IIntermediate dimensions - a survey
Kenneth J. Falconer
Mathematical Institute, University of St Andrews,St Andrews, Fife KY16 9SS, UKE-mail: [email protected]
Abstract
This article surveys the θ -intermediate dimensions that were introduced recentlywhich provide a parameterised continuum of dimensions that run from Hausdorffdimension when θ = 0 to box-counting dimensions when θ = 1. We bring togetherdiverse properties of intermediate dimensions which we illustrate by examples. Mathematics Subject Classification 2010 : primary: 28A80; secondary: 37C45.
Key words and phrases : Hausdorff dimension, box-counting dimension, intermediatedimensions.
Many interesting fractals, for example many self-affine carpets, have differing box-countingand Hausdorff dimensions. A smaller value for Hausdorff dimension can result becausecovering sets of widely ranging scales are permitted in the definition, whereas box-countingdimensions essentially come from counting covering sets that are all of the same size. In-termediate dimensions were introduced in [9] in 2019 to provide a continuum of dimensionsbetween Hausdorff and box-counting; this is achieved by restricting the families of allow-able covers in the definition of Hausdorff dimension by requiring that | U | ≤ | V | θ for allsets U, V in an admissible cover, where θ ∈ [0 ,
1] is a parameter. When θ = 1 only coversusing sets of the same size are allowable and we recover box-counting dimension, andwhen θ = 0 there are no restrictions giving Hausdorff dimension.This article brings together what is currently known about intermediate dimensionsfrom a number of sources, especially [2, 3, 9, 19]. We first consider basic properties of θ -intermediate dimensions, notably continuity when θ ∈ (0 , R n throughout, although much of the theory easily extendsto more general metric spaces. To avoid problems of definition, we assume throughoutthis account that all the sets F ⊂ R n whose dimensions are considered are non-emptyand bounded. 1 a r X i v : . [ m a t h . M G ] N ov hilst Hausdorff dimension dim H is usually defined via Hausdorff measure, it mayalso be defined directly, see [4, Section 3.2]. For F ⊂ R n we write | F | for the diameter diameter of F and say that a finite or countable collection of subsets { U i } of R n is a cover of F if F ⊂ (cid:83) i U i . Then the Hausdorff dimension of F is given by:dim H F = inf (cid:8) s ≥ ε > { U i } of F such that (cid:80) i | U i | s ≤ ε (cid:9) . (Lower) box-counting dimension dim B may be expressed in a similar manner except thathere we require the covering sets all to be of equal diameter. For bounded F ⊂ R n ,dim B F = inf (cid:8) s ≥ ε > { U i } of F such that | U i | = | U j | for all i, j and (cid:80) i | U i | s ≤ ε (cid:9) . From this viewpoint, Hausdorff and box-counting dimensions may be regarded as extremecases of the same definition, one with no restriction on the size of covering sets, and theother requiring them all to have equal diameters; one might regard these two definitionsas the extremes of a continuum of dimensions with increasing restrictions on the relativesizes of covering sets. This motivates the definition of intermediate dimensions where thecoverings are restricted by requiring the diameters of the covering sets to lie in a geometricrange δ /θ ≤ | U i | ≤ δ (or equivalently δ ≤ | U i | ≤ δ θ ) where 0 ≤ θ ≤ Definition 1.1.
Let F ⊂ R n . For ≤ θ ≤ the lower θ -intermediate dimension of F isdefined by dim θ F = inf (cid:8) s ≥ ε > δ >
0, there exists 0 < δ ≤ δ and a cover { U i } of F such that δ /θ ≤ | U i | ≤ δ and (cid:80) | U i | s ≤ ε (cid:9) . Analogously the upper θ -intermediate dimension of F is defined by dim θ F = inf (cid:8) s ≥ ε > δ > < δ ≤ δ ,there is a cover { U i } of F such that δ /θ ≤ | U i | ≤ δ and (cid:80) | U i | s ≤ ε (cid:9) . [Note that, apart from when θ = 0, these definitions are unchanged if δ /θ ≤ | U i | ≤ δ isreplaced by δ ≤ | U i | ≤ δ θ .]It is immediate thatdim H F = dim F = dim F, dim B F = dim F and dim B F = dim F, where dim B is upper box-counting dimension. Furthermore, for a bounded F ⊂ R n and θ ∈ [0 , ≤ dim H F ≤ dim θ F ≤ dim θ F ≤ dim B F ≤ n and 0 ≤ dim θ F ≤ dim B F ≤ n. As with box-counting dimensions we often have dim θ F = dim θ F in which case we justwrite dim θ F = dim θ F = dim θ F for the θ -intermediate dimension of F .We remark that a continuum of dimensions of a different form, known as the Assouadspectrum , has also been investigated recently, see [12, 13, 15]; this provides a parameterisedfamily of dimensions which interpolate between upper box-counting dimension and quasi-Assouad dimension, but we do not pursue this here.2
Properties of intermediate dimensions
We start by indicating some basic properties of intermediate dimensions of a type thatare familiar in many definitions of dimension.1.
