On uniform approximation to real numbers

Abstract

Let $n\ge 2$ be an integer and $\xi$ a transcendental real number. We establish several new relations between the values at $\xi$ of the exponents of Diophantine approximation $w_n,w_n^{\ast },{\hat{w}}_n$, and ${\hat{w}}_n^{\ast }$. Combining our results with recent estimates by Schmidt and Summerer allows us to refine the inequality ${\hat{w}}_n(\xi )\le 2n-1$ proved by Davenport and Schmidt in 1969.

15 pages. Former version of Theorems 2.2, 2.4 improved