Abstract view
Exponents of Diophantine Approximation in Dimension Two


Published:20090201
Printed: Feb 2009
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 $\Omega(\Theta) =\Omega$.