On the discrepancy between best and uniform approximation

TL;DR

This paper investigates the difference between two classical Diophantine exponents for transcendental real numbers, providing new bounds and extending previous results through the study of parametric geometry of numbers.

Contribution

The paper unconditionally extends previous bounds on the discrepancy between Diophantine exponents, introducing an additional exponent and refining the understanding of their relationship.

Findings

Established an unconditioned inequality relating the exponents

Extended previous results to a broader setting without additional assumptions

Introduced an additional exponent in the bounds, relevant in specific cases

Abstract

For $\zeta$ a transcendental real number, we consider the classical Diophantine exponents $w_{n}(\zeta)$ and $\widehat{w}_{n}(\zeta)$. They measure how small $| P(\zeta)|$ can be for an integer polynomial $P$ of degree at most $n$ and naive height bounded by $X$, for arbitrarily large and all large $X$, respectively. The discrepancy between the exponents $w_{n}(\zeta)$ and $\widehat{w}_{n}(\zeta)$ has attracted interest recently. Studying parametric geometry of numbers, W. Schmidt and L. Summerer were the first to refine the trivial inequality $w_{n}(\zeta)\geq \widehat{w}_{n}(\zeta)$. Y. Bugeaud and the author found another estimation provided that the condition $w_{n}(\zeta)>w_{n-1}(\zeta)$ holds. In this paper we establish an unconditioned version of the latter result, which can be regarded as a proper extension. Unfortunately, the new contribution involves an additional exponent and is of interest only in certain cases.