Consider the classical problem of rational simultaneous approximation to a point in \({\mathbb {R}}^{n}\). The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to the points where the gap is close to this minimal value. Combining the latter result with parametric geometry of numbers, we effectively bound the dual linear form exponents in the described situation. As an application, we slightly improve the upper bound for the classical exponent of uniform Diophantine approximation \({\widehat{\lambda }}_{n}(\xi )\), for even \(n\ge 4\). Unfortunately our improvements are small, for \(n=4\) only in the fifth decimal digit. However, the underlying method in principle can be improved with more effort to provide better bounds. We indeed establish reasonably stronger results for numbers which almost satisfy equality in the estimate by Marnat and Moshchevitin. We conclude with consequences on the classical problem of approximation to real numbers by algebraic numbers/integers of uniformly bounded degree.

考虑有理同时逼近\({\mathbb {R}}^{n}\) 中的一个点的经典问题。Marnat 和 Moshchevitin 在 2018 年建立了诱导普通近似指数和均匀近似指数之间差距的最佳下界。最近 Nguyen、Poels 和 Roy 提供了关于差距接近这个最小值的点的最佳近似有理向量的信息价值。将后一个结果与数字的参数几何相结合，我们在所描述的情况下有效地限制了对偶线性形式的指数。作为一个应用，我们稍微改进了均匀丢番图近似\({\widehat{\lambda }}_{n}(\xi )\)的经典指数的上限，即使是\(n\ge 4\). 不幸的是，我们的改进很小，因为\(n=4\)仅在第五位十进制数字中。但是，原则上可以通过更多的努力来改进底层方法以提供更好的边界。对于几乎满足 Marnat 和 Moshchevitin 估计中的相等性的数字，我们确实建立了相当强的结果。我们以代数数/均匀有界度整数逼近实数的经典问题作为结论。