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}, thendimH⁡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).