Projection theorems for intermediate dimensions

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Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, relying on this, we show that the intermediate dimensions of the projection of a set E ⊂ Rn onto almost all m-dimensional subspaces depend only on m and E, that is, they are almost surely independent of the choice of subspace. Our approach is based on ‘intermediate dimension profiles’ which are expressed in terms of capacities. We discuss several applications at the end of the paper, including a surprising result that relates the boxdimensions of the projections of a set to the Hausdorff dimension of the set.
Original languageEnglish
Pages (from-to)95-116
Number of pages22
JournalJournal of Fractal Geometry
Issue number2
Early online date30 Apr 2021
Publication statusPublished - 1 May 2021


  • Intermediate dimensions
  • Marstrand theorem
  • Projections
  • Capacity


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