On uniform polynomial approximation

Abstract

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