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A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets |
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| Author: | Wu, Meng1 |
| Organizations: |
1Department of Mathematical Sciences, University of Oulu, Oulu, Finland |
| Format: | article |
| Version: | accepted version |
| Access: | open |
| Online Access: | PDF Full Text (PDF, 0.4 MB) |
| Persistent link: | http://urn.fi/urn:nbn:fi-fe2019101532732 |
| Language: | English |
| Published: |
Mathematics Department, Princeton University,
2019
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| Publish Date: | 2019-10-15 |
| Description: |
AbstractWe prove the following conjecture of Furstenberg (1969): if A, B ⊂ [0, 1] are closed and invariant under ⨯p mod 1 and ⨯q mod 1, respectively, and if log p/log q ∉ ℚ, then for all real numbers u and v, dimH(uA + v) ∩ B ≤ max{0, dimH A + dimH B − 1}. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on ℝ. 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. see all
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| Series: |
Annals of mathematics |
| ISSN: | 0003-486X |
| ISSN-E: | 1939-8980 |
| ISSN-L: | 0003-486X |
| Volume: | 189 |
| Issue: | 3 |
| Pages: | 707 - 751 |
| DOI: | 10.4007/annals.2019.189.3.2 |
| OADOI: | https://oadoi.org/10.4007/annals.2019.189.3.2 |
| Type of Publication: |
A1 Journal article – refereed |
| Field of Science: |
111 Mathematics |
| Subjects: | |
| Funding: |
We acknowledge the postdoc fellowships supported by Academy of Finland (Centre of Excellence in Analysis and Dynamics Research) and ERC grant 306496. |
| Copyright information: |
© 2019 Department of Mathematics, Princeton University. Published in this repository with the kind permission of the publisher. |