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 © 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|θfor all setsU,V used in a particular cover, whereθ ∈ [0,1]is a parameter. Thus, whenθ =1 only covers using sets of the same size are allowable, and we recover the box dimensions, and whenθ =0 there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function ofθ), 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 ofRn throughout, although much of what we establish also holds in more general metric spaces. We denote thediameterof a setFby|F|, and when we refer to a cover{Ui}of a setFwe mean thatF⊆
iUiwhere{Ui}is a finite or countable collection
of sets.
B
Jonathan M. Fraser [email protected] Kenneth J. Falconer [email protected] Tom KemptonRecall that Hausdorff dimension dimHmay be defined without introducing Hausdorff measures, but using Hausdorff content. ForF⊆Rn,
dimHF=inf
s≥0: for allε >0 there exists a cover{Ui}ofFsuch that
|Ui|s≤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 boundedF⊆Rn,
dimBF=inf
s≥0: for allε >0 there exists a cover{Ui}ofF
such that|Ui| = |Uj|for alli,jand
|Ui|s ≤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 geometric rangeδ1/θ ≤ |Ui| ≤δfor some 0≤θ≤1.
Definition 1.1 LetF⊆Rnbe bounded. For 0≤θ ≤1 we define thelowerθ-intermediate dimensionofFby
dimθF=inf
s≥0: for allε >0 and allδ0>0, there exists 0< δ≤δ0
and a cover{Ui}ofFsuch thatδ1/θ≤ |Ui| ≤δand
|Ui|s ≤ε
.
Similarly, we define theupperθ-intermediate dimensionofFby
dimθF=inf
s≥0: for allε >0 there existsδ0>0 such that for all 0< δ≤δ0,
there is a cover{Ui}ofFsuch thatδ1/θ ≤ |Ui| ≤δand
|Ui|s≤ε
.
With these definitions,
dim0F= dim0F=dimHF, dim1F=dimBF and dim1F=dimBF,
where dimB is the upper box dimension. Moreover, it follows immediately that, for a bounded setFandθ∈ [0,1],
dimHF≤dimθF≤dimθF≤dimBF and dimθF≤dimBF.
It is also immediate that dimθFand dimθFare increasing inθ, though as we shall see they need not be strictly increasing. Furthermore, dimθis finitely stable, that is dimθ(F1∪F2)= max{dimθF1,dimθF2}, and, forθ ∈ (0,1], both dimθF and dimθF are unchanged on replacingFby its closure.
This paper is devoted to understandingθ-intermediate dimensions. The hope is that dimθF 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 dim0Fand dim0Fare 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] where a new dimension function was introduced to interpolate between the box dimension and the Assouad dimension. In this case the dimension function was called theAssouad spectrum, denoted by dimθAF(θ∈(0,1)).
2 Properties of intermediate dimensions
2.1 Continuity
The first natural question is whether, for a fixed bounded setF, dimθF and dimθF vary continuously forθ ∈ [0,1]. We show this is the case, except possibly atθ =0. We provide simple examples exhibiting discontinuity atθ =0 in Sect.3.2. However, for many natural setsFwe find that the intermediate dimensionsarecontinuous 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 ofRnand let0≤θ < φ≤1. Then
dimθF ≤ dimφF ≤ dimθF+
1−θ
φ (n−dimθF). (2.1)
and
dimθF ≤ dimφF ≤ dimθF+
1−θ
φ (n−dimθF). (2.2)
In particular,θ →dimθF andθ →dimθF are continuous forθ ∈(0,1].
Proof We will only prove (2.1) since (2.2) is similar. The left-hand inequality of (2.1) is just the monotonicity of dimθF. The right-hand inequality is trivially satisfied when dimθF=n, so we assume that 0≤dimθF<n. Suppose that 0≤θ < φ≤1 and that 0≤dimθF<s<n. Then, givenε >0, we may find arbitrarily smallδ >0 and countable or finite covers{Ui}i∈I
ofFsuch that
i∈I
|Ui|s < ε and δ≤ |Ui| ≤δθ for alli∈I. (2.3)
Let
For eachi∈I1we may splitUiinto subsets of small coordinate cubes to get sets{Ui,j}j∈Ji
such thatUi ⊆
j∈JiUi,j, with|Ui,j| ≤δφ and cardJi ≤ cn|Ui|nδ−φn ≤ cnδn(θ−φ),
wherecn =4nnn/2.
