标题： 最佳与一致丢番图逼近 显示英文标题

标题： Best versus uniform Diophantine approximatio

摘要： 设 $0<m<n$ 为整数，并令 $K_w$ 表示数域 $K$ 在非平凡位 $w$ 处的完备化。 For each non-zero $\textbf{u}\in K_w^n$, let $\omega_{m-1}(\textbf{u})$ denote the exponent of best approximation to $\textbf{u}$ by vector subspaces of $K_w^n$ of dimension $m$ defined over $K$, and let $\widehat{\omega}_{m-1}(\textbf{u})$ denote the corresponding exponent of uniform approximation. 最后，令 $S_{m,n}$ 表示所有对 $(\widehat{\omega}_{m-1}(\textbf{u}),\omega_{m-1}(\textbf{u}))$ 的集合，其中 $\textbf{u}$ 遍历 $K_w^n$ 中的所有点，且这些点的坐标在 $K$ 上线性无关。 本文中我们利用参数几何数论来研究这个谱集$S_{m,n}$，首先注意到它与选择 $K$和 $w$无关。 因此我们可以假设 $K=\mathbb{Q}$和$K_w=\mathbb{R}$。 在此背景下，Schmidt 和 Summerer 对$S_{1,n}$和$S_{n-1,n}$提出了猜想性描述，这些描述被 Marnat 和 Moshchevitin 对每个$n\ge 2$都予以证实。 我们基于第一作者的博士论文给出了他们结果的一个替代证明，并强调了这两个谱之间的对偶性。 在第一作者的论文中，他将这一猜想推广到了任意一对$(m,n)$，并证明其对$S_{2,4}$也成立。 我们也展示了这一点，但表明这一自然猜想对于$S_{3,5}$不成立。 此外，我们在这里成功计算出的$S_{3,5}$的一部分表明该集合的边界可能非常复杂且未必是半代数的。 我们还对一般二元组$(m,n)$给出了$S_{m,n}$的定性描述。 显示英文摘要

摘要： Let $0<m<n$ be integers, and let $K_w$ denote the completion of a number field $K$ at a non-trivial place $w$. For each non-zero $\textbf{u}\in K_w^n$, let $\omega_{m-1}(\textbf{u})$ denote the exponent of best approximation to $\textbf{u}$ by vector subspaces of $K_w^n$ of dimension $m$ defined over $K$, and let $\widehat{\omega}_{m-1}(\textbf{u})$ denote the corresponding exponent of uniform approximation. Finally, let $S_{m,n}$ denote the set of all pairs $(\widehat{\omega}_{m-1}(\textbf{u}),\omega_{m-1}(\textbf{u}))$ where $\textbf{u}$ runs through all points of $K_w^n$ with linearly independent coordinates over $K$. In this paper we use parametric geometry of numbers to study this spectrum $S_{m,n}$, noting at first that it is independent of the choice of $K$ and $w$. We may thus assume that $K=\mathbb{Q}$ and $K_w=\mathbb{R}$. In this context, Schmidt and Summerer proposed conjectural descriptions for $S_{1,n}$ and $S_{n-1,n}$ which were confirmed by Marnat and Moshchevitin for each $n\ge 2$. We give an alternative proof of their result based on the PhD thesis of the first author, highlighting the duality between the two spectra. In his thesis, the first author generalized the conjecture to any pair $(m,n)$ and proved it to be true also for $S_{2,4}$. We present this as well, but show that this natural conjecture fails for $S_{3,5}$. Moreover, the part of $S_{3,5}$ that we succeed to compute here suggests a complicated boundary for that set, possibly not semialgebraic. We also give a qualitative description of $S_{m,n}$ for a general pair $(m,n)$.