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| le |V|^theta for all sets U, V used in a particular cover, where theta in [0,1] is a parameter. Thus, when theta =1 only covers using sets of the same size are allowable, and we recover the box dimensions, and when theta =0 there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function of theta ), 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.
Highlights
We work with subsets of Rn throughout, much of what we establish 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
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 δ1/θ ≤ |Ui | ≤ δ for some 0 ≤ θ ≤ 1
Summary
We work with subsets of Rn throughout, much of what we establish holds in more general metric spaces. 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. Intermediate dimensions provide an insight into the distribution of the diameters of covering sets needed when estimating the Hausdorff dimensions of sets whose Hausdorff and box dimensions differ. They have concrete applications to well-studied problems. In this case the dimension function was called the Assouad spectrum, denoted by dimθA F (θ ∈ (0, 1))
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