## 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.

### MSC:

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

### Citations:

Zbl 0097.03503; Zbl 0155.09503; Zbl 0226.10032
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