Monotonicity.
For all θ ∈ [0 ,
1] if E ⊂ F then dim θ E ≤ dim θ F and dim θ E ≤ dim θ F .2. Finite stability.
For all θ ∈ [0 ,
1] if
E, F ⊂ R n then dim θ E ∪ F = max { dim θ E, dim θ F } .Note that, analoguously with box-counting dimensions, dim θ is not finitely stable,and neither dim θ or dim θ are countably stable.3. Monotonicity in θ . For all bounded F , dim θ F and dim θ F are monotonicallyincreasing in θ ∈ [0 , Closure.
For all θ ∈ (0 , θ F = dim θ F and dim θ F = dim θ F where F is theclosure of F . (This follows since for θ ∈ (0 ,
1] it is enough to consider finite coversof closed sets in the definitions of intermediate dimensions.)5.
Lipschitz and H¨older properties.
Let f : F → R m be an α -H¨older map, i.e. | f ( x ) − f ( y ) | ≤ c | x − y | α for α ∈ (0 ,
1] and c >
0. Then for all θ ∈ [0 , θ f ( F ) ≤ α dim θ F and dim θ f ( F ) ≤ α dim θ F. (To see this, if { U i } is a cover of F with δ ≤ | U i | ≤ δ θ consider the cover of f ( F ) bythe sets { f ( U i ) } if cδ α ≤ | f ( U i ) | and by sets V i ⊇ f ( U i ) with | V i | = cδ α otherwise.)In particular, if f : F → f ( F ) ⊂ R m is bi-Lipschitz then dim θ f ( F ) = dim θ F anddim θ f ( F ) = dim θ F , i.e. dim θ and dim θ are bi-Lipschitz invariants. A natural question is whether, for a fixed bounded set F , dim θ F and dim θ F vary con-tinuously for θ ∈ [0 , θ = 0 wherethe intermediate dimensions may or may not be continuous, see the examples in Section4. Continuity on (0 ,
1] follows immediately from the following inequalities which relatedim θ F , respectively dim θ F , for different values of θ . Proposition 2.1.
Let F be a bounded subset of R n and let < θ < φ ≤ . Then dim θ F ≤ dim φ F ≤ φθ dim θ F (2.1) and dim θ F ≤ dim φ F ≤ dim θ F + (cid:16) − θφ (cid:17) ( n − dim θ F ) , (2.2) with corresponding inequalities where dim θ and dim φ are replaced by dim θ and dim φ . roof. We include the proof of (2.1) since it does not appear elsewhere. The left-handinequality is just monotonicity of dim θ F .With 0 < θ < φ ≤ t > φθ dim θ F and choose s such that dim θ F < s < θφ t . Given ε >
0, for all sufficiently small 0 < δ < { U i } i ∈ I of F such that (cid:88) i ∈ I | U i | s < ε and δ ≤ | U i | ≤ δ θ for all i ∈ I. (2.3)Let I = { i ∈ I : δ ≤ | U i | < δ θ/φ } and I = { i ∈ I : δ θ/φ ≤ | U i | ≤ δ θ } . For each i ∈ I let V i be a set with V i ⊃ U i and | V i | = δ θ/φ . Let 0 < s < tθ/φ ≤ n . Then { W i } i ∈ I := { V i } i ∈ I ∪ { U i } i ∈ I is a cover of F by sets with diameters in the range [ δ θ/φ , δ θ ].Taking sums with respect to this cover: (cid:88) i ∈ I | W i | t = (cid:88) i ∈ I | V i | t + (cid:88) i ∈ I | U i | t = (cid:88) i ∈ I δ t θ/φ + (cid:88) i ∈ I | U i | t ≤ (cid:88) i ∈ I | U i | t θ/φ + (cid:88) i ∈ I | U i | t θ/φ = (cid:88) i ∈ I | U i | t θ/φ ≤ (cid:88) i ∈ I | U i | s < ε. (2.4)Thus for all t > φθ dim θ F , for all ε >
0, for all sufficiently small δ (equivalently, for allsufficiently small δ θ ) there is a cover { W i } i of F by sets with ( δ θ ) /φ ≤ | W i | ≤ δ θ satisfying(2.4), so dim φ F ≤ φθ dim θ F .The analogue of (2.1) for dim θ follows by exactly the same argument by choosingcovers of F with δ ≤ | U i | ≤ δ θ for arbitrarily small δ .The proof of (2.2) is given in [9]: essentially given a cover of F by sets { U i } with δ ≤ | U i | ≤ δ θ one breaks up those U i with δ φ ≤ | U i | ≤ δ θ into smaller pieces to get a coverof F by sets with diameters in the range [ δ, δ φ ].Note that the right hand inequality of (2.1) is stronger than that in (2.2) preciselywhen θφ ≤ n dim φ F −
1, which is the case for all 0 < θ < φ ≤ φ F ≤ n ; similarlyfor lower dimensions.Also note that (2.1) implies that dim θ Fθ and dim θ Fθ are monotonic decreasing in θ ∈ (0 , θ (cid:55)→ dim θ F and θ (cid:55)→ dim θ F (0 < θ ≤
1) are starshapedwith respect to the origin.The following corollary is immediate.
