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.


11J04 Homogeneous approximation to one number
11J13 Simultaneous homogeneous approximation, linear forms
11J82 Measures of irrationality and of transcendence
11J68 Approximation to algebraic numbers
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