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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Hausdorff dimension of microsets
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by Jonathan M. Fraser, Douglas C. Howroyd, Antti Käenmäki and Han Yu PDF
Proc. Amer. Math. Soc. 147 (2019), 4921-4936 Request permission

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 $\mathcal {F}_\sigma$ set $\Delta \subseteq [0,d]$ 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 $\Delta$. Our work is motivated by the general programme of determining what geometric information about a set can be determined at the level of tangents.
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Additional Information
  • Jonathan M. Fraser
  • Affiliation: School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, United Kingdom
  • MR Author ID: 946983
  • Email: jmf32@st-andrews.ac.uk
  • Douglas C. Howroyd
  • Affiliation: School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, United Kingdom
  • MR Author ID: 1213286
  • Email: dch8@st-andrews.ac.uk
  • Antti Käenmäki
  • Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
  • Email: antti.kaenmaki@uef.fi
  • Han Yu
  • Affiliation: School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, United Kingdom
  • MR Author ID: 1223262
  • Email: hy25@st-andrews.ac.uk
  • Received by editor(s): September 27, 2018
  • Received by editor(s) in revised form: October 28, 2018, October 29, 2018, and February 27, 2019
  • Published electronically: June 10, 2019
  • Additional Notes: The first author was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1).
    The second author was financially supported by an EPSRC Doctoral Training Grant (EP/N509759/1).
    The third author was financially supported by the Finnish Center of Excellence in Analysis and Dynamics Research, the Finnish Academy of Science and Letters, and the Väisälä Foundation.
    The fourth author was financially supported by the University of St Andrews.
  • Communicated by: Jeremy Tyson
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4921-4936
  • MSC (2010): Primary 28A80; Secondary 28A78
  • DOI: https://doi.org/10.1090/proc/14613
  • MathSciNet review: 4011524