Upper bounds for the uniform simultaneous Diophantine exponents

Abstract

We give several upper bounds for the uniform simultaneous Diophantine exponent ${\hat{\lambda }}_n(\xi )$ of a transcendental number $\xi \in \mathbb{R}$. The most important one relates ${\hat{\lambda }}_n(\xi )$ 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 ${\hat{\lambda }}_n(\xi )$ 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 ${\hat{\lambda }}_n(\xi )$ for even values of $n$.