Corollary 2.2.
The maps θ (cid:55)→ dim θ F and θ (cid:55)→ dim θ F are continuous for θ ∈ (0 , . By setting φ = 1 in Proposition 2.1 and rearranging we get useful comparisons withbox-counting dimensions. Corollary 2.3.
Let F be a bounded subset of R n . Then dim θ F ≥ n − (cid:0) n − dim B F (cid:1) θ (2.5)4 nd dim θ F ≥ θ dim B F, (2.6) with corresponding inequalities where dim θ and dim B are replaced by dim θ and dim B . Again (2.6) gives a better lower bound than (2.5) if and only if θ ≤ n dim B F − θ ∈ (0 ,
1] if dim B F ≤ n , and similarly for lower dimensions.Intermediate dimensions may or may not be continuous when θ = 0, see Section 4.2for examples. Indeed, determining whether a given set has intermediate dimensions thatare continuous at θ = 0, which relates to the distribution of scales of covering sets forHausdorff and box dimensions, is one of the key question in this subject. As with other notions of dimension, there are some basic techniques that are useful forstudying intermediate dimensions and calculating them in specific cases.
The mass distribution principle is frequently used for finding lower bounds for Hausdorffdimension by considering local behaviour of measures supported on the set, see [4, Prin-ciple 4.2]. Here are the natural analogues for dim θ and dim θ which are proved using aneasy modification of the standard proof for Hausdorff dimensions. Proposition 3.1. [9, Proposition 2.2]
Let F be a Borel subset of R n and let ≤ θ ≤ and s ≥ . Suppose that there are numbers a, c > such that for arbitrarily small δ > we can find a Borel measure µ δ supported on F such that µ δ ( F ) ≥ a , and with µ δ ( U ) ≤ c | U | s for all Borel sets U ⊂ R n with δ ≤ | U | ≤ δ θ . (3.7) Then dim θ F ≥ s . Alternatively, if measures µ δ with the above properties can be found forall sufficiently small δ , then dim θ F ≥ s . Note that in Proposition 3.1 a different measure µ δ is used for each δ , but it is essentialthat they all assign mass at least a > F . In practice µ δ is often a finite sum of pointmasses. Frostman’s lemma is another powerful tool in fractal geometry which is a sort of dual toProposition 3.1. We state here a version for intermediate dimensions. As usual B ( x, r )denotes the closed ball of centre x and radius r . Proposition 3.2. [9, Proposition 2.3]
Let F be a compact subset of R n , let < θ ≤ ,and let < s < dim θ F . Then there exists c > such that for all δ ∈ (0 , there is aBorel probability measure µ δ supported on F such that for all x ∈ R n and δ /θ ≤ r ≤ δ , µ δ ( B ( x, r )) ≤ cr s . (3.8)5onathan Fraser has pointed out a nice alternative proof of (2.1) using the Frostman’slemma and the mass distribution principle. Briefly, let 0 < θ < φ ≤
1. if s < dim φ F ,Proposition 3.2 gives probability measures µ δ on F (which we may take to be compact)such that µ δ ( B ( x, r )) ≤ cr s for δ /φ ≤ r ≤ δ . If δ /θ ≤ r ≤ δ /φ then µ δ ( B ( x, r )) ≤ µ δ ( B ( x, δ /φ )) ≤ c δ s/φ ≤ c r sθ/φ , so µ δ ( B ( x, r )) ≤ c r sθ/φ for all δ /θ ≤ r ≤ δ . Using Proposition 3.1 dim θ F ≥ sφ/θ . Thisis true for all s < dim φ F so dim θ F ≥ θφ dim φ F . Assouad dimension has been studied intensively in recent years, see the books [13, 24]and paper [11]. Although Assouad dimension does not a priori seem closely related tointermediate dimensions, it turns out that information about the Assouad dimension of aset can inform calculations of intermediate dimensions and under certain conditions implydiscontinuity at θ = 0.The Assouad dimension of F ⊂ R n is defined bydim A F = inf (cid:110) s ≥ C > N r ( F ∩ B ( x, R )) ≤ C (cid:16) Rr (cid:17) s for all x ∈ F and all 0 < r < R (cid:111) , where N r ( A ) denotes the smallest number of sets of diameter at most r that can cover aset A . In general dim B F ≤ dim B F ≤ dim A F ≤ n , but equality of these three dimensionsoften occurs, even if the Hausdorff dimension and box-counting dimension differ, forexample if the box-counting dimension is equal to the ambient spatial dimension. Thefollowing proposition gives useful lower bounds for intermediate dimensions in terms ofAssouad dimensions, in particular near θ = 1. Proposition 3.3. [9, Proposition 2.4]
For a bounded set F ⊂ R n and θ ∈ (0 , , dim θ F ≥ dim A F − dim A F − dim B Fθ , with a similar inequality for uper dimensions. In particular, if dim B F = dim A F (whichis always the case if dim B F = n ), then dim θ F = dim θ F = dim B F = dim A F for all θ ∈ (0 , . One consequence of Proposition 3.3 is that if dim H F < dim B F = dim A F , then theintermediate dimensions dim θ F and dim θ F are constant on (0 ,
1] and discontinuous at θ = 0. This will help us analyse examples that exhibit a range of behaviours in Section4.2 . It is natural to relate dimensions of products of sets to those of the sets themselves. Thefollowing product formulae for intermediate dimensions are of interest in their own rightand are also useful in constructing examples.6 roposition 3.4. [9, Proposition 2.5]
Let E ⊂ R n and F ⊂ R m be bounded and let θ ∈ [0 , . Then dim θ E + dim θ F ≤ dim θ ( E × F ) ≤ dim θ ( E × F ) ≤ dim θ E + dim B F. (3.9) Sketch proof.
