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 dimHK∈[0,2]∪{3}, thendimHπe(K)≥dimHK for H1 almost every e∈S1. This was known earlier if dimHK∈[0,1].The proofs for dimHK∈[0,2] and dimHK=3 are based on different techniques. For dimHK∈[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 dimHK=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 dimHK∈(5/2,3).
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