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Abstract. Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith. 140 (2009), 67-91], approximation exponents ϕ i , ϕ i , (i = 1, 2, 3), attached to points ξ = (ξ 1 , ξ 2 ) in R 2 , give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math. 61 (2009), 165-189] had described all possible quadruples (ϕ 1 , ϕ 1 , ϕ 3 , ϕ 3 ) arising in this way. Our emphasis here will be on ϕ 2 , ϕ 2 and the construction of suitable "3-systems".

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