Abstract This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m-measure. In addition, if A is a measurable set of Hausdorff dimension greater than m, then there is a set B ⊂ ℝ2n with dim B ⩽ m such that for all x ∈ ℝ2n\B there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dim A – m. These results are then applied to obtain analogous results on the nth Heisenberg group.

中文翻译：

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各向同性格拉斯曼和海森堡群的投影和切片定理的交集

摘要 本文研究了ℝ2n 的子集的各向同性投影的交集的豪斯多夫维数，以及集合与各向同性平面的交集的维数。结果表明，如果A和B是维数大于m的ℝ2n的Borel子集，那么对于各向同性m平面的正测度集，A和B在这些平面上正交投影下的图像的交集具有正Hausdorff m -措施。此外，如果 A 是一个大于 m 的 Hausdorff 维的可测集合，那么存在一个集合 B ⊂ ℝ2n，其中暗 B ⩽ m 使得对于所有 x ∈ ℝ2n\B 有一个正的各向同性 m 平面测度集：每个这样的平面的正交补的 x 平移，与 A 在一组维度 dim A – m 上相交。然后将这些结果应用于第 n 个海森堡群的类似结果。