Lets<t≤n. Then{Ui}i∈I0∪ {Ui,j}i∈I1,j∈Jiis a cover ofFsuch thatδ≤ |Ui|,|Ui,j| ≤
δφ. Taking sums with respect to this cover:
i∈I0
|Ui|t+
i∈I1
j∈Ji
|Ui,j|t ≤
i∈I0
|Ui|t+
i∈I1
δφtc
n|Ui|nδ−φn
≤
i∈I0
|Ui|t+cn
i∈I1
|Ui|s|Ui|n−sδφ(t−n)
≤
i∈I0
|Ui|s+cn
i∈I1
|Ui|sδθ(n−s)δφ(t−n)
≤
i∈I0
|Ui|s+cnδφ[t−(nφ+θ(s−n))/φ]
i∈I1
|Ui|s
≤(1+cn)
i∈I
|Ui|s < (1+cn)ε
ift ≥(nφ+θ(s−n))/φ, from (2.3). This holds for some cover for arbitrarily smallεand alls>dimθF, giving dimφF≤n+θ(dimθF−n)/φ, which rearranges to give (2.1).
Finally, note that (2.2) follows by exactly the same argument noting that the assumption dimθF<sgives rise toδ0>0 such that for allδ∈(0, δ0)we can find covers{Ui}i∈IofF
satisfying (2.3).
2.2 A mass distribution principle for dimÂand dimÂ
Themass distribution principleis 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]. We present natural analogues for dimθand dimθ.
Proposition 2.2 Let F be a Borel subset ofRnand let0≤θ ≤1and s≥0. Suppose that there are numbers a,c, δ0>0such that for all0< δ≤δ0we can find a Borel measureμδ supported by F withμδ(F)≥a, and with
μδ(U)≤c|U|s for all Borel sets U ⊆Rnwithδ≤ |U| ≤δθ. (2.4)
ThendimθF≥s. Moreover, if measuresμδwith the above properties can be found only for a sequence ofδ→0, then the conclusion is weakened todimθF≥s.
Proof Let{Ui}be a cover ofFsuch thatδ≤ |Ui| ≤δθfor alli. Then
a ≤ μδ(F) ≤ μδ
i Ui
≤
i
μδ(Ui) ≤ c
i |Ui|s,
so thati|Ui|s≥a/c>0 for every admissible cover and therefore dimθF≥s.
The weaker conclusion regarding the upper intermediate dimension is obtained similarly.
μδis only required to describe a range of scales, in practice one can often use finite sums of point masses. Whilst the measuresμδmay vary, it is essential that they all assign mass at leasta>0 toF.
2.3 A Frostman type lemma for dimÂ
Frostman’s lemmais another powerful tool in fractal geometry, which asserts the existence of measures of the type considered by the mass distribution principle, see [2, page 77] or [8, page 112]. The following analogue of Frostman’s lemma holds for intermediate dimensions and is a useful dual to Proposition2.2.
Proposition 2.3 Let F be a compact subset ofRn, let0 < θ ≤ 1, and suppose0 < s <
dimθF . There exists a constant c > 0 such that for allδ ∈ (0,1) we can find a Borel probability measureμδsupported on F such that for all x∈Rnandδ1/θ ≤r≤δ,
μδ(B(x,r))≤crs.
Moreover,μδcan be taken to be a finite collection of atoms.
Proof This proof follows the proof of the classical version of Frostman’s lemma given in [8, pages 112–114]. Form≥0 letDmdenote the familiar partition of[0,1]nconsisting of
2nm pairwise disjoint half-open dyadic cubes of sidelength 2−m, that is cubes of the form [a1,a1+2−m)× · · · × [an,an+2−m). By translating and rescaling we may assume without
loss of generality thatF ⊆ [0,1]n and that Fis not contained in any Q ∈D1. It follows
from the definition of dimθFthat there existsε >0 such that for allδ∈(0,1)and for all covers{Ui}i ofFsatisfyingδ1/θ ≤ |Ui| ≤δ,
i
|Ui|s> ε. (2.5)
Givenδ∈(0,1), letm ≥0 be the unique integer satisfying 2−m−1 < δ1/θ ≤2−mand let
μmbe a measure defined on Fas follows: for each Q ∈ Dm such thatQ∩ F = ∅, then
choose an arbitrary pointxQ∈Q∩Fand let
μm=
Q∈Dm:Q∩F=∅
2−msδxQ
whereδxQis a point mass atxQ. Modifyμmto form a measureμm−1, supported on the same
finite set, defined by
μm−1|Q=min{1,2−(m−1)sμm(Q)−1}μm|Q
for allQ ∈Dm−1, whereν|Edenotes the restriction ofνtoE. The purpose of this
modifi-cation is to reduce the mass of cubes which carry too much measure. This is done since we are ultimately trying to construct a measure which we can estimate uniformly from above. Continuing inductively,μm−k−1is obtained fromμm−kby
μm−k−1|Q=min{1,2−(m−k−1)sμm−k(Q)−1}μm−k|Q
for allQ∈Dm−k−1. We terminate this process when we defineμm−lwherelis the largest
integer satisfying 2−(m−l)n1/2 ≤δ. (We may assume thatl ≥0 by choosingδsufficiently small to begin with.) In particular, cubes Q ∈ Dm−l satisfy|Q| = 2−(m−l)n1/2 ≤ δ. By
construction we have
for allk=0, . . . ,landQ∈Dm−k. Moreover, for allx∈F, there is at least onek∈ {0, . . . ,l}
andQ∈Dm−kwithx∈Qsuch that the inequality in (2.6) is an equality. This is because all
cubes at levelmsatisfy the equality forμmand if a cubeQsatisfies the equality forμm−k,
then eitherQor its parent cube satisfies the equality forμm−k−1. For eachx∈F, choosing
the largest suchQyields a finite collection of cubesQ1, . . . ,Qt which coverFand satisfy
δ1/θ≤ |Q
i| ≤δfori=1, . . . ,t. Therefore, using (2.5),
μm−l(F)= t
i=1
μm−l(Qi)= t
i=1
|Qi|sn−s/2> εn−s/2.
