Sums of two homogeneous cantor sets

Yuki Takahashi

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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 L2-density.

Original languageEnglish
Pages (from-to)1817-1832
Number of pages16
JournalTransactions of the American Mathematical Society
Issue number3
Publication statusPublished - 2019 Aug 1

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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