Approximation to real numbers by cubic algebraic integers. II

It has been conjectured for some time that, for any integer n > 2, any real number e > 0 and any transcendental real number ξ, there would exist infinitely many algebraic integers a of degree at most n with the property that |ξ-α| ≤ H(α) -n+e , where H(α) denotes the height of a. Although this is true for n = 2, we show here that, for n = 3, the optimal exponent of approximation is not 3 but (3 + √5)/2 ≃ 2.618.