ON SIMULTANEOUS RATIONAL APPROXIMATIONS TO A REAL NUMBER, ITS SQUARE, AND ITS CUBE
Damien Roy · 2012
Abstract. We show that, for any transcendental real number ξ, the uniform exponent of simultaneous approximation of the triple (ξ, ξ 2, ξ 3) by rational numbers with the same denominator is at most (1 + 2γ − √ 1 + 4γ 2)/2 ∼ = 0.4245 where γ = (1 + √ 5)/2 stands for the golden ratio. As a consequence, we get a lower bound on the exponent of approximation of such a number ξ by algebraic integers of degree at most 4. 1.