Exponents of Diophantine Approximation in dimension two

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Let Θ=(α,β) be a point in ², with 1,α,β linearly independent over. 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 Jarnık, 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∈ ² for which Ω(Θ) =Ω.