Measures of algebraic approximation to Markoff extremal numbers
Damien Roy, Dmitrij Zelo · Journal of the London Mathematical Society · 2011
Let ξ be a real number which is neither rational nor quadratic over ℚ. Based on work of Davenport and Schmidt, Bugeaud and Laurent have shown that, for any real number θ, there exist a constant c > 0 and infinitely many non-zero polynomials P ∈ ℤ[T] of degree at most 2 such that |θ−P(ξ)| ⩽ c ‖P‖− γ where γ = ( 1 + 5 ) / 2 denotes for the golden ratio and where the norm ‖P‖ of P stands for the largest absolute value of its coefficients. In the present paper, we show conversely that there exists a class of transcendental numbers ξ for which the above estimates are optimal up to the value of the constant c when one takes θ = R(ξ) for a polynomial R ∈ ℤ[T] of degree d ∈ {3, 4, 5}, but curiously not for degree d = 6, even with θ = 2ξ6.