A note on Kakeya sets of horizontal and SL(2) lines
Fässler, K., & Orponen, T. (2023). A note on Kakeya sets of horizontal and SL(2) lines. Bulletin of the London Mathematical Society, 55(5), 2195-2204. https://doi.org/10.1112/blms.12844
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Bulletin of the London Mathematical SocietyDate
2023Copyright
© 2023 The Author(s). The Bulletin of the London Mathematical Society is copyright © London Mathematical Society.
We consider unions of S L (2) lines in R3. These are lines of the form L=(a,b,0)+span(c,d,1) where ab-cd=1. We show that if L is a Kakeya set of S L (2) lines, then the union U L has Hausdorff dimension 3. This answers a question of Wang and Zahl. The S L (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.
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Wiley-BlackwellISSN Search the Publication Forum
0024-6093Keywords
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https://converis.jyu.fi/converis/portal/detail/Publication/183150025
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Research Council of FinlandFunding program(s)
Academy Research Fellow, AoFAdditional information about funding
Academy of Finland, Grant/AwardNumbers: 321696, 321896License
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