Upper bounds for the uniform simultaneous Diophantine exponents

Abstract

We give several upper bounds for the uniform simultaneous Diophantine exponent $\widehatλ_n(ξ)$ of a transcendental number $ξ\in\mathbb{R}$. The most important one relates $\widehatλ_n(ξ)$ and the ordinary simultaneous exponent $ω_k(ξ)$ in the case when $k$ is substantially smaller than $n$. In particular, in the generic case $ω_k(ξ)=k$ with a properly chosen $k$, the upper bound for $\widehatλ_n(ξ)$ 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 $\widehatλ_n(ξ)$ for even values of $n$.