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On the Hausdorff dimension of microsets


Authors: Jonathan M. Fraser, Douglas C. Howroyd, Antti Käenmäki and Han Yu
Journal: Proc. Amer. Math. Soc. 147 (2019), 4921-4936
MSC (2010): Primary 28A80; Secondary 28A78
DOI: https://doi.org/10.1090/proc/14613
Published electronically: June 10, 2019
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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 $ \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
Email: jmf32@st-andrews.ac.uk

Douglas C. Howroyd
Affiliation: School of Mathematics & Statistics, University of St Andrews, St Andrews, KY16 9SS, United Kingdom
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
Email: hy25@st-andrews.ac.uk

DOI: https://doi.org/10.1090/proc/14613
Keywords: Weak tangent, microset, Hausdorff dimension, Assouad type dimensions
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
Article copyright: © Copyright 2019 American Mathematical Society