On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane
Abstract
Let 0 s 1 and 0 t 2. An .s;t/-Furstenberg set is a set K R2 with the following property: there exists a line set L of Hausdorff dimension dimH L t such that dimH.K \ `/ s for all ` 2 L. We prove that for s 2 .0;1/ and t 2 .s;2, the Hausdorff dimension of .s;t/-Furstenberg sets in R2 is no smaller than 2s C , where >0 depends only on s and t. For s > 1=2 and t D 1, this is an -improvement over a result of Wolff from 1999. The same method also yields an -improvement to Kaufman’s projection theorem from 1968. We show that if s 2 .0;1/, t 2 .s;2, and K R2 is an analytic set with dimH K D t, then dimH ® e 2 S1 W dimH e.K/ s ¯ s ; where >0 depends only on s and t. Here e is the orthogonal projection to the line in direction e.
Main Authors
Format
Articles
Research article
Published
2023
Series
Subjects
Publication in research information system
Publisher
Duke University Press
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202403052274Use this for linking
Review status
Peer reviewed
ISSN
0012-7094
DOI
https://doi.org/10.1215/00127094-2022-0103
Language
English
Published in
Duke Mathematical Journal
Citation
- Orponen, T., & Shmerkin, P. (2023). On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane. Duke Mathematical Journal, 172(18), 3559-3632. https://doi.org/10.1215/00127094-2022-0103
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Additional information about funding
Orponen’s work was partially supported by Academy of Finland grants 309365, 314172, and 321896 via the projects “Quantitative rectifiability in Euclidean and nonEuclidean spaces” and “Incidences on fractals.” Shmerkin’s work was partially supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) discovery grant.
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