A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets

Wu, Meng (2019-05-01)
 
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https://doi.org/10.4007/annals.2019.189.3.2

Wu, Meng
Mathematics Department
01.05.2019

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 https://www.jstor.org/stable/10.4007/annals.2019.189.3.2

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© 2019 Department of Mathematics, Princeton University. Published in this repository with the kind permission of the publisher.
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doi:https://doi.org/10.4007/annals.2019.189.3.2
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Abstract

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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