Exponents of Diophantine Approximation in Dimension Two

Laurent, Michel

Canadian journal of mathematics, Tome 61 (2009) no. 1, pp. 165-189

Résumé

Let $\Theta \,=\,\left( \alpha ,\,\beta\right)$ be a point in ${{\text{R}}^{2}}$ , with $1,\,\alpha ,\,\beta $ linearly independent over $\mathrm{Q}$ . We attach to $\Theta $ a quadruple $\Omega \left( \Theta\right)$ of exponents that measure the quality of approximation to $\Theta $ both by rational points and by rational lines. The two “uniform” components of $\Omega \left( \Theta\right)$ are related by an equation due to Jarník, and the four exponents satisfy two inequalities that refine Khintchine's transference principle. Conversely, we show that for any quadruple $\Omega $ fulfilling these necessary conditions, there exists a point $\Theta \,\in \,{{\text{R}}^{2}}$ for which $\Omega \left( \Theta\right)\,=\,\Omega $ .

Détail

DOI : 10.4153/CJM-2009-008-2

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