2022/10/18 by Katrin Fässler, Fässler, Katrin, Tuomas Orponen +1 · 2 citations
Mathematics · #28A78 #28A80 #Advanced Algebra and Geometry #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2210.09955
openalex publication_date 2022/10/18 · openalex created_date 2022/10/21 · openalex updated_date 2026/07/31
We consider unions of SL(2) lines in ℝ3. These are lines of the form L = (a,b,0) + span(c,d,1), where ad - bc = 1. We show that if L is a Kakeya set of SL(2) lines, then the union ∪ L has Hausdorff dimension 3. This answers a question of Wang and Zahl. The SL(2) lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point-line duality principle between horizontal and conical lines in ℝ3, combined with recent work on restricted families of projections to planes, due to Gan, Guo, Guth, Harris, Maldague, and Wang. Our result also has a corollary for Nikodym sets associated with horizontal lines, which answers a special case of a question of Kim.