Exponents of Diophantine Approximation in Dimension Two

Source abstract

Abstract. Let Θ = (α, β) be a point in R 2 , with 1, α, β linearly independent over Q . We attach to Θ a quadruple Ω (Θ) of exponents that 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 Jarník, and the four exponents satisfy two inequalities that refine Khintchine's transference principle. Conversely, we show that for any quadruple Ω fulfilling these necessary conditions, there exists a point Θ ∈ R 2 for which Ω (Θ) = Ω.