On uniform approximation to real numbers
Volume 175 / 2016
Acta Arithmetica 175 (2016), 255-268
MSC: Primary 11J04; Secondary 11J13, 11J82.
DOI: 10.4064/aa8372-7-2016
Published online: 23 September 2016
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 $w_n$, $w_{n}^{\ast}$, $\widehat{w}_{n}$, and $\widehat{w}_{n}^{\ast}$. Combining our results with recent estimates by Schmidt and Summerer allows us to refine the inequality $\widehat{w}_{n}(\xi) \le 2n-1$ proved by Davenport and Schmidt in 1969.