Letμδ = μm−l(F)−1μm−l, which is clearly a probability measure supported on a finite
collection of points. Moreover, for all x ∈ Rn andδ1/θ ≤ r ≤ δ, B(x,r) is certainly contained in at mostcn cubes inDm−kwherekis chosen to be the largest integer satisfying
0≤k ≤land 2−(m−k+1)<r, andcn is a constant depending only onn. Therefore, using
(2.6),
μδ(B(x,r))≤cnμm−l(F)−12−(m−k)s≤cnε−1ns/22srs
which completes the proof, settingc=cnε−1ns/22s.
2.4 General bounds
Here we consider general bounds which rely on the Assouad dimension and which have interesting consequences for continuity. Namely, they provide numerous examples where the intermediate dimensions arediscontinuous atθ =0 and also provide another proof that the intermediate dimensions are continuous atθ=1. TheAssouad dimensionofF⊆Rnis
defined by
dimAF=inf
s≥0 : there existsC >0 such that for allx ∈F,
and for all 0<r<R, we haveNr(F∩B(x,R))≤C
R r
s
whereNr(A)denotes the smallest number of sets of diameter at mostrrequired to cover a
setA. In general dimBF ≤dimBF≤dimAF ≤n, but equality of these three dimensions occurs in many cases, even if the Hausdorff dimension and box dimension are distinct, for example if the box dimension is equal to the ambient spatial dimension. See [4,11] for more background on the Assouad dimension. The following proposition gives lower bounds for the intermediate dimensions in terms of Assouad dimensions.
Proposition 2.4 Given any non-empty bounded F⊆Rnandθ ∈(0,1),
dimθF≥dimAF−
dimAF−dimBF
θ ,
and
dimθF≥dimAF−
dimAF−dimBF
θ .
In particular, ifdimBF = dimAF , thendimθF = dimθF = dimBF = dimAF for all
Proof We will prove the lower bound for dimθF, the proof for dimθF is similar. Fixθ ∈
(0,1)and assume that dimBF>0, otherwise the result is trivial. Let 0<b<dimBF≤dimAF<d<∞
andδ ∈ (0,1)be given. By the definition of lower box dimension, there exists a uniform constantC0, depending only onFandb, such that there is aδ-separated set of points in F of cardinality at leastC0δ−b. Letμδbe a uniformly distributed probability measure on these points, i.e. a sum ofC0δ−bpoint masses each with massC0−1δb. We use our mass distribution principle with this measure to prove the proposition.
LetU ⊆ Rn be a Borel set with|U| = δγ for someγ ∈ [θ,1]. By the definition of Assouad dimension there exists a uniform constantC1, depending only on Fandd, such thatU intersects at mostC1(δγ/δ)d points in the support ofμδ. Therefore
μδ(U)≤C1δ(γ−1)dC0−1δb=C1C0−1|U|(γd−d+b)/γ ≤C1C0−1|U|(θd−d+b)/θ which, using Proposition2.2, implies that
dimθF≥(θd−d+b)/θ =d−d−b
θ .
Lettingd→dimAFandb→dimBFyields the desired result.
This proposition implies that for bounded sets with dimHF < dimBF = dimAF, the intermediate dimensions dimθF and dimθF are necessarily discontinuous at θ = 0. In fact the intermediate dimensions are constant on(0,1]in this case. On the other hand, this gives an alternative demonstration that dimθFand dimθFarealwayscontinuous atθ =1. Moreover, the proposition provides a quantitative lower bound nearθ =1. In Sect.3.2we will use Proposition2.4to construct examples exhibiting a range of behaviours.
2.5 Product formulae
A well-studied problem in dimension theory is how dimensions of product sets behave. The following product formulae for intermediate dimensions may be of interest in their own right, but in Sect.3.2they will be used to construct examples.
Proposition 2.5 Let E⊆Rnand F⊆Rmbe bounded andθ∈ [0,1]. Then
dimθE+dimθF ≤ dimθ(E×F) ≤ dimθ(E×F) ≤ dimθE+dimBF.
