Intersections of Projections and Slicing Theorems for the Isotropic Grassmannian and the Heisenberg group

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ 2 n , as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ 2 n 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 ⊂ ℝ 2 n with dim B ⩽ m such that for all x ∈ ℝ 2 n \ 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 n th Heisenberg group.

Abstract licence: CC BY

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