Skip to content

Research at St Andrews

On the Hausdorff dimension of microsets

Research output: Contribution to journalArticle

DOI

Open Access permissions

Open

Author(s)

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

School/Research organisations

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.
Close

Details

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

    Research areas

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

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. Intermediate dimensions

    Falconer, K. J., Fraser, J. & Kempton, T. M. W., 6 Nov 2019, (Accepted/In press) In : Mathematische Zeitschrift. 19 p.

    Research output: Contribution to journalArticle

  2. Projection theorems for intermediate dimensions

    Burrell, S. A., Falconer, K. J. & Fraser, J. M., 26 Oct 2019, (Accepted/In press) In : Journal of Fractal Geometry. 16 p.

    Research output: Contribution to journalArticle

  3. Almost arithmetic progressions in the primes and other large sets

    Fraser, J. M., May 2019, In : The American Mathematical Monthly. 126, 6, p. 553-558 6 p.

    Research output: Contribution to journalArticle

  4. The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra

    Fraser, J. M., Hare, K. E., Hare, K. G., Troscheit, S. & Yu, H., 17 Jan 2019, In : Annales Academiae Scientiarum Fennicae-Mathematica. 44, 1, p. 379-387

    Research output: Contribution to journalArticle

Related by journal

  1. On The Lq dimensions of measures on Heuter-Lalley type self-affine sets

    Fraser, J. M. & Kempton, T., Jan 2018, In : Proceedings of the American Mathematical Society. 146, 1, p. 161-173

    Research output: Contribution to journalArticle

  2. Finite presentability and isomorphism of Cayley graphs of monoids

    Awang, J. S., Pfeiffer, M. J. & Ruskuc, N., Nov 2017, In : Proceedings of the American Mathematical Society. 145, 11, p. 4585-4593

    Research output: Contribution to journalArticle

  3. The Assouad dimension of self-affine carpets with no grid structure

    Fraser, J. M. & Jordan, T., 16 Jun 2017, In : Proceedings of the American Mathematical Society. 145, p. 4905-4918

    Research output: Contribution to journalArticle

  4. An exploration of normalish subgroups of R. Thompson's groups F and T

    Bleak, C., 10 Mar 2016, (Submitted) In : Proceedings of the American Mathematical Society. p. 1-4 4 p.

    Research output: Contribution to journalArticle

ID: 258177256

Top