2023
DOI: 10.1112/blms.12844
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A note on Kakeya sets of horizontal and SL(2)$SL(2)$ lines

Abstract: We consider unions of lines in . These are lines of the form where . We show that if is a Kakeya set of lines, then the union has Hausdorff dimension 3. This answers a question of Wang and Zahl. The 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 , combined with r… Show more

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