On the Hausdorff dimension of microsets
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.
Citation
Fraser , J M , Howroyd , D C , Käenmäki , A & Yu , H 2019 , ' On the Hausdorff dimension of microsets ' , Proceedings of the American Mathematical Society , vol. 147 , no. 11 , pp. 4921-4936 . https://doi.org/10.1090/proc/14613
Publication
Proceedings of the American Mathematical Society
Status
Peer reviewed
ISSN
0002-9939Type
Journal article
Rights
© 2019, American Mathematical Society. This work has been made available online in accordance with the publisher's policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1090/proc/14613
Description
Funding: Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1) (JMF); EPSRC Doctoral Training Grant (EP/N509759/1) (DCH).Collections
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