A proof of Furstenberg's conjecture on the intersections of ⨯p and ⨯qinvariant sets 

Author:  Wu, Meng^{1} 
Organizations: 
^{1}Department 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:fife2019101532732 
Language:  English 
Published: 
Mathematics Department, Princeton University,
2019

Publish Date:  20191015 
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, dim_{H}(uA + v) ∩ B ≤ max{0, dim_{H} A + dim_{H} B − 1}. We obtain this result as a consequence of our study on the intersections of incommensurable selfsimilar sets on ℝ. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar selfsimilar sets satisfying SSC and certain natural irreducible conditions. see all

Series: 
Annals of mathematics 
ISSN:  0003486X 
ISSNE:  19398980 
ISSNL:  0003486X 
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. 