On uniform polynomial approximation

TL;DR

This paper improves the upper bounds on the uniform polynomial approximation exponent for transcendental real numbers, introducing new techniques that tighten previous bounds from 2n-2 to approximately 2n minus a fractional correction.

Contribution

The authors develop novel methods to refine the upper bounds on the uniform polynomial approximation exponent, surpassing previous results for all sufficiently large n.

Findings

Established an improved upper bound of approximately 2n - (1/3)n^{1/3} + O(1).

Extended the understanding of polynomial approximation exponents for transcendental numbers.

Provided new techniques that could influence future research in Diophantine 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 $2n-\frac{1}{3}n^{1/3}+\mathcal{O}(1)$.