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 Davenport, Schmidt and Laurent concerning the maximum value of the exponent ${\hat{\lambda }}_n(\zeta )$ among all real transcendental $\zeta$. In particular we refine the estimation ${\hat{\lambda }}_n(\zeta )\le \lceil n/2{\rceil }^{-1}$ due to M. Laurent by ${\hat{\lambda }}_n(\zeta )\le {\hat{w}}_{\lceil n/2\rceil }(\zeta {)}^{-1}$ for all $n\ge 1$, and for even $n$ we replace the bound $2/n$ for ${\hat{\lambda }}_n(\zeta )$ first found by Davenport and Schmidt by roughly $\frac{2}{n}-\frac{1}{2n^3}$, which provides the currently best known bounds when $n\ge 6$.

arXiv comment18 pages; Two minor inaccuracies in the proof of Lemma 3.2 were corrected (false in printed version!). (66) was slightly changed, and in the last highlighted equuality the index should be m instead of n