Exponents of Diophantine Approximation in dimension two

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 which 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{\'ı}k, and the four exponents satisfy two inequalities which refine Khintchine's transference principle. Conversely, we show that for any quadruple $\Omega$ fulfilling these necessary conditions, there exists a point $Θ\in \bR^2$ for which $\Omega (\Theta )=\Omega$.

Cited by in corpus (3)

• Contribution to Vojtech Jarnik
• Exponents for three-dimensional simultaneous Diophantine approximations
• On Diophantine exponents in dimension 4