Vertical projections in the Heisenberg group via cinematic functions and point-plate incidences

Abstract

Let {πe : H → We : 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).