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Generalised intermediate dimensions

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posted on 2023-11-07, 15:56 authored by Amlan BanajiAmlan Banaji
We introduce a family of dimensions, which we call the Φ-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by restricting the relative sizes of the covering sets in a way that allows for greater refinement than in the definition of the intermediate dimensions. We also extend the theory from Euclidean space to a wider class of metric spaces. We prove that these dimensions can be used to 'recover the interpolation' between the Hausdorff and box dimensions of compact subsets for which the intermediate dimensions are discontinuous at θ=0, thus providing finer geometric information about such sets. We prove continuity-like results involving the Assouad and lower dimensions, which give a sharp general lower bound for the intermediate dimensions that is positive for all θ∈(0,1] for sets with positive box dimension. We also prove Hölder distortion estimates, a mass distribution principle, and a Frostman type lemma, which we use to study dimensions of product sets.

Funding

Leverhulme Trust Research Project Grant (RPG-2019-034)

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Monatshefte fuer Mathematik

Volume

202

Issue

3

Pages

465–506

Publisher

Springer

Version

  • VoR (Version of Record)

Rights holder

© The Author(s)

Publisher statement

This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Acceptance date

2023-06-26

Publication date

2023-07-18

Copyright date

2023

ISSN

0026-9255

eISSN

1436-5081

Language

  • en

Depositor

Amlan Banaji. Deposit date: 9 July 2023

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