The cases θ = 0 , θ the lefthand inequality follows by using Proposition 3.2 to put measures on E and F satisfyinginequalities of the form (3.8) and then applying Proposition 3.1 to the product of thesetwo measures.The middle inequality is trivial. For the right hand inequality let s > dim θ E and d > dim B F . We can find a cover of E by sets { U i } with δ /θ ≤ | U i | ≤ δ for all i andwith (cid:80) i | U i | s ≤ ε . Then, for each i , we find a cover { U i,j } j of F by at most | U i | − d setswith diameters | U i,j | = | U i | for all j . Thus E × F ⊂ (cid:83) i (cid:83) j (cid:0) U i × U i,j (cid:1) where δ /θ ≤| U i × U i,j | ≤ √ δ for all i, j . A simple estimate gives (cid:80) i (cid:80) j | U i × U i,j | s + d ≤ ( s + d ) / ε ,leading to the right hand inequality. (cid:50) The following basic examples in R or R serve to give a feel for intermediate dimensionsand indicate some possible behaviours of dim θ and dim θ as θ varies. The p th power sequence for p > F p = (cid:110) , p , p , p , . . . (cid:111) . (4.1)Since F p is countable, dim H F p = 0 and a standard exercise showa that dim B F p = 1 / ( p +1),see [4, Chapter 2]. We obtain the intermediate dimensions of F p . Proposition 4.1. [9, Proposition 3.1]
For p > and ≤ θ ≤ , dim θ F p = dim θ F p = θp + θ . (4.2) Sketch proof.
This is clearly valid when θ = 0. Otherwise, to bound dim θ F p from above,let 0 < δ < M = (cid:100) δ − ( s + θ (1 − s )) / ( p +1) (cid:101) . Take a covering U of F p consisting of the M intervals B ( k − p , δ/
2) of length δ for 1 ≤ k ≤ M together with (cid:100) M − p /δ θ (cid:101) ≤ M − p /δ θ + 1intervals of length δ θ that cover the left hand interval [0 , M − p ] . Then (cid:88) U ∈U | U | s ≤ M δ s + δ θs (cid:16) M p δ θ + 1 (cid:17) (4.3) ≤ δ ( θ ( s − sp ) / ( p +1) + δ s + δ θs → δ → s ( θ + p ) > θ . Thus dim θ F p ≤ θ/ ( p + θ ). [Note that M was chosen essentiallyto minimise the expression (4.3) for given δ .]7or the lower bound we put a suitable measure on F p and apply Proposition 3.1. Let s = θ/ ( p + θ ) and 0 < δ < M = (cid:100) δ − ( s + θ (1 − s )) / ( p +1) (cid:101) .Define µ δ as the sum of point masses on the points 1 /k p (1 ≤ k < ∞ ) with µ δ (cid:16)(cid:110) k p (cid:111)(cid:17) = (cid:26) δ s if 1 ≤ k ≤ M M + 1 ≤ k < ∞ . (4.4)Then µ δ ( F p ) = M δ s ≥ δ − ( s + θ (1 − s )) / ( p +1) δ s = 1by the choice of s . To check (3.7), note that the gap between any two points of F p carrying mass is at least p/M p +1 . A set U such that δ ≤ | U | ≤ δ θ , intersects at most1 + | U | / ( p/M p +1 ) = 1 + | U | M p +1 /p of the points of F p which have mass δ s . Hence µ δ ( U ) ≤ δ s + 1 p | U | δ s δ − ( s + θ (1 − s )) ≤ (cid:16) p (cid:17) | U | s , Proposition 3.1 gives dim θ F p ≥ s = θ/ ( p + θ ). (cid:50) Here is a generalisation of Proposition 4.1 to sequences with ‘decreasing gaps’. Let a ∈ R and let f : [ a, ∞ ) → (0 ,
1] be continuously differentiable with f (cid:48) ( x ) negative andincreasing and f ( x ) → x → ∞ . Considering integer values, the mean value theoremgives that f ( n ) − f ( n + 1) is decreasing, so the sequence { f ( n ) } n is a ‘decreasing sequencewith decreasing gaps’. Proposition 4.2.
