intersections of projections and slicing theorems for the isotropic grassmannian and the heisenberg group

• DOI: 10.48550/arxiv.1907.07218
• Publication Date: 2020-01-01
• Authors: Fernando Román-Ga...
• Keywords: QA299.6-433; orthogonal projections; isotropic grassmanian; Metric Geometry (math.MG); 01 natural sciences; Mathematics - Metric Geometry; Mathematics - Classical Analysis and ODEs; intersection of planes and sets; Classical Analysis and ODEs (math.CA); FOS: Mathematics; 28a75; 28A75; 0101 mathematics; Analysis; integral geometry

AbstractThis 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 ifAandBare Borel subsets of ℝ2nof dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images ofAandBunder orthogonal projections onto these planes have positive Hausdorffm-measure. In addition, ifAis a measurable set of Hausdorff dimension greater thanm, then there is a setB⊂ ℝ2nwith dimB⩽msuch that for allx∈ ℝ2n\Bthere is a positive measure set of isotropic m-planes for which the translate byxof the orthogonal complement of each such plane, intersectsAon a set of dimension dimA – m. These results are then applied to obtain analogous results on thenthHeisenberg group.