Exponents of Diophantine Approximation in dimension two

Abstract

Let $Θ=(α,β)$ be a point in $\bR^2$, with $1,α,β$ linearly independent over $\bQ$. We attach to $Θ$ a quadruple $Ω(Θ)$ of exponents which measure the quality of approximation to $Θ$ both by rational points and by rational lines. The two ``uniform'' components of $Ω(Θ)$ 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 $Ω$ fulfilling these necessary conditions, there exists a point $Θ\in \bR^2$ for which $Ω(Θ) =Ω$.