With f as above, let F = (cid:8) , f (1) , f (2) , . . . (cid:9) . Suppose that xf (cid:48) ( x ) f ( x ) → − p as x → ∞ , where ≤ p ≤ ∞ . Then for all < θ ≤ , dim θ F = dim θ F = θp + θ , taking this expression to be when p = ∞ . This may be proved in a similar way to Proposition 4.1 using that xf (cid:48) ( x ) /f ( x ) is closeto, rather than equal to, − p when x is large.For example, taking f ( x ) = 1 / log( x + 1), the sequence F log = (cid:110) , , , , . . . (cid:111) (4.5)has dim θ F log = 1 if θ ∈ (0 ,
1] and dim F log = 0, so there is a discontinuity at 0. On theother hand, with f ( x ) = e − x , F exp = (cid:8) , e − , e − , e − , . . . (cid:9) has dim θ F exp = 0 for all θ ∈ [0 , θ F for the three examples in Section 4.2 Using the examples above together with tools from Section 3 we can build up simpleexamples of sets exhibiting various behaviours as θ ranges over [0 , Example 1: Continuous at , part constant, then strictly increasing. Let F = F ∪ E where F is as in (4.1) and let E ⊂ R be any compact set with dim H E = dim B E = 1 / θ F = max (cid:110) θ θ , / (cid:111) ( θ ∈ [0 , . This follows using (4.2) and the finite stability of upper intermediate dimensions.
Example 2: Discontinuous at , part constant, then strictly increasing. Let F = F ∪ E where this time E ⊂ R is any closed countable set with dim B E = dim A E = 1 /
4. UsingProposition 3.3 and finite stability of upper intermediate dimensions,dim θ F = max (cid:110) θ θ , / (cid:111) ( θ ∈ (0 , , and dim F log = 0. Note that the intermediate dimensions are exactly as in Example 1except when θ = 0 and a discontinuity occurs. Example 3: Discontinuous at , smooth and strictly increasing. Consider the countableset F = F × F log ⊂ R . Then dim F = dim H F = 0 anddim θ F = θ θ + 1 ( θ ∈ (0 , , noting that dim θ F log = dim B F log = dim A F log = 1 for θ ∈ (0 ,
1] using (4.5) and Proposi-tions 3.3 and 3.4. 9igure 2: A 2 × × Self affine carpets are a well-studied class of fractals where the Hausdorff and box-countingdimensions generally differ; this is a consequence of the alignment of the componentrectangles in the iterated construction. The dimensions of planar self-affine carpets werefirst investigated by Bedford [1] and McMullen [22] independently, see also [23], andthese carpets have been widely studied and generalised, see [5, 14] and references therein.Finding the intermediate dimensions of these carpets gives information about the rangeof scales of covering sets needed to realise their Hausdorff and box-counting dimensions.Deriving exact formulae seems a major challenge, but some lower and upper boundshave been obtained, in particular enough to demonstrate continuity of the intermediatedimensions at θ = 0 and that they attain a strict minimum when θ = 0.Bedford-McMullen carpets are attractors of iterated function systems of a set of affinecontractions, all translates of each other which preserve horizontal and vertical directions.More precisely, for integers n > m ≥
2, an m × n -carpet is defined in the following way. Let I = { , . . . , m − } and J = { , . . . , n − } and let D ⊂ I × J be a digit set with at leasttwo elements. For each ( p, q ) ∈ D we define the affine contraction S ( p,q ) : [0 , → [0 , by S ( p,q ) ( x, y ) = (cid:18) x + pm , y + qn (cid:19) . Then (cid:8) S ( p,q ) (cid:9) ( p,q ) ∈ D is an iterated function system so there exists a unique non-emptycompact set F ⊂ [0 , satisfying F = (cid:91) ( p,q ) ∈ D S ( p,q ) ( F )called a Bedford-McMullen self-affine carpet , see Figure 2 for examples. The carpet canalso be thought of as the set constructed using a ‘template’ consisting of the selectedrectangles (cid:8) S ( p,q ) ([0 , ) (cid:9) ( p,q ) ∈ D by repeatedly substituting affine copies of the templatein each of the selected rectangles.Bedford [1] and McMullen [22] showed that the box-counting dimension of F exists10ith dim B F = log M log m + log N − log M log n (5.6)where N is the total number of selected rectangles and M is the number of p such thatthere is a q with ( p, q ) ∈ D , that is the number of columns of the template containing atleast one rectangle. They also showed thatdim H F = log (cid:0) (cid:80) mp =1 N log n mp (cid:1) log m , (5.7)where N p (1 ≤ p ≤ m ) is the number of q such that ( p, q ) ∈ D , that is the number ofrectangles in the p th column of the template. The Hausdorff and box-counting dimensionsof F are equal if and only if the number of selected rectangles in every non-empty columnis