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Exponents of diophantine approximation. (English) Zbl 1229.11098

Zannier, Umberto (ed.), Diophantine geometry. Selected papers of a the workshop, Pisa, Italy, April 12–July 22, 2005. Pisa: Edizioni della Normale (ISBN 978-88-7642-206-5/pbk). Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series (Nuova Serie) 4, 101-121 (2007).
[To describe the content of the paper, the reviewer finds as most appropriate to cite (more or less directly) a good part of the authors’ introduction.]
The Dirichlet theorem asserts that for any irrational real number \(\xi\) and any real number \(Q\geq 1\), there exist integers \(p\) and \(q\) with \(1\leq p\leq Q\) and \[ |q\xi-p|\leq Q^{-1}. \] As known, there is no \(\xi\) for which the exponent of \(Q\) can be lowered. However, for any \(\omega >1\), there exist real numbers \(\xi\) for which, for arbitrarily large \(Q\), the equation \[ |q\xi-p|\leq Q^{-\omega} \] has a solution in integers \(p\) and \(q\) with \(1\leq q\leq Q\).
The Dirichlet theorem extends well to rational simultaneous approximation, and to simultaneous approximation of linear forms. In Section 2, the authors define exponents of Diophantine approximation related to these questions, and survey known results on them. The two-dimensional case results are displayed in Section 3. In order to extend possibly these results to higher dimensions, in Section 4 the authors define geometrically more exponents of Diophantine approximation, in connection with the work of Schmidt (1967).
The questions of Diophantine approximation are in general much more difficult when the quantities we approximate are dependent. Classical examples include the simultaneous rational approximation of the first \(n\) powers of a transcendental number, and the approximation of linear forms whose coefficients are precisely the first \(n\) powers of a transcendental number. These are considered in Section 5. Section 6 is devoted to Roy’s recent results for \(n=2\) and some of their extensions.
The present paper is a survey, but it contains also new theorems.
For the entire collection see [Zbl 1113.11003].

MSC:

11J04 Homogeneous approximation to one number
11J13 Simultaneous homogeneous approximation, linear forms
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