On the Hausdorff dimension of microsets

Jonathan MacDonald Fraser, Douglas Charles Howroyd, Antti Käenmäki, Han Yu

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Abstract

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can analogously be obtained as the minimal dimension of a microset. In particular, the maximum and minimum exist. We also show that for an arbitrary Fσ set ∆ ⊆ [0, d] containing its infimum and supremum there is a compact set in [0,1]d for which the set of Hausdorff dimensions attained by its microsets is exactly equal to the set ∆. Our work is motivated by the general programme of determining what geometric information about a set can be determined at the level of tangents.
Original languageEnglish
Pages (from-to)4921-4936
Number of pages16
JournalProceedings of the American Mathematical Society
Volume147
Issue number11
Early online date10 Jun 2019
DOIs
Publication statusPublished - Nov 2019

Keywords

  • Weak tangent
  • Microset
  • Hausdorff dimension
  • Assouad type dimensions

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