transcendental over $$\mathbb Q$$ and a positive constant $$c_1$$ such that, for any algebraic integer $$\alpha$$ of degree at most $$3$$ over $$\mathbb Q$$, we have $| \xi-\alpha| \geq c_1 H(\alpha)^{-\gamma^2}.$
If, for a positive integer $$n$$, one defines $$\tau_n$$ as the supremum of all real numbers $$\tau$$ with the property that any transcendental number $$\xi$$ admits infinitely many approxiamtions by algebraic integers $$\alpha$$ of degree at most $$n$$ with $$| \xi-\alpha| \leq H(\alpha)^{-\tau}$$, then it was known that $$\tau_2=2$$ and conjectured that $$\tau_n=n$$ for all $$n$$. The present theorem shows that $$\tau_3=\gamma^2$$, so that the conjecture is false for $$n=3$$. For $$n>3$$ the exponent $$\tau_n$$ remains unknown.
The highly ingenious proof is based on previous results of the author [see Part I, Proc. Lond. Math. Soc. (3) 88, No. 1, 42–62 (2004; Zbl 1035.11028)]. It uses the notion of extremal number introduced in this reference. Explicit extremal numbers satisfying the Theorem are constructed.

##### MSC:
 11J04 Homogeneous approximation to one number 11J82 Measures of irrationality and of transcendence 11J13 Simultaneous homogeneous approximation, linear forms
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