Intermediate dimensions
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Mathematische Zeitschrift
Intermediate dimensions Kenneth J. Falconer1 · Jonathan M. Fraser1 · Tom Kempton2 Received: 21 March 2019 / Accepted: 6 November 2019 © The Author(s) 2019
Abstract We introduce a continuum of dimensions which are ‘intermediate’ between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that |U | ≤ |V |θ for all sets U , V used in a particular cover, where θ ∈ [0, 1] is a parameter. Thus, when θ = 1 only covers using sets of the same size are allowable, and we recover the box dimensions, and when θ = 0 there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function of θ ), including proving that it is continuous on (0, 1] but not necessarily continuous at 0, as well as establishing appropriate analogues of the mass distribution principle, Frostman’s lemma, and the dimension formulae for products. We also compute, or estimate, the intermediate dimensions of some familiar sets, including sequences formed by negative powers of integers, and Bedford–McMullen carpets. Keywords Hausdorff dimension · Box dimension · Self-affine carpet Mathematics Subject Classification Primary 28A80; Secondary 37C45
1 Intermediate dimensions: definitions and background We work with subsets of Rn throughout, although much of what we establish also holds in more general metric spaces. We denote the diameter of a set F by |F|, and when we refer to a cover {Ui } of a set F we mean that F ⊆ i Ui where {Ui } is a finite or countable collection of sets.
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Jonathan M. Fraser [email protected] Kenneth J. Falconer [email protected] Tom Kempton [email protected]
1
Mathematical Institute, University of St Andrews, St Andrews, UK
2
School of Mathematics, University of Manchester, Manchester, UK
123
K. J. Falconer et al.
Recall that Hausdorff dimension dimH may be defined without introducing Hausdorff measures, but using Hausdorff content. For F ⊆ Rn , |Ui |s ≤ ε , dimH F = inf s ≥ 0 : for all ε > 0 there exists a cover {Ui } of F such that see [2, Section 3.2]. (Lower) box dimension dimB may be expressed in a similar manner, by forcing the covering sets to be of the same diameter. For bounded F ⊆ Rn , dimB F = inf s ≥ 0 : for all ε > 0 there exists a cover {Ui } of F such that |Ui | = |U j | for all i, j and
|Ui | ≤ ε . s
see [2, Chapter 2]. Expressed in this way, Hausdorff and box dimensions may be regarded as extreme cases of the same definition, one with no restriction on the size of covering sets, and the other requiring them all to have equal diameters. With this in mind, one might regard them as the extremes of a continuum of dimensions with increasing restrictions on the relative sizes of covering sets. This is the main idea of this paper, which we formalise by considering restricted coverings where the diameters of the smallest and largest covering sets lie in a geometric range δ 1/θ ≤ |Ui
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