University of Oulu

Wu, M. (2019). A proof of Furstenberg’s conjecture on the intersections of ⨯p- and ⨯q-invariant sets. Annals of Mathematics, 189(3), 707-751. Retrieved from

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)
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Language: English
Published: Mathematics Department, Princeton University, 2019
Publish Date: 2019-10-15


We 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.

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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
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
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.