On uniform approximation to successive powers of a real number

Abstract

We establish new inequalities involving classical exponents of Diophantine approximation. This allows for improving on the work of H. Davenport, W.M. Schmidt and M. Laurent concerning the maximum value of the exponent λ̂n(ζ) among all real transcendental ζ. In particular we refine the estimation λ̂n(ζ)≤⌈n/2⌉−1 due to M. Laurent by λ̂n(ζ)≤ŵ⌈n/2⌉(ζ)−1 for all n≥1, and for even n we replace the bound 2/n for λ̂n(ζ) first found by Davenport and Schmidt by roughly 2n−4n3, which provides the currently best known bound when n≥6.