Approximation to real numbers by cubic algebraic integers II

It has been conjectured for some time that, for any integer n\ge 2, any real number \epsilon >0 and any transcendental real number \xi, there would exist infinitely many algebraic integers \alpha of degree at most n with the property that |\xi-\alpha| < H(\alpha)^{-n+\epsilon}, where H(\alpha) denotes the height of \alpha. Although this is true for n=2, we show here that, for n=3, the optimal exponent of approximation is not 3 but (3+\sqrt{5})/2 = 2.618…