This was a long-standing question since 90s whether one-point intersection property for a self-similar set implies open set condition (OSC). We answer this question negatively. We give an example of a totally disconnected self-similar set K ⊂ ℝ which does not have OSC and has minimal overlap of its pieces, that is, all intersections of its pieces Ki∩Kj, i≠j are empty except only one, which is a single point.

Original languageEnglish
Pages (from-to)408-416
Number of pages9
Issue number1
Publication statusPublished - 1 Jan 2020


  • general position theorem
  • Hausdorff dimension
  • open set condition
  • self-similar set
  • weak separation property

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