On uniform approximation to real numbers

TL;DR

This paper investigates new relationships between various exponents of Diophantine approximation for transcendental real numbers and refines a classical inequality, enhancing understanding of approximation quality.

Contribution

It establishes new relations among Diophantine exponents and improves a longstanding inequality using recent estimates, advancing the theoretical framework.

Findings

New relations between Diophantine exponents $w_n, w_{n}^{ ext{*}}, \, \hat{w}_n, \hat{w}_n^{\text{*}}$

Refinement of the inequality $\hat{w}_n(\xi) \le 2n-1$

Enhanced bounds for approximation exponents

Abstract

Let $n \ge 2$ be an integer and $\xi$ a transcendental real number. We establish several new relations between the values at $\xi$ of the exponents of Diophantine approximation $w_n, w_{n}^{\ast}, \hat{w}_{n}$, and $\hat{w}_{n}^{\ast}$. Combining our results with recent estimates by Schmidt and Summerer allows us to refine the inequality $\hat{w}_{n}(\xi) \le 2n-1$ proved by Davenport and Schmidt in 1969.