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Intermediate Dimensions Mathematische Zeitschrift (2020) 296:813–830 https://doi.org/10.1007/s00209-019-02452-0 Mathematische Zeitschrift Intermediate dimensions Kenneth J. Falconer1 · Jonathan M. Fraser1 · Tom Kempton2 Received: 21 March 2019 / Accepted: 6 November 2019 / Published online: 26 December 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 = 0there 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 thediameter 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. B 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 123 814 K. J. Falconer et al. Recall that Hausdorff dimension dimH may be defined without introducing Hausdorff measures, but using Hausdorff content. For F ⊆ Rn, s dimH F =inf s ≥0 : for all ε>0 there exists a cover {Ui } of F such that |Ui | ≤ε , 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, = ≥ : ε> { } dimB F inf s 0 for all 0 there exists a cover Ui of F s such that |Ui |=|U j | for all i, j and |Ui | ≤ ε . 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 1/θ geometric range δ ≤|Ui |≤δ for some 0 ≤ θ ≤ 1. Definition 1.1 Let F ⊆ Rn be bounded. For 0 ≤ θ ≤ 1wedefinethelower θ-intermediate dimension of F by dim θ F = inf s ≥ 0 : for all ε>0andallδ0 > 0, there exists 0 <δ≤ δ0 1/θ s and a cover {Ui } of F such that δ ≤|Ui |≤δ and |Ui | ≤ ε . Similarly, we define the upper θ-intermediate dimension of F by dim θ F = inf s ≥ 0 : for all ε>0 there exists δ0 > 0 such that for all 0 <δ≤ δ0, 1/θ s there is a cover {Ui } of F such that δ ≤|Ui |≤δ and |Ui | ≤ ε . With these definitions, = = , = = , dim0 F dim0 F dimH F dim1 F dimB F and dim1 F dimB F where dimB is the upper box dimension. Moreover, it follows immediately that, for a bounded set F and θ ∈[0, 1], ≤ ≤ ≤ ≤ . dimH F dim θ F dim θ F dimB F and dim θ F dimB F It is also immediate that dim θ F and dim θ F are increasing in θ, though as we shall see they need not be strictly increasing. Furthermore, dim θ is finitely stable, that is dim θ (F1 ∪ F2) = max{dim θ F1, dim θ F2}, and, for θ ∈ (0, 1], both dim θ F and dim θ F are unchanged on replacing F by its closure. < = In many situations, even if dimH F dimB F, we still have dimB F dimB F and dim θ F = dim θ F for all θ ∈[0, 1]. In this case we refer to the box dimension dimB F = = θ = = dimB F dimB F and the -intermediate dimension dimθ F dim θ F dim θ F. 123 Intermediate dimensions 815 This paper is devoted to understanding θ-intermediate dimensions. The hope is that dimθ F will interpolate between the Hausdorff and box dimensions in a meaningful way, a rich and robust theory will be discovered, and interesting further questions unearthed. We first derive useful properties of intermediate dimensions, including that dim0 F and dim0 F are contin- uous on (0, 1] but not necessarily at 0, as well as proving versions of the mass distribution principle, Frostman’s lemma and product formulae. We then examine a range of examples illustrating different types of behaviour including sequences formed by negative powers of integers, and self-affine Bedford–McMullen carpets. Intermediate dimensions provide an insight into the distribution of the diameters of cov- ering sets needed when estimating the Hausdorff dimensions of sets whose Hausdorff and box dimensions differ. They also have concrete applications to well-studied problems. For example, since the intermediate dimensions are preserved under bi-Lipschitz mappings, they provide another invariant for Lipschitz classification of sets. A very specific variant was used in [6] to estimate the singular sets of partial differential equations. A related approach to ‘dimension interpolation’ was recently considered in [5]wherea new dimension function was introduced to interpolate between the box dimension and the Assouad dimension. In this case the dimension function was called the Assouad spectrum, θ (θ ∈ ( , )) denoted by dimA F 0 1 . 2 Properties of intermediate dimensions 2.1 Continuity The first natural question is whether, for a fixed bounded set F,dimθ F and dim θ F vary continuously for θ ∈[0, 1]. We show this is the case, except possibly at θ = 0. We provide simple examples exhibiting discontinuity at θ = 0 in Sect. 3.2.However,formanynatural sets F we find that the intermediate dimensions are continuous at 0 (and thus on [0, 1]), for example for self-affine carpets, see Sect. 4. Proposition 2.1 Let F be a non-empty bounded subset of Rn and let 0 ≤ θ<φ≤ 1.Then θ F ≤ F ≤ F + − (n − F). dim θ dim φ dim θ 1 φ dim θ (2.1) and θ θ F ≤ φ F ≤ θ F + − (n − θ F). dim dim dim 1 φ dim (2.2) In particular, θ → dim θ F and θ → dim θ F are continuous for θ ∈ (0, 1]. Proof Wewill only prove (2.1)since(2.2) is similar. The left-hand inequality of (2.1) is just the monotonicity of dim θ F. The right-hand inequality is trivially satisfied when dim θ F = n,so we assume that 0 ≤ dim θ F < n. Suppose that 0 ≤ θ<φ≤ 1andthat0≤ dim θ F < s < n. Then, given ε>0, we may find arbitrarily small δ>0 and countable or finite covers {Ui }i∈I of F such that s θ |Ui | <ε and δ ≤|Ui |≤δ for all i ∈ I . (2.3) i∈I Let φ φ θ I0 ={i ∈ I : δ ≤|Ui |≤δ } and I1 ={i ∈ I : δ < |Ui |≤δ }. 123 816 K. J. Falconer et al. ∈ { } For each i I1 we may split Ui into subsets of small coordinate cubes to get sets Ui, j j∈Ji φ −φ (θ−φ) such that U ⊆ U , , with |U , |≤δ and cardJ ≤ c |U |nδ n ≤ c δn , i j∈Ji i j i j i n i n n n/2 where cn = 4 n . < ≤ { } ∪{ } δ ≤| |, | |≤ Let s t n.Then Ui i∈I0 Ui, j i∈I1, j∈Ji is a cover of F such that Ui Ui, j δφ. Taking sums with respect to this cover: t t t φt n −φn |Ui | + |Ui, j | ≤ |Ui | + δ cn|Ui | δ i∈I i∈I j∈J i∈I i∈I 0 1 i 0 1 t s n−s φ(t−n) ≤ |Ui | + cn |Ui | |Ui | δ ∈ ∈ iI0 iI1 s s θ(n−s) φ(t−n) ≤ |Ui | + cn |Ui | δ δ ∈ ∈ iI0 i I1 s φ[t−(nφ+θ(s−n))/φ] s ≤ |Ui | + cnδ |Ui | ∈ ∈ i I0 i I1 s ≤ (1 + cn) |Ui | <(1 + cn)ε i∈I if t ≥ (nφ + θ(s − n))/φ, from (2.3). This holds for some cover for arbitrarily small ε and all s > dim θ F, giving dim φ F ≤ n + θ(dim θ F − n)/φ, which rearranges to give (2.1). Finally, note that (2.2) follows by exactly the same argument noting that the assumption dim θ F < s gives rise to δ0 > 0 such that for all δ ∈ (0,δ0) we can find covers {Ui }i∈I of F satisfying (2.3). 2.2 A mass distribution principle for dim  and dim  The mass distribution principle is a powerful tool in fractal geometry and provides a useful mechanism for estimating the Hausdorff dimension from below by considering measures supported on the set, see [2, page 67].
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