On uniform approximation to successive powers of a real number

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Contribution

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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 $\hat{\lambda}_{n}(\zeta)\leq \lceil n/2\rceil^{-1}$ due to M. Laurent by $\hat{\lambda}_{n}(\zeta)\leq \hat{w}_{\lceil n/2\rceil}(\zeta)^{-1}$ for all $n\geq 1$, 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 $n\geq 6$.