A proof of Furstenbergโ€™s conjecture on the intersections of ร—๐‘- and ร—๐‘ž-invariant sets

Meng Wu Department of Mathematical Sciences, University of Oulu, Oulu, Finland

TBD mathscidoc:2203.43023

Annals of Mathematics, 189, 707-751, 2019.5
We prove the following conjecture of Furstenberg (1969): if ๐ด,๐ตโŠ‚[0,1] are closed and invariant under ร—๐‘ mod1 and ร—๐‘ž mod1, respectively, and if log๐‘/log๐‘žโˆ‰โ„š, then for all real numbers ๐‘ข and ๐‘ฃ, dim_H (๐‘ข๐ด + ๐‘ฃ) โˆฉ ๐ต โ‰ค max {0, dim_H ๐ด + dim_H ๐ต โˆ’ 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on โ„. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.
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@inproceedings{meng2019a,
  title={A proof of Furstenbergโ€™s conjecture on the intersections of ร—๐‘- and ร—๐‘ž-invariant sets},
  author={Meng Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316102010723057970},
  booktitle={Annals of Mathematics},
  volume={189},
  pages={707-751},
  year={2019},
}
Meng Wu. A proof of Furstenbergโ€™s conjecture on the intersections of ร—๐‘- and ร—๐‘ž-invariant sets. 2019. Vol. 189. In Annals of Mathematics. pp.707-751. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316102010723057970.
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