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.


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


Zbl 1035.11028
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