To any k-dimensional subspace of $\mathbb {Q}^n$ one can naturally associate a point in the Grassmannian $\mathrm {Gr}_{n,k}(\mathbb {R})$ and two shapes of lattices of rank k and $n-k$ , respectively. These lattices originate by intersecting the k-dimensional subspace and its orthogonal with the lattice $\mathbb {Z}^n$ . Using unipotent dynamics, we prove simultaneous equidistribution of all of these objects under congruence conditions when $(k,n) \neq (2,4)$ .

中文翻译：

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有理子空间及其形状的均匀分布

去任何k的维子空间 $\mathbb {Q}^n$ 人们可以很自然地将格拉斯曼方程中的一个点联系起来 $\mathrm {Gr}_{n,k}(\mathbb {R})$ 和两种形状的等级格子k和 $nk$ ， 分别。这些晶格起源于相交k维子空间及其与格子的正交 $\mathbb {Z}^n$ 。使用单能动力学，我们证明了在全等条件下所有这些对象的同时均分布 $(k,n) \neq (2,4)$ 。