2022/10/02 by Katrin Fässler, Fässler, Katrin, Tuomas Orponen +1 · 1 citation
Mathematics · #28A78 #28A80 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2210.00458
openalex publication_date 2022/10/02 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Let \πe \colon ℍ → \mathbbWe : e ∈ S1\ be the family of vertical projections in the first Heisenberg group ℍ. We prove that if K ⊂ ℍ is a Borel set with Hausdorff dimension dimℍ K ∈ [0,2] ∪ \3\, then dimℍ πe(K) ≥ dimℍ K for H1 almost every e ∈ S1. This was known earlier if dimℍ K ∈ [0,1]. The proofs for dimℍ K ∈ [0,2] and dimℍ K = 3 are based on different techniques. For dimℍ K ∈ [0,2], we reduce matters to a Euclidean problem, and apply the method of cinematic functions due to Pramanik, Yang, and Zahl. To handle the case dimℍ K = 3, we introduce a point-line duality between horizontal lines and conical lines in ℝ3. This allows us to transform the Heisenberg problem into a point-plate incidence question in ℝ3. To solve the latter, we apply a Kakeya inequality for plates in ℝ3, due to Guth, Wang, and Zhang. This method also yields partial results for Borel sets K ⊂ ℍ with dimℍ K ∈ (5/2,3).