2024/07/16 by Terence L. J. Harris, Harris, Terence L. J.
Mathematics · Physics and Astronomy · #28A78 #28A80 #Advanced Differential Geometry Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Morphological variations and asymmetry #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2407.11475
openalex publication_date 2024/07/16 · openalex created_date 2024/10/26 · openalex updated_date 2026/07/29
It is shown that if A is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying 2< dim A < 3, then the packing dimensions of vertical projections of A are almost surely not less than dim A, where both packing and Hausdorff dimensions are defined with respect to the Korányi metric. For the Hausdorff dimension of the projections, a weaker almost sure lower bound is obtained which improves the known bound in the range 2 < dim A < (1)/(8)( 17 + √(33)) ≈ 2.84. The bound is slightly larger than 1+(1)/(2) dim A and behaves similarly near dim A =2. Both proofs rely on a variable coefficient local smoothing inequality.