Proof Fixθ ∈(0,1)throughout, noting that the cases whenθ =0,1 are well-known, see [2, Chapter 7]. We begin by demonstrating the left-hand inequality. We may assume that dimθE,dimθF >0 as otherwise the conclusion follows by monotonicity. Moreover, since E,Fare bounded we may assume they are compact since all the dimensions considered are unchanged under taking closure. Let 0<s<dimθEand 0<t <dimθF. It follows from Proposition2.3that there exist constantsCs,Ct >0 such that for allδ∈(0,1)there exist
Borel probability measuresμδsupported onEandνδsupported onFsuch that for allx ∈Rn andδ1/θ≤r≤δ,
μδ(B(x,r))≤Csrs and νδ(B(x,r))≤Ctrt.
Consider the product measureμδ×νδwhich is supported onE×F. Forz∈Rn×Rmand
δ1/θ≤r≤δ,
and Proposition2.2yields dimθ(E×F)≥s+t; lettings→dimθEandt→dimθFgives the desired inequality.
The middle inequality is trivial and so it remains to prove the right-hand inequality. Let s>dimθEandd>dimBF. From the definition of dimBFthere exists a constantδ1∈(0,1) such that for all 0 <r < δ1there is a cover of Fby at mostr−d sets of diameterr. Let
ε >0. By the definition of dimθEthere existsδ0∈(0, δ1)such that for all 0< δ < δ0there is a cover ofEby sets{Ui}iwithδ1/θ ≤ |Ui| ≤δfor alliand
i
|Ui|s ≤ε.
Given such a cover ofE, for eachilet{Ui,j}jbe a cover of Fby at most|Ui|−d sets with
diameters|Ui,j| = |Ui|for all j. Then
E×F ⊆ i
j
Ui×Ui,j
,
with
i
j
|Ui×Ui,j|s+d ≤
i |Ui|−d
√
2|Ui|
s+d
= 2(s+d)/2
i
|Ui|s ≤ 2(s+d)/2ε.
Sinceδ1/θ ≤ |U
i×Ui,j| ≤ √
2δfor alli,j, each setUi×Ui,j may be covered by at most cn+m sets{Vi,j,k}kwith diametersδ1/θ ≤ |Vi,j,k| ≤min{|Ui ×Ui,j|, δ} ≤δ, wherecq is
the least number such that every set inRq of diameter√2 can be covered by at mostcqsets
of diameter 1. Hence
i
j
k
|Vi,j,k|s+d ≤ cn+m2(s+d)/2ε.
Asε may be taken arbitrarily small, dimθ(E ×F) ≤ s+d; lettings → dimθE and d→dimBFcompletes the proof.
3 Examples
In this section we construct several simple examples where the intermediate dimensions exhibit a range of phenomena. All of our examples are compact subsets ofRorR2and in all examples the upper and the lower intermediate dimensions coincide for allθ ∈ [0,1].
3.1 Convergent sequences
Letp>0 and
Fp=
0, 1 1p,
1 2p,
1 3p, . . .
.
SinceFp is countable, dimHFp = 0. It is well-known that dimBFp = 1/(p+1), see [2,
Chapter 2]. We obtain the intermediate dimensions ofFp.
Proposition 3.1 For p>0and0≤θ≤1,
Proof We first bound dimθFp above. Let 0 < δ < 1 and letM = δ−(s+θ(1−s))/(p+1).
WriteB(x,r)for the closed interval (ball) of centrex and length 2r. Take a coveringUof Fp consisting of Mintervals B(k−p, δ/2)of lengthδ for 1 ≤k ≤ MandM−p/δθ ≤ M−p/δθ+1 intervals of lengthδθthat cover[0,M−p].Then
U∈U
|U|s ≤Mδs+δθs
1 Mpδθ +1
=Mδs+δ
θ(s−1)
Mp +δθ s
≤(δ−(s+θ(1−s))/(p+1)+1)δs+δθ(s−1)δ(s+θ(1−s))p/(p+1)+δθs =2δ(θ(s−1)+sp)/(p+1)+δs+δθs → 0
asδ→0 ifs(θ+p) > θ. Thus dimθFp ≤θ/(p+θ).
For the lower bound we put suitable measures on Fp and apply Proposition2.2. Fix s=θ/(p+θ). Let 0 < δ <1 and again letM = δ−(s+θ(1−s))/(p+1). Defineμδas the sum of point masses on the points 1/kp(1≤k<∞)with
μδ
1 kp =
δs if 1≤k≤M
0 if M+1≤k<∞ . (3.1)
Then
μδ(Fp)=Mδs
≥δ−(s+θ(1−s))/(p+1)δs =δ(ps+θ(s−1))/(p+1)=1
by the choice ofs.
