Geometry of planar curves intersecting many lines at a few points
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- by D. Vardakis and A. Volberg
- St. Petersburg Math. J. 33 (2022), 1047-1062
- DOI: https://doi.org/10.1090/spmj/1742
- Published electronically: October 31, 2022
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Abstract:
The local Lipschitz property is shown for the graphs avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand’s well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is $\sigma$-finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each “piece” is found.References
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Bibliographic Information
- D. Vardakis
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
- Email: jimvardakis@gmail.com
- A. Volberg
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, MI. 48823; and Hausdorff Center for Mathematics, Bonn, Germany
- Email: volberg@msu.edu
- Received by editor(s): March 15, 2021
- Published electronically: October 31, 2022
- © Copyright 2022 American Mathematical Society
- Journal: St. Petersburg Math. J. 33 (2022), 1047-1062
- MSC (2020): Primary 42B20, 42B35, 47A30
- DOI: https://doi.org/10.1090/spmj/1742