Simultaneous rational approximation to the successive powers of a real number

Abstract

Let n be an integer ≥ 1 and let θ be a real number which is not an algebraic number of degree ≤ [n/2]. We show that there exist ϵ > 0 and arbitrary large real numbers X such that the system of linear inequalities |x0| ≤ X and |x0θj − xj| ≤ ϵX−1/[n/2] for 1 < j < n, has only the zero solution in rational integers x0,…, xn. This result refines a similar statement due to H. Davenport and W. M. Schmidt, where the upper integer part [n/2] is replaced everywhere by the integer part [n/2]. As a corollary, we improve slightly the exponent of approximation to 0 by algebraic integers of degree n + 1 over Q obtained by these authors.