To see that (2.4) is satisfied, note that if 2 ≤ k ≤ M then, by a mean value theorem estimate,
1
(k−1)p −
1 kp ≥
p kp+1 ≥
p Mp+1;
thus the gap between any two points ofFpcarrying mass is at leastp/Mp+1. LetUbe such
thatδ≤ |U| ≤δθ. ThenU intersects at most 1+ |U|/(p/Mp+1)=1+ |U|Mp+1/pof the points ofFpwhich have massδs. Hence
μδ(U)≤δs+ 1 p|U|δ
sδ−(s+θ(1−s))
=δs+ 1 p|U|δ
(θ(s−1))
≤ |U|s+ 1 p|U||U|
s−1
=
1+ 1 p |U|
s.
3.2 Simple examples exhibiting different phenomena
We use the convergent sequencesFpfrom the previous section, the product formulae from
Proposition2.5, and that theupperintermediate dimensions are finitely stable to construct examples displaying a range of different features which are illustrated in Fig.1.
The natural question which began this investigation is ‘does dimθ vary continuously between the Hausdorff and lower box dimension?’. This indeed happens for the convergent sequences considered in the previous section, but turns out to be false in general. The first example in this direction is provided by another convergent sequence.
Example 1: Discontinuous at 0, otherwise constant.Let
Flog= {0,1/log 2,1/log 3,1/log 4, . . .}.
This sequence converges slower than any of the polynomial sequencesFp and it is
well-known and easy to prove that dimBFlog =dimAFlog =1. It follows from Proposition2.4 that
dimθFlog=dimθFlog=1, θ∈(0,1].
Since dim0Flog=dimHFlog=0 there is a discontinuity atθ =0.
[image:10.439.75.362.304.601.2]Example 2: Continuous at 0, part constant, part strictly increasing.In the opposite direction, it is possible that dimθF =dimHF <dimBFfor someθ >0. Indeed, letF = F1∪ E where F1 = {0,1/1,1/2,1/3, . . .} as before, and let E ⊂ R be any compact set with dimHE =dimBE=1/3 (for example an appropriately chosen self-similar set). Then it is straightforward to deduce that
dimθF=dimθF=max
θ
1+θ,1/3
, θ ∈ [0,1].
It is also possible for dimθFto approach a value strictly in between dimHFand dimBFas
θ→0. This is the subject of the next two examples.
Example 3: Discontinuous at 0, part constant, part strictly increasing.For an example that is constant on an interval adjacent to 0, letF=E∪F1where this timeE⊂Ris any closed countable set with dimBE=dimAE=1/4. It is immediate that
dimθF=dimθF=max
θ
1+θ,1/4
, θ∈(0,1],
with dimHF=0 and dimBF=1/2.
Example 4: Discontinuous at 0, strictly increasing.Finally, for an example where dimθFis smooth, strictly increasing but not continuous atθ =0, let
F=F1×Flog⊂R2.
Here dimHF=0 and dimBF=3/2 and Proposition2.5gives
dimθF=dimθF= θ
1+θ +1, θ ∈(0,1],
noting that dimθFlog = dimθFlog = dimBFlog = dimBFlog = dimAFlog = 1 forθ ∈
(0,1].
4 Bedford–McMullen carpets
A well-known class of fractals where the Hausdorff and box dimensions differ are the self-affine carpets; this is a consequence of the alignment of the component rectangles in the iterated construction. The first studies of planar self-affine carpets were by Bedford [1] and McMullen [9] independently, see also [10] and these Bedford–McMullen carpets have been widely studied and generalised, see for example [3] and references therein. Indeed, the question of the distribution of scales of covering sets for Hausdorff and box dimensions of Bedford–McMullen carpets was one of our motivations for studying intermediate dimensions. Finding an exact formula for the intermediate dimensions of Bedford–McMullen carpets seems a difficult problem, so here we obtain some lower and upper bounds, which in particular establish continuity atθ =0 and that the intermediate dimensions take a strict minimum at
θ=0.
We introduce the notation we need, generally following that of McMullen [9]. Choose integersn>m≥2. LetI = {0, . . . ,m−1}andJ = {0, . . . ,n−1}. Choose a fixed digit setD⊆ I×J with at least two elements. For(p,q)∈ Dwe define the affine contraction S(p,q): [0,1]2→ [0,1]2by
S(p,q)(x,y)=
x+p m ,
There exists a unique non-empty compact setF⊆ [0,1]2satisfying
F=
(p,q)∈D
S(p,q)(F);
that isFis the attractor of the iterated function system{S(p,q)}(p,q)∈D. We call such a setF
aBedford–McMullen self-affine carpet. It is sometimes convenient to denote pairs inDby
=(p,q).
We model our carpetFvia the symbolic spaceDN, which consists of all infinite words overDand is equipped with the product topology. We writei≡(i1,i2, . . .)for elements of DNand(i1, . . . ,ik)for words of lengthkinDk, whereij∈D. Then the canonical projection
τ :DN→ [0,1]2is defined by
{τ(i)} ≡ {τ(i1,i2, . . .)} =
k∈N
Si1◦ · · · ◦Sik([0,1]
2).