constant.Virtually all work on these carpets depends on dividing the iterated rectangles into‘approximate squares’. The box-counting dimension result (5.6) is then a straightfor-ward counting argument. The Hausdorff dimension (5.6) argument is more involved;McMullen’s approach defined a Bernoulli-type measure µ on F via the iterated rectanglesand obtained an upper bound for the local upper density of µ that is valid everywhere anda lower bound valid µ -almost everywhere. These ideas have been adapted and extendedfor estimating intermediate dimensions, but with the considerable complication that oneseeks good density estimates that are valid over a restricted range of scales, but evengetting close estimates for the intermediate dimensions seems a considerable challenge.The best upper bounds known at the time of writing are:dim θ F ≤ dim H F + (cid:18) m n ) log a log n (cid:19) − log θ (cid:0) < θ < (log n m ) (cid:1) , (5.8)proved in [9]. The − / log θ term makes this a very poor upper bound as θ increases awayfrom 0, but at least it implies that dim θ F and dim θ F are continuous at θ = 0 and so arecontinuous on [0 , θ that is better except close to 0 was given in[19]: dim θ F ≤ dim B F − ∆ ( θ )log n (1 − θ ) < dim B F (log n m < θ < , (5.9)where ∆ ( θ ) is the solution an equation involving a large deviation rate term which canbe found numerically in particular cases. This upper bound is strictly decreasing near 1and by monotonicity also gives a constant upper bound if 0 < θ < log n m .A reasonable lower bound that is linear in θ isdim θ F ≥ dim H F + θ log | D | − H ( µ )log n (0 ≤ θ ≤ , (5.10)where H ( µ ) is the entropy of McMullen’s measure µ ; this was essentially proved in [9], butsee [19] for a note on the constant. In particular this implies that there is a strict minimumfor the intermediate dimensions at θ = 0. An alternative lower bound depending on aoptimising a certain function was given by [19]:dim θ F ≥ sup t> ψ ( t, θ ) (0 ≤ θ ≤
1) (5.11)11ere ψ ( t, θ ) depends on entropies of linear interpolants of probability measures of theform θ t (cid:101) p + (1 − θ t ) (cid:98) p and θ t (cid:101) q + (1 − θ t ) (cid:98) q where (cid:101) p , (cid:101) q and (cid:98) p , (cid:98) q are measures that occurnaturally in the calculations for, respectively, the box-counting and Hausdorff dimensionsof the carpets. Of course, the general lower bounds given by Corollary 2.3 in terms of thebox-counting dimension also apply here.Many questions on the intermediate dimensions of Bedford-McMullen carpets remain,most notably finding the exact forms of dim θ F and dim θ F . Towards that we would atleast conjecture that the lower and upper intermediate dimensions are equal and strictlymonotonic. The potential-theoretic approach for estimating Hausdorff dimensions goes back to Kauf-man [18]. More recently box-counting dimensions have been defined in terms of energiesand potentials with respect to suitable kernels and these have been used to obtain resultson the box-counting dimensions of projections of sets in terms of ‘dimension profiles’, see[6, 7]. In particular the box-counting dimension of the projection of a Borel set F ⊂ R n onto m -dimensional subspaces is constant for almost all subspaces (with respect to thenatural invariant measure), generalising the long-standing results of Marstrand [20] andMattila [21] for Hausdorff dimensions.As with Hausdorff and box-counting dimensions, it turns out that θ -intermediate di-mensions can be characterised in terms of capacities with respect to certain kernels, andthis can be extremely useful as will be seen in Section 7. Let θ ∈ (0 ,
1] and 0 < m ≤ n ( m is often an integer, though it need not be so). For 0 ≤ s ≤ m and 0 < r <
1, definethe kernels φ s,mr,θ ( x ) = ≤ | x | < r (cid:0) r | x | (cid:1) s r ≤ | x | < r θr θ ( m − s )+ s | x | m r θ ≤ | x | ( x ∈ R n ) . (6.12)If s = m this reduces to φ m,mr,θ ( x ) = (cid:40) ≤ | x | < r (cid:0) r | x | (cid:1) m r ≤ | x | ( x ∈ R n ) , (6.13)which are the kernels φ mr ( x ) used in the context of box-counting dimensions [6, 7]. Notethat φ s,mr,θ ( x ) is continuous in x and monotonically decreasing in | x | . Let M ( F ) denotethe set of Borel probability measures supported on a compact F ⊂ R n . The energy of µ ∈ M ( F ) with respect to φ s,mr,θ is (cid:90) (cid:90) φ s,mr,θ ( x − y ) dµ ( x ) dµ ( y ) (6.14)and the potential of µ at x ∈ R n is (cid:90) φ s,mr,θ ( x − y ) dµ ( y ) . (6.15)12he capacity C s,mr,θ ( F ) of F