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# Exponents of Diophantine Approximation in Dimension Two

Published:2009-02-01
Printed: Feb 2009
• Michel Laurent
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## Abstract

Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha, \beta$ linearly independent over $\bQ$. We attach to $\Theta$ a quadruple $\Omega(\Theta)$ of exponents that measure the quality of approximation to $\Theta$ both by rational points and by rational lines. The two uniform'' components of $\Omega(\Theta)$ are related by an equation due to Jarn\'\i k, 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 $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.
 MSC Classifications: 11J13 - Simultaneous homogeneous approximation, linear forms 11J70 - Continued fractions and generalizations [See also 11A55, 11K50]