This allows us to switch between symbolic and geometric notation since
τ(DN)=F.
Bedford [1] and McMullen [9] showed that
dimBF= logm0
logm +
log|D| −logm0
logn (4.1)
wherem0is the number of psuch that there is aqwith(p,q)∈ D, that is the number of columns of the array containing at least one rectangle. They also showed that
dimHF=log
⎛
⎝m
p=1 nlognm
p
⎞
⎠logm, (4.2)
wherenp(1 ≤ p ≤ m)is the number ofq such that (p,q) ∈ D, that is the number of
selected rectangles in thepth column of the array.
For each∈Dwe letabe the number of rectangles ofDin the same column as. Then, writingd=dimHF, (4.2) may be written as
md = m
p=1 nlognm
p =
∈D
a(lognm−1)
, (4.3)
where equality of the sums follows from the definitions ofnpanda.
We assume that the non-zeronpare not all equal, otherwise dimHF=dimBF; in par-ticular this implies thata:=max∈Da≥2.
We denote thekth-level iterated rectangles by
Rk(i1, . . . ,ik) = Si1◦ · · · ◦Sik([0,1]
2).
We also writeRk(i)≡Rk(i1,i2, . . .)for this rectangle when we wish to indicate thekth-level iterated rectangle containing the pointτ(i)=τ(i1,i2, . . .).
We will associate a probability vector {b}∈DwithDand letμ=
k∈N(
ik∈Dbikδik)
be the natural Borel product probability measure onDN, whereδis the Dirac measure on Dconcentrated at. Then the measure
is a self-affine measure supported onF. Following McMullen [9] we set
b=a(lognm−1)
/md (∈D), (4.4)
noting that∈Db=1, to get a measureμonF; thus the measures of the iterated rectangles are
μRk(i1, . . . ,ik)
= bi1. . .bik = m− kd(a
i1. . .aik)(
lognm−1). (4.5)
Approximate squares are well-known tools in the study of self-affine carpets. Givenk∈N letl(k)= klognmso that
klognm ≤ l(k) ≤klognm + 1 (4.6)
For suchkandi=(i1,i2, . . .)∈ DNwe define theapproximate squarecontainingτ(i)as the union ofm−k×n−krectangles:
Qk(i)=Qk(i1,i2, . . .)= Rk(i1, . . . ,ik):pij =pij for j=1, . . . ,k
andqi
j =qij forj =1, . . . ,l(k)
,
recalling that=(p,q). This approximate square has sidesm−k×n−l(k)wheren−1m−k≤ n−l(k)≤m−k.
Note that, by virtue of self-affinity and since the sequence(pi1, . . .pik)is the same for
all level-krectanglesRk(i1, . . . ,ik)in the same approximate square, theail(k)+1. . .aik level-krectangles that comprise the approximate square Qk(i)all have equalμ-measure. Thus,
writingL=lognm,
μ(Qk(i))=m−kdai(1L−1). . .a
(L−1)
ik ×ail(k)+1. . .aik (4.7)
=m−kdaiL1. . .aikL×ai−1
1 . . .a
−1
il(k). (4.8)
We now obtain an upper bound for dimθFwhich implies continuity atθ =0 and so on [0,1]. Recall thata=max∈Da≥2.
Proposition 4.1 Let F be the Bedford–McMullen carpet as above. Then for0 < θ < 1
4(lognm)2,
dimθF ≤ dimHF+
2 log(logmn)loga logn
1
−logθ. (4.9)
In particular,dimθF anddimθF are continuous atθ=0and so are continuous on[0,1].
Proof Fori=(i1,i2, . . .), rewriting (4.8) gives
μQk(i)
=m−kd
(ai1. . .aik)
1/k
(ai1. . .ail(k))1/l(k)
Lk
ai1. . .ail(k)
(k L/l(k))−1
. (4.10)
We consider the two bracketed terms on the right-hand side of (4.10) in turn. We show that the first term cannot be too small for too many consecutivek and that the second term is bounded below by 1.
Fork>1 withl(k)= klognmas usual, define
fk(i) ≡ fk(i1,i2, . . .) =
(ai1. . .aik)
1/k
(ai1. . .ail(k))1/l(k)
We claim that for allK ≥ L/(1−L)and alli = (i1,i2, . . .) ∈ DN, there existsk with K ≤k≤K/θsuch that
fk(i) ≥ alogL/log(L/2θ). (4.12)
Suppose this is false for some(i1,i2, . . .)andK ≥1/L(L−1), so for allK ≤k ≤K/θ, fk(i) < λ:=alogL/log(L/2θ), that is
(ai1ai2. . .aik)
1/k < λ(a
i1ai2. . .ail)
1/l(k). (4.13)
Define a sequence of integerskr(r = 0,1,2, . . .)inductively byk0 = K and forr ≥1 takingkr to be the least integer such thatkrlognm =kr−1. Thenkr ≤kr−1/L+1, and a simple induction shows that
kr ≤ L−r
K+L/(1−L)−L/(1−L) ≤ L−rK+L/(1−L) (r≥0).
