Sums of two homogeneous Cantor sets
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- by Yuki Takahashi PDF
- Trans. Amer. Math. Soc. 372 (2019), 1817-1832 Request permission
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.References
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Additional Information
- Yuki Takahashi
- Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan, 5290002, Israel
- MR Author ID: 1165224
- Email: takahashi@math.biu.ac.il
- 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).
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 1817-1832
- MSC (2010): Primary 28A75, 28A80
- DOI: https://doi.org/10.1090/tran/7649
- MathSciNet review: 3976578