On the approximation exponents for subspaces of ℝn

Abstract

This paper follows the generalisation of the classical theory of Diophantine approximation to subspaces of $\mathbb{R}^n$ established by W. M. Schmidt in 1967. Let $A$ and $B$ be two subspaces of $\mathbb{R}^n$ of respective dimensions $d$ and $e$ with $d+e\leqslant n$. The proximity between $A$ and $B$ is measured by $t=\min(d,e)$ canonical angles $0\leqslant \theta_1\leqslant \cdots\leqslant \theta_t\leqslant \pi/2$; we set $\psi_j(A,B)=\sin\theta_j$. If $B$ is a rational subspace, his complexity is measured by its height $H(B)=\mathrm{covol}(B\cap\mathbb{Z}^n)$. We denote by $\mu_n(A\vert e)_j$ the exponent of approximation defined as the upper bound (possibly equal to $+\infty$) of the set of $\beta>0$ such that the inequality $\psi_j(A,B)\leqslant H(B)^{-\beta}$ holds for infinitely many rational subspaces $B$ of dimension $e$. We are interested in the minimal value $\mathring{\mu}_n(d\vert e)_j$ taken by $\mu_n(A\vert e)_j$ when $A$ ranges through the set of subspaces of dimension $d$ of $\mathbb{R}^n$ such that for all rational subspaces $B$ of dimension $e$ one has $\dim (A\cap B)

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关于子空间的近似指数ℝn

本文推广了W.M.Schmidt于1967年建立的对$\mathbb｛R｝^n$子空间的丢番图近似的经典理论。设$A$和$B$是分别具有$d$和$e$维度的$\mathbb｛R｝^n$的两个子空间，具有$d+e\leqslant n$。$A$和$B$之间的接近度通过$t=\min（d，e）$标准角$0\leqslant\theta_1\leqslant\cdots\leqslant_theta\t\leqslant\pi/2$来测量；我们设置$\psi_j（A，B）=\sin\theta_j$。如果$B$是有理子空间，则其复杂度由其高度$H（B）=\mathrm｛covol｝（B\cap\mathbb｛Z｝^n）$来度量。我们用$\mu_n（A\vert e）_j$表示近似指数，该指数定义为$\beta>0$集合的上界（可能等于$+\infty$），使得不等式$\psi_j（A，B）\leqslant H（B）^｛-\beta}$对于维数$e$的无穷多有理子空间$B$成立。当$A$的范围通过$\mathbb｛R｝^n$的维度$d$的子空间集时，$\mathring｛\mu｝_n（d\vert e）_j$所取的最小值$\mathring｛\mu_n（A\vert e）_j$$$e，使得对于维度$e$的所有有理子空间$B$，一个具有$\dim（A\cap B）＜j$。我们证明了$\mathring｛\mu｝_4（2\vert2）_1=3$，$\mathring｛\mu｝_5（3\vert2）_1 \le 6$和$\mathering｛\ mu｝_｛2d｝（d\vert\ell）_1\le qslant 2d^2/（2d-\ell）$。我们还证明了一般情况下的下界，这意味着$\mathring｛\mu｝_n（d\vert d）_d\xrightarrow[n\to+\infty]｛｝1/d$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。