FixN to be the greatest integer such thatkN ≤K/θ. Then
L−(N+1)K+L(1−L) ≥ kN+1 > K/θ
so rearranging, providedK ≥L/(1−L),
LN < θ
K +L(1−L)
L K ≤
2θ L,
that is
N > log(2θ/L)
logL . (4.14)
From (4.13)
(ai1ai2. . .aikr)
1/kr < λ(a
i1ai2. . .aikr−1)
1/kr−1 (1≤r≤N)
so iterating
1 ≤ (ai1ai2. . .aikN)
1/kN < λN(a
i1ai2. . .aiK)
1/K ≤ λNa.
Combining with (4.14) we obtain
λ > a−1/N ≥alogL/log(L/2θ)
which contradicts the definition ofλ. Thus the claim (4.12) is established. For the second bracket on the right-hand side of (4.10) note that
0 ≤ (k L/l(k))−1 = klognm− klognm l(k) ≤
1 l(k)
so that
1 ≤ ai1. . .ail(k)
(k L/l(k))−1
≤a. (4.15)
Putting together (4.10), (4.12) and (4.15), we conclude that for allK ≥ L/(1−L)and alli∈DNthere existsK ≤k≤K/θsuch that
μQk(i)
≥ m−dkak LlogL/log(L/2θ) ≥ m−dka2k LlogL/log(1/θ) = m−k(d+ε(θ)),
asθ ≤L2/4, where
ε(θ) = −(loga)2LlogL logmlog(1/θ) =
2 log(logmn)loga logn
Geometrically this means that for K ≥ L/(1−L) every pointz ∈ F belongs to at least one approximate square, Qk(z) say, with K ≤ k(z) ≤ K/θ and with μ(Qk(z)) ≥ m−k(d+ε(θ)). Since the approximate squares form a nested hierarchy we may choose a subset {Qk(zn)}nN=1⊂ {Qk(z):z∈F}that is disjoint (except possibly at boundaries of approximate squares) and which coverF. Thus
1 = μ(F) = N
n=1
μ(Qk(zn)) ≥ N
n=1
m−k(zn)(d+ε(θ)) ≥ N
n=1
(2−1/2|Qk(zn)|)( d+ε(θ))
where|Qk|denotes the diameter of the approximate squareQk, noting that|Qk| ≤21/2m−k.
It follows that dimθF≤d+ε(θ)as claimed.
The following lemma brings together some basic estimates that we will need to obtain a lower bound for the intermediate dimensions ofF.
Lemma 4.2 Letε >0. There exists K0∈Nand a set E ⊂ F withμ(E)≥12 such that for alliwithτ(i)∈E and k≥K0,
μ(Qk(i))≤m−k(d−ε) (4.16)
and
μ(Rk(i))≥exp(−k(H(μ)+ε)), (4.17)
where d=dimHF and H(μ)∈(0,log|D|)is the entropy of the measureμ.
Proof McMullen [9, Lemmas 3,4(a)] shows that forμ-almost alli∈DN
lim
k→∞μ(Qk(i))
1/k→m−d.
Thus by Egorov’s theorem we may find a setE1⊂DNwithμ(E1)≥ 34, andK1∈Nsuch that (4.16) holds for alli∈E1andk ≥K1.
Furthermore, it is immediate from the Shannon–MacMillan–Breimann Theorem and (4.5) that forμ-almost alli∈DN,
lim
k→∞μ(Rk(i)))
1/k→exp(−H(μ)),
and again by Egorov’s theorem there is a setE2 ⊂DNwithμ(E2)≥ 34, andK2∈Nsuch that (4.17) holds for alli ∈ E2 andk ≥ K2. The conclusion of the lemma follows taking E=τ(E1∩E2)andK0 =max{K1,K2}.
We now obtain a lower bound for dimθF, showing in particular that dimθF >dimHF for allθ >0.
