Skip to content

Research at St Andrews

A capacity approach to box and packing dimensions of projections of sets and exceptional directions

Research output: Contribution to journalArticle

Author(s)

School/Research organisations

Abstract

Dimension profiles were introduced in [8,11] to give a formula for the box-counting and packing dimensions of the orthogonal projections of a set E in ℝn onto almost all m-dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of E with respect to certain kernels, which lead to the box-counting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.
Close

Details

Original languageEnglish
Number of pages20
JournalJournal of Fractal Geometry
Publication statusAccepted/In press - 12 Mar 2019

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. Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

    Falconer, K. & Jin, X., 15 Aug 2019, In : Transactions of the American Mathematical Society. 372, 4, p. 2921-2957 37 p.

    Research output: Contribution to journalArticle

  4. Marstrand's Theorem revisited: projecting sets of dimension zero

    Beresnevich, V., Falconer, K., Velani, S. & Zafeiropoulos, A., 15 Apr 2019, In : Journal of Mathematical Analysis and Applications. 472, 2, p. 1820-1845 26 p.

    Research output: Contribution to journalArticle

  5. Self-stabilizing processes based on random signs

    Falconer, K. J. & Lévy Véhel, J., 29 Sep 2018, In : Journal of Theoretical Probability. First Online, 19 p.

    Research output: Contribution to journalArticle

Related by journal

  1. Journal of Fractal Geometry (Journal)

    Kenneth John Falconer (Member of editorial board)
    1 Jun 2013 → …

    Activity: Publication peer-review and editorial work typesEditor of research journal

Related by journal

  1. 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

  2. Dimensions of equilibrium measures on a class of planar self-affine sets

    Fraser, J. M., Jordan, T. & Jurga, N., 11 Jun 2018, (Accepted/In press) In : Journal of Fractal Geometry.

    Research output: Contribution to journalArticle

  3. Uniform scaling limits for ergodic measures

    Fraser, J. M. & Pollicott, M., 2017, In : Journal of Fractal Geometry. 4, 1, p. 1-19

    Research output: Contribution to journalArticle

  4. On the dimensions of attractors of random self-similar graph directed iterated function systems

    Troscheit, S., 29 Feb 2016, (Accepted/In press) In : Journal of Fractal Geometry.

    Research output: Contribution to journalArticle

ID: 257593123

Top