is the reciprocal of the minimum energy achieved by probabilitymeasures on F , that is C s,mr,θ ( F ) = (cid:18) inf µ ∈M ( E ) (cid:90) (cid:90) φ s,mr,θ ( x − y ) dµ ( x ) dµ ( y ) (cid:19) − . (6.16)Since φ s,mr,θ ( x ) is continuous in x and strictly positive and F is compact, C s,mr,θ ( F ) is positiveand finite. For general bounded sets we take the capacity of a set to be that of its closure.The existence of energy minimising measures and the relationship between the minimalenergy and the corresponding potentials is standard in classical potential theory, see [6,Lemma 2.1] and [3] in this setting. In particular, there exists an equilibrium measure µ ∈ M ( E ) for which the energy (6.14) attains a minimum value, say γ . Moreover,the potential (6.15) of this equilibrium measure is at least γ for all x ∈ F (otherwiseperturbing µ by a point mass where the potential is less than γ reduces the energy) withequality for µ -almost all x ∈ F . These properties turn out to be key in expressing thesedimensions in terms of capacities.Let F ⊂ R n be compact, m ∈ (0 , n ], θ ∈ (0 ,
1] and r ∈ (0 , C s,mr,θ ( F ) − log r − s (6.17)is continuous in s and decreases monotonically from positive when s = 0 to negative or0 when s = m . Thus there is a unique s for which (6.17) equals 0. Moreover, the rateof decrease of (6.17) is bounded away from 0 and from −∞ uniformly for r ∈ (0 , r → m ∈ (0 , n ] define the lower θ -intermediate dimension profile of F ⊂ R n asdim mθ F = the unique s ∈ [0 , m ] such that lim inf r → log C s,mr,θ ( F ) − log r = s (6.18)and the upper θ -intermediate dimension profile asdim mθ F = the unique s ∈ [0 , m ] such that lim sup r → log C s,mr,θ ( F ) − log r = s. (6.19)Since the kernels φ t,mr,θ ( x ) are decreasing in m the intermediate dimension profiles (6.18)and (6.19) are increasing in m .The reason for introducing (6.18) and (6.19) is that they not only permit an equivalentdefinition of θ -intermediate dimensions but also give the intemediate dimensions of theimages of sets under certain mappings, as we will see in Section 7. The following theoremstates the equivalence between intermediate dimensions when defined by sums of powersof diameters as in Definition 1.1 and using this capacity formulation. Theorem 6.1.
Let F ⊂ R n be bounded and θ ∈ (0 , . Then dim θ F = dim nθ F and dim θ F = dim nθ F. s -power sums of diam-eters of covering balls of F with diameters in the required range, using a decompositioninto annuli to relate this to the kernels, see [3, Section4].We defined the intermediate dimension profiles dim mθ F and dim mθ F for F ⊂ R n butTheorem 6.1 refers just to the case when m = n . The significance of these dimensionprofiles when 0 < m < n will become clear in the next section. The relationship between the dimensions of a set F ⊂ R n and its orthogonal projections π V ( F ) onto subspaces V ∈ G ( n, m ), where G ( n, m ) is the Grassmanian of m -dimensionalsubspaces of R n and π V : R n → V denotes orthogonal projection, goes back to thefoundational work on Hausdorff dimension by Marstrand [20] for G (2 ,
1) and Mattila [21]for general G ( n, m ). They showed that for a Borel set F ⊂ R n dim H π V ( F ) = min { dim H F, m } (7.20)for almost all m -dimensional subspaces V with respect to the natural invariant probabilitymeasure γ n,m on G ( n, m ), where dim H denotes Hausdorff dimension. Later Kaufman [18]gave a potential-theoretic proof of these results. See, for example, [8] for a survey ofthe many generalisations, specialisations and consequences of these projection results.In particular, there are theorems that guarantee that the lower and upper box-countingdimensions and the packing dimensions of the projections π V ( F ) are constant for almostall V ∈ G ( n, m ), see [10, 16]. This constant value is not the direct analogue of (7.20) butrather it is given by a dimension profile of F . More recently these profiles were definedin terms of capacities of F with respect to certain energy kernels, see [6, 7].Thus a natural question is whether there is a Marstrand-Mattila-type theorem forintermediate dimensions, and it turns out that this is the case with the θ -intermediatedimension profiles dim mθ F and dim mθ F defined in (6.18) and (6.19) providing the almostsure values for orthogonal projections from R n onto m -dimensional subspaces. Intuitively,we think of dim mθ F and dim mθ F as the intermediate dimensions of F when regarded froman m -dimensional viewpoint. Theorem 7.1.
