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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sums of two homogeneous Cantor sets


Author: Yuki Takahashi
Journal: Trans. Amer. Math. Soc. 372 (2019), 1817-1832
MSC (2010): Primary 28A75, 28A80
DOI: https://doi.org/10.1090/tran/7649
Published electronically: March 20, 2019
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Abstract: We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis conjecture. We also consider self-similar sets with overlaps on the real line (not necessarily homogeneous) and show that one can create an interval by applying arbitrary small perturbations if the uniform self-similar measure has $ L^2$-density.


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Additional Information

Yuki Takahashi
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel
Email: takahashi@math.biu.ac.il

DOI: https://doi.org/10.1090/tran/7649
Received by editor(s): November 8, 2017
Received by editor(s) in revised form: May 11, 2018, and May 30, 2018
Published electronically: March 20, 2019
Additional Notes: The author was supported in part by NSF grant DMS-1301515 (PI: A. Gorodetski) and by the Israel Science Foundation grant 396/15 (PI: B. Solomyak).
Article copyright: © Copyright 2019 American Mathematical Society