On the Hausdorff dimension of microsets

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

doi:10.1090/proc/14613

On the Hausdorff dimension of microsets

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

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

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 F_σset ∆ ⊆ [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 language	English
Pages (from-to)	4921-4936
Number of pages	16
Journal	Proceedings of the American Mathematical Society
Volume	147
Issue number	11
Early online date	10 Jun 2019
DOIs	https://doi.org/10.1090/proc/14613
Publication status	Published - Nov 2019

Keywords

Weak tangent
Microset
Hausdorff dimension
Assouad type dimensions

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10.1090/proc/14613

microsets_PAMS_revision
© 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
Accepted author manuscript, 545 KB

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard
Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
The Leverhulme Trust
1/09/16 → 30/06/18
Project: Fellowship

Cite this

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title = "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.",

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note = "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).",

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AB - 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.

KW - Weak tangent

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KW - Hausdorff dimension

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On the Hausdorff dimension of microsets

Abstract

Keywords

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Projects

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Fractal Geometry and Dimension: Fractal Geometry and dimension theory

Cite this