Proposition 4.3 Let F be the Bedford–McMullen carpet as above. Then for0≤θ ≤lognm,
dimθF≥dimHF+θlog|D| −H(μ)
logm . (4.18)
Proof Fixθ ∈(0,1), letE⊂FandK0be given by Lemma4.2, and letK ≥K0. We define a measureνK which assigns equal mass to all level-K rectangles, and then subdivides this
gives a measure to which we can apply the mass distribution principle, Lemma2.2. Thus for k≥K, writingb=a(lognm−1)
/mdas in (4.4),
νK
Rk(i1, . . . ,ik)
:=|D|−KbiK+1. . .bik=|D|−
Km−(k−K)d(a
iK+1. . .aik)
L−1. (4.19)
We now consider an approximate squareQk(i)containing the pointi. This approximate
square is a union of rectangles Rk(j) which, as explained in our comment before (4.7),
each have the sameμmeasure equal toμ(Rk(i)). The same argument gives that for any Rk(j)⊂Qk(i)
νK(Rk(j))=νK(Rk(i))=μ(Rk(i)) | D|−K
μ(RK(i))
where the final equality holds since the formula forνK differs from that ofμonly in the
mass it assigns according to the firstK letters. Putting this together allows one to express theνK-measure of an approximate square of side lengthm−kin relation to theμ-measure
of such a square. Forτ(i)∈Eandk ≥K, the approximate squareQk(i)that contains the
pointihasνK-measure
νK(Qk(i))= | D|−K
μ(RK(i))μ( Qk(i))
≤ |D|−Km−k(d−ε)
exp(−K(H(μ)+ε)), (4.20)
using Lemma4.2. (Alternatively (4.20) may be verified directly using (4.5), (4.8) and (4.19).) Then
νK(Qk(i))≤m−k(d−ε)−Klogm
|D|exp(−H(μ)−ε)
≤m−k
d−ε+Kk(log|D|−H(μ)−ε)/logm. (4.21)
We need bounds that are valid for allk ∈ [K,K/θ]corresponding to approximate squares of sides between (approximately)m−K andm−K/θ. The exponent in (4.21) is maximised whenk=K/θ, so that
νK(Qk(i)) ≤ m−k
d−ε+θ(log|D|−H(μ)−ε)/logm
(4.22)
for alliwithτ(i)∈Eand integersk∈ [K,K/θ], whereμ(E)≥ 12.
To use our mass distribution principle we need equation (4.22) to hold on a set ofiof largeνK mass, whereas currently we have that it holds on a setEof largeμmass. Let
E=τ{i:inequality (4.22) is satisfied for allk∈ [K(logn/logm),K/θ]}.
Firstly we observe thatQk(i)depends only on(i1, . . .ik), and we are dealing withk≤K/θ, so
the question of whetherτ(i)∈Eis independent of(iK/θ+1,iK/θ+2, . . .). Secondly,Qk(i)
is a union of rectanglesRk(i), whereνK(Rk(i))is independent of(i1, . . . ,iK). Furthermore,
since k ≥ K(logn/logm), the number of rectangles Rk(i) ∈ Qk(i) is independent of
(i1, . . . ,iK). Thus the question of whetherτ(i) ∈ Eis independent of(i1, . . . ,iK). Thus
we can write
E=
iK+1...iK/θ∈I
⎛
⎝
i1...iK∈DK
RK/θ(i1, . . . ,iK/θ)
for some setI⊂DK/θ−K. But using (4.19) gives
νK
⎛
⎝
i1...iK∈DK
RK/θ(i1, . . . ,iK/θ)
⎞
⎠=
i1...iK∈DK
1
|D|KbiK+1. . .biK/θ =biK+1. . .biK/θ
and (4.5) gives
μ ⎛
⎝
i1...iK∈DK
RK/θ(i1, . . . ,iK/θ)
⎞
⎠=
i1...iK∈DK
bi1. . .biKbiK+1. . .biK/θ
=
i1...iK∈DK
bi1. . .biK biK+1. . .biK/θ
=biK+1. . .biK/θ, asi∈Dbi =1. Since these quantities are equal we conclude that
νK(E) = μ(E) ≥ μ(E) ≥ 12
as required.
Since (4.22) holds for all i ∈ E, a straightforward variant on our mass distribution principle, where we use approximate squares instead of balls, gives
dimθF ≥ dimHF−ε+θ
log|D| −H(μ)−ε
logm .
Sinceεcan be chosen arbitrarily small, (4.18) follows.
Note that, in (4.18),
H(μ) = −m−d
∈D
(aL−1(L−1)loga−dlogm ≤ log|D|
with equality if and only ifμgives equal mass to all cylinders of the same length, which happens if and only if each column in our construction contains the same number of rectangles. This happens exactly when the box and Hausdorff dimension coincide. Thus our lower bounds give that dimθF>dimHFwheneverθ >0, provided that the Hausdorff and box dimensions ofFare different.
Since the measures that we have constructed to give lower bounds on dimθFare rather crude, it is unlikely that our lower bound for dimθFconverges to dimBFasθ →1. However, a lower bound whichdoesapproach dimBFasθ →1 is given by Proposition2.4, noting that dimAF>dimBF=dimBFprovided dimBF>dimHF, see [4,7].
Many questions on the intermediate dimensions of Bedford–McMullen carpets remain, most notably finding the exact forms of dimθF and dimθF. In that direction we would at least conjecture that these intermediate dimensions are equal and strictly monotonic. One might hope to get better estimates using alternative definitions ofμin Proposition4.1and
νK in Proposition4.3, but McMullen’s measure and our modifications seemed to work best
when optimising mass distribution type estimates acrossFand over the required range of scales.
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