Upper bounds for the uniform simultaneous Diophantine exponents

TL;DR

This paper establishes new upper bounds for the uniform simultaneous Diophantine exponents of transcendental numbers, improving upon previous bounds and relating these exponents to the ordinary exponents in specific cases.

Contribution

It introduces several new upper bounds for the uniform exponents, especially relating them to the ordinary exponents when one is much smaller than the other, with significant improvements in certain generic cases.

Findings

New upper bounds for $\widehat{\lambda}_n(\xi)$ are derived.

In the generic case, the bounds are significantly tighter than previous results.

Improved unconditional bounds for even values of n.

Abstract

We give several upper bounds for the uniform simultaneous Diophantine exponent $\widehat{\lambda}_n(\xi)$ of a transcendental number $\xi\in\mathbb{R}$. The most important one relates $\widehat{\lambda}_n(\xi)$ and the ordinary simultaneous exponent $\omega_k(\xi)$ in the case when $k$ is substantially smaller than $n$. In particular, in the generic case $\omega_k(\xi)=k$ with a properly chosen $k$, the upper bound for $\widehat{\lambda}_n(\xi)$ becomes as small as $\frac{3}{2n} + O(n^{-2})$ which is substantially better than the best currently known unconditional bound of $\frac{2}{n} + O(n^{-2})$. We also improve an unconditional upper bound on $\widehat{\lambda}_n(\xi)$ for even values of $n$.