Let F ⊂ R n be bounded. Then, for all V ∈ G ( n, m )dim θ π V F ≤ dim mθ F and dim θ π V F ≤ dim mθ F (7.21) for all θ ∈ (0 , . Moreover, for γ n,m -almost all V ∈ G ( n, m ) , dim θ π V F = dim mθ F and dim θ π V F = dim mθ F (7.22) for all θ ∈ (0 , . The upper bounds in (7.21) utilise the fact that orthogonal projection does not increasedistances, so does not increase the values taken by the kernels, that is φ s,mr,θ ( π V x − π V y ) ≥ φ s,mr,θ ( x − y ) ( x, y ∈ R n ) . By comparing the energy of the equilibrium measure on F with its projections ontoeach π V F it follows that C s,mr,θ ( π V F ) ≥ C s,mr,θ ( F ) and using (6.18) or (6.19) gives the θ -intermediate dimensions of π V F as a subset of the m -dimensional space V .14he almost sure lower bounds in (7.22) essentially depend on the relationship betweenthe kernels and on R n and on their averages over V ∈ G ( n, m ). More specifically, for m ∈ { , . . . , n − } and 0 ≤ s < m there is a constant a >
0, depending only on n, m and s , such that for all x ∈ R n , θ ∈ (0 ,
1) and 0 < r < , (cid:90) φ s,mr,θ ( π V x − π V y ) dγ n,m ( V ) ≤ a φ s,mr,θ ( x − y ) log r | x − y | . Using this for a sequence r = 2 − k with a Borel-Cantelli argument gives (7.22). Full detailsmay be found in [3, Section 5].Theorem 7.1 has various consequences, firstly concerning continuity at θ = 0. Corollary 7.2.
Let F ⊂ R n be such that dim θ F is continuous at θ = 0 . Then dim θ π V F is continuous at θ = 0 for almost all V . A similar result holds for the upper intermediatedimensions.Proof. If dim H F ≥ m then for almost all V , dim H π V ( F ) = m = dim θ π V F for all θ ∈ [0 , V and all θ ∈ [(0 , H F = dim H π V ( F ) ≤ dim θ π V F ≤ dim θ F → dim H F as θ →
0, where we have used (7.20) and (7.21).For example, taking F ⊂ R to be an m × n Bedford-McMullen carpet (see Section 5),it follows from (5.8) and Corollary (7.2) that the intermediate dimensions of projectionsof F onto almost all lines are continuous at 0. In fact more is true: if log m/ log n / ∈ Q then dim θ π V F and dim θ π V F are continuous at 0 for projections onto all lines V , see [3,Corollaries 6.1 and 6.2] for more details.The following curious corollary shows that continuity of intermediate dimensions of aset at 0 has implications for the box-counting dimensions of the projections of a set. Corollary 7.3.
Let F ⊂ R n be a bounded set such that dim θ F is continuous at θ = 0 .Then dim B π V F = m for almost all V ∈ G ( n, m ) if and only if dim H F ≥ m. A similar result holds on replacing lower by upper dimensions.Proof.
The ‘if’ direction is clear even without the continuity assumption, since if dim H F ≥ m , then m ≥ dim B π V F ≥ dim H π V F ≥ m for all V using (7.20).On the other hand, suppose that dim B π V F = m for almost all V . The final statementof Proposition 3.3 gives that dim θ π V F = m for all θ ∈ (0 ,
1] for almost all V . As dim θ F is assumed continuous at θ = 0, Corollary 7.2 implies that dim θ π V F is continuous at 0for almost all V and so dim H F = dim H π V F = dim π V F = m for almost all V , using(7.20). 15n interesting example is given by products of the sequence sets F p of (4.1) for p > B F p = θ/ ( θ + p ) so by Proposition 3.4dim θ ( F p × F p ) = 2 θθ + p ( θ ∈ [0 , , which is continuous at θ = 0. Since dim H ( F p × F p ) = 0, Corollary 7.3 implies thatdim B π V ( F p × F p ) < V . This is particularly striking when p is close to 0, since dim B ( F p × F p ) =2 / (1 + p ) is close to 2 but still the box-counting dimensions of its projections never reach1. In fact, a calculation not unlike that in Proposition 4.1 shows that for all projectionsonto lines V , apart from the horizontal and vertical projections,dim B π V ( F p × F p ) = 1 − (cid:18) pp + 1 (cid:19) , Analogous ideas using dimension profiles can be be used to find dimensions of images ofa given set F under other parameterised families of mappings. These include images undercertain stochastic processes (which are parameterised by points in the probability space).For example, let B α : R → R m be index- α fractional Brownian motion where 0 < α < Theorem 7.4.
Let F ⊂ R n be compact. Then, almost surely dim θ B α ( F ) = 1 α dim mαθ F and dim θ B α ( F ) = 1 α dim mαθ F (7.23)The proof of this is along the same lines as for projections, see [2] for details. Theupper bound uses that for all ε > | B α ( x ) − B α ( y ) | ≤ M | x − y | / − ε for x, y ∈ F , where M is a randomconstant. The almost sure lower bound uses that E (cid:0) φ smr,θ ( B α ( x ) − B α ( y )) (cid:1) ≤ c φ smr,θ ( x − y )where c depends only on m and s . Acknowledgements
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