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**Approximation to real numbers by cubic algebraic integers. I.**
*(English)*
Zbl 1035.11028

The study of approximation of real numbers by algebraic numbers of bounded degree \(d\) began in 1961 with a paper of E. Wirsing [J. Reine Angew. Math. 206, 67–77 (1961; Zbl 0097.03503)]. In 1967, H. Davenport and W. Schmidt [Acta Arith. 13, 169–176 (1967; Zbl 0155.09503)] studied the case \(d=2\) and in 1969 the approximation by algebraic integers of bounded degree and obtained an optimal result for \(d=2\) [Sympos. Math., Roma 1969, 4, 113–139 (1970; Zbl 0226.10032)]. In this paper, the author studies the case \(d=3\), the first case for which the optimal exponent of approximation was not known.

Let \(\gamma=(1+\sqrt 5)/2\). Davenport and Schmidt showed that, for any real number \(\xi\) which is neither rational nor quadratic irrational, there exist a positive constant \(c=c(\xi)\) and arbitrary large real numbers \(X\) such that the inequalities \[ | x_0| \leq X, \quad | x_0\xi -x_1| \leq cX^{-1/\gamma}, \quad | x_0\xi^2 -x_2| \leq cX^{-1/\gamma} \tag{1} \] have no solutions in non trivial triples \((x_0,x_1,x_2)\) of rational integers. This result implies that, for such a number \(\xi\), there exist some positive constant \(c'\) and infinitely many algebraic integers \(\alpha\) of degree at most \(3\) which satisfy \[ 0<| \xi-\alpha| \leq c' H(\alpha)^{-\gamma^2}, \tag{2} \] where \(H(\alpha)\) is the naive height.

The author obtains in particular the two following results:

– There exists a real number \(\xi\) which is neither rational nor quadratic irrational such that, for a suitable positive constant \(c\), the inequalities (1) have a non trivial solution \((x_0,x_1,x_2)\in \mathbb{Z}^3\) for any \(X\geq 1\). Such a \(\xi\) is called an extremal number, it is transcendental and the set of extremal numbers is countable. [Note that this result proves that the conditions in (1) are optimal.]

Let \(\xi\) be an extremal number. There exists a positive constant \(c=c(\xi)\) such that for any \(X\geq 1\) the inequalities \[ | x_0| \leq X, \quad | x_1| \leq X , \quad | x_0\xi^2 +x_1\xi+x_2| \leq cX^{- \gamma^2} \tag{3} \] have a non trivial solution \((x_0,x_1,x_2)\in \mathbb{Z}^3\). He also proves results of approximation of extremal numbers by rational numbers, by quadratic irrationalities and by algebraic integers of degree at most three.

Let \(\gamma=(1+\sqrt 5)/2\). Davenport and Schmidt showed that, for any real number \(\xi\) which is neither rational nor quadratic irrational, there exist a positive constant \(c=c(\xi)\) and arbitrary large real numbers \(X\) such that the inequalities \[ | x_0| \leq X, \quad | x_0\xi -x_1| \leq cX^{-1/\gamma}, \quad | x_0\xi^2 -x_2| \leq cX^{-1/\gamma} \tag{1} \] have no solutions in non trivial triples \((x_0,x_1,x_2)\) of rational integers. This result implies that, for such a number \(\xi\), there exist some positive constant \(c'\) and infinitely many algebraic integers \(\alpha\) of degree at most \(3\) which satisfy \[ 0<| \xi-\alpha| \leq c' H(\alpha)^{-\gamma^2}, \tag{2} \] where \(H(\alpha)\) is the naive height.

The author obtains in particular the two following results:

– There exists a real number \(\xi\) which is neither rational nor quadratic irrational such that, for a suitable positive constant \(c\), the inequalities (1) have a non trivial solution \((x_0,x_1,x_2)\in \mathbb{Z}^3\) for any \(X\geq 1\). Such a \(\xi\) is called an extremal number, it is transcendental and the set of extremal numbers is countable. [Note that this result proves that the conditions in (1) are optimal.]

Let \(\xi\) be an extremal number. There exists a positive constant \(c=c(\xi)\) such that for any \(X\geq 1\) the inequalities \[ | x_0| \leq X, \quad | x_1| \leq X , \quad | x_0\xi^2 +x_1\xi+x_2| \leq cX^{- \gamma^2} \tag{3} \] have a non trivial solution \((x_0,x_1,x_2)\in \mathbb{Z}^3\). He also proves results of approximation of extremal numbers by rational numbers, by quadratic irrationalities and by algebraic integers of degree at most three.

Reviewer: Maurice Mignotte (Strasbourg)