Markoff-Lagrange spectrum and extremal numbers

Abstract

Let gamma denote the golden ratio. H. Davenport and W. M.Schmidt showed in 1969 that, for each non-quadratic irrational real number xi, there exists a constant c>0 with the property that, for arbitrarily large values of X, the inequalities |x_0| \le X, |x_0*xi - x_1| \le cX^{-1/gamma}, |x_0*xi^2 - x_2| \le cX^{-1/gamma} admit no non-zero integer solution (x_0,x_1,x_2). Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers xi such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X\ge 1. Such extremal numbers are transcendental and their set is stable under the action of GL_2(Z) by linear fractional transformations. In this paper, it is shown that there exists extremal numbers xi for which the Lagrange constant is 1/3, the largest possible value for a non-quadratic number, and that there is a natural bijection between the GL_2(Z)-equivalence classes of such numbers and the non-trivial solutions of Markoff's equation.