2002/10/24 by Damien Roy
Mathematics · #math.NT #msc:11J04 #msc:11J82
published as Annals of Mathematics 158 (2003), 1081-1087. · 7 pages; major simplification of the original proof
arxiv created 2002/10/24 · arxiv updated 2009/11/30
It has been conjectured for some time that, for any integer n≥ 2, any real number ε>0 and any transcendental real number ξ, there would exist infinitely many algebraic integers αof degree at most n with the property that |ξ-α| < H(α)-n+ε, where H(α) denotes the height of α. 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...