SIMULTANEOUS RATIONAL APPROXIMATION TO THE SUCCESSIVE POWERS OF A REAL NUMBER by Michel LAURENT

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 >0 and arbitrary large real numbers X such that the system of linear inequalities |x0|≤X and |x0θ−xj |≤ X−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 θ by algebraic integers of degree n+1 over Q obtained by these authors.