Upper bounds for the uniform simultaneous Diophantine exponents

We give several upper bounds for the uniform simultaneous Diophantine exponent lambda n(xi)$\widehat{\lambda }_n(\xi )$ of a transcendental number xi is an element of R$\xi \in \mathbb {R}$. The most important one relates lambda n(xi)$\widehat{\lambda }_n(\xi )$ and the ordinary simultaneous exponent omega k(xi)$\omega _k(\xi )$ in the case when k is substantially smaller than n. In particular, in the generic case omega k(xi)=k$\omega _k(\xi )=k$ with a properly chosen k, the upper bound for lambda n(xi)$\widehat{\lambda }_n(\xi )$ becomes as small as 32n+O(n-2)$\frac{3}{2n} + O(n<^>{-2})$ which is substantially better than the best currently known unconditional bound of 2n+O(n-2)$\frac{2}{n} + O(n<^>{-2})$. We also improve an unconditional upper bound on lambda n(xi)$\widehat{\lambda }_n(\xi )$ for even values of n.