We generalize the construction of Roy's Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation ω2(ξ), ω2(ξ), λ2(ξ), λ2(ξ) associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new information on the joint spectra of the above exponents as well as a new family of numbers for which it is possible to construct the sequence of the best rational approximations.

Let Z be a quadratic hypersurface of double-struckPnfalse(double-struckRfalse) defined over double-struckQ containing points whose coordinates are linearly independent over double-struckQ. We show that, among these points, the largest exponent of uniform rational approximation is the inverse 1/ρ of an explicit Pisot number ρ<2 depending only on n if the Witt index (over double-struckQ) of the quadratic form q defining Z is at most 1, and that it is equal to 1 otherwise. Furthermore, there are points of Z which realize this maximum. They constitute a countably infinite set in the first case, and an uncountable set in the second case. The proof for the upper bound 1/ρ uses a recent transference inequality of Marnat and Moshchevitin. In the case n=2, we recover results of the second author while for n>2, this completes recent work of Kleinbock and Moshchevitin.