Approximation to real numbers by cubic algebraic integers. II.

*(English)*Zbl 1044.11061The 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.

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.

Reviewer: Maurice Mignotte (Strasbourg)