Uniform Diophantine approximation and best approximation polynomials

TL;DR

This paper introduces a new method using parametric geometry of numbers to bound the uniform approximation exponent of transcendental numbers, improving existing bounds unconditionally and providing stronger conditional bounds.

Contribution

The authors develop a novel approach based on parametric geometry of numbers to derive new bounds for uniform polynomial approximation exponents, both unconditionally and conditionally.

Findings

Unconditional bound:  \u2264 2n-2+o(1) for large n

Improved bound over previous results by Bugeaud and the author

Conditional bounds suggest the exponent is smaller than current upper bounds

Abstract

Let $\zeta$ be a real transcendental number. We introduce a new method to find upper bounds for the classical exponent $\widehat{w}_{n}(\zeta)$ concerning uniform polynomial approximation. Our method is based on the parametric geometry of numbers introduced by Schmidt and Summerer, and transference of the original approximation problem in dimension $n$ to suitable higher dimensions. For large $n$, we can provide an unconditional bound of order $\widehat{w}_{n}(\zeta)\leq 2n-2+o(1)$. While this improves the bound of order $2n-\frac{3}{2}+o(1)$ due to Bugeaud and the author, it is unfortunately slightly weaker than what can be obtained when incorporating a recently proved conjecture of Schmidt and Summerer. However, the method also enables us to establish significantly stronger conditional bound upon a certain presumably weak assumption on the structure of the best approximation polynomials. Thereby we provide serious evidence that the exponent should be reasonably smaller than the known upper bounds.