On uniform polynomial approximation

Abstract

Let $n$ be a positive integer and $\hat{I}¾$ a transcendental real number. We are interested in bounding from above the uniform exponent of polynomial approximation $\hat{\ddot{I}}{‰}_n(\hat{I}¾)$. Davenport and Schmidt's original 1969 inequality $\hat{\ddot{I}}{‰}_n(\hat{I}¾)\le 2n-1$ was improved recently, and the best upper bound known to date is $2n-2$ for each $n\ge 10$. In this paper, we develop new techniques leading us to the improved upper bound $2n-\frac{1}{3}{n}^{1/3}+\mathcal{O}(1)$.

41 pages, 1 figure