Exponents of Diophantine Approximation in Dimension Two

http://dx.doi.org/10.4153/CJM-2009-008-2
Canad. J. Math. 61(2009), 165-189

  Published:2009-02-01
 Printed: Feb 2009

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 $\Omega(\Theta) =\Omega$.