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.
Categories: math.NT