arXiv 1911.032152019-11math.NT

关于数的几何与Veronese曲线的一致有理逼近

On geometry of numbers and uniform rational approximation to the Veronese curve

Johannes Schleischitz

备注：: 18 pages. Some inaccuracies in proof of Thm 1.2 corrected

摘要：

考虑$\mathbb{R}^{n}$中有理同时逼近一点的经典问题。Marnat和Moshchevitin在2018年建立了诱导的普通和一致逼近指数之间的差距的最佳下限。最近，阮，Poels和罗伊提供的信息，最好的逼近合理的向量点的差距是接近这个极小值。结合后一个结果和参数数的几何，我们有效地限制了对偶线性形式指数在所描述的情况。作为应用，我们对均匀丢番图逼近$\widehatλ_{n}(ξ)$的经典指数的上界作了改进，对于偶数的$n\geq 4$ .不幸的是，我们的改进很小，对于$n=4$只有第五位十进制数。然而，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。我们确实建立了合理的更强的结果，几乎满足平等的估计，由Marnat和Moshchevitin。我们的结论与后果的经典问题的逼近实数的代数数/整数的一致有界度。

英文摘要：

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 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 $\widehatλ_{n}(ξ)$, for even $n\geq 4$. 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.

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摘要：

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