GGeneralised intermediate dimensions
Amlan Banaji ∗ Abstract
We introduce a family of dimensions, which we call the Φ-intermediate dimensions, thatlie between the Hausdorff and box dimensions and generalise the intermediate dimensionsintroduced by Falconer, Fraser and Kempton. We do this by restricting the allowable covers inthe definition of Hausdorff dimension, but in a wider variety of ways than in the definition of theintermediate dimensions. We also extend the theory from Euclidean space to a wider class ofmetric spaces. We investigate relationships between the Φ-intermediate dimensions and othernotions of dimension, and study many analytic and geometric properties of the dimensions.We prove continuity-like results which improve similar results for the intermediate dimensionsand give a sharp general lower bound for the intermediate dimensions that is positive for all θ ∈ (0 ,
1] for sets with positive box dimension. We prove H¨older distortion estimates whichimply bi-Lipschitz stability for the Φ-intermediate dimensions. We prove a mass distributionprinciple and Frostman type lemma, and use these to study dimensions of product sets, and toshow that the lower versions of the dimensions, unlike the the upper versions, are not finitelystable. We show that for any compact subset of an appropriate space, these dimensions canbe used to ‘recover the interpolation’ between the Hausdorff and box dimensions of sets forwhich the intermediate dimensions are discontinuous at θ = 0, thus providing more refinedgeometric information about such sets. Mathematics Subject Classification 2020 : 28A80 (Primary); 28A78 (Secondary)
Key words and phrases : intermediate dimensions, Φ-intermediate dimensions, Hausdorffdimension, box dimension, dimension interpolation
When studying the geometry of fractal subsets of a metric space such as R n , it is common toconsider different notions of dimension, which attempt to quantify the extent to which the subsetfills up space at small scales. Two of the most common notions of dimension are the Hausdorff andbox dimensions. In [6], Falconer, Fraser and Kempton noted that these “may be regarded as twoextreme cases of the same definition, one with no restriction on the size of covering sets, and theother requiring them all to have equal diameters,” as is clear from (2.2) and (2.1). They defined afamily of dimensions, the intermediate dimensions , by restricting the sizes of the covering sets ina particular way, explained below. As the name suggests, these lie between the Hausdorff and boxdimensions, and have been studied in [1, 2, 3, 5, 6, 9, 19, 26], with a specific variant used in [20]to study the singular sets of certain partial differential equations.When defining the intermediate dimensions, Falconer, Fraser and Kempton restricted the sizesof the covering sets in the definition of Hausdorff dimension to lie in intervals of the particularform [ δ /θ , δ ]. For any set, the intermediate dimensions are continuous at each θ ∈ (0 , θ = 0, and so fully interpolatebetween the Hausdorff and box dimensions, which means that the difference between the Hausdorffand box dimensions is captured by restricting the covering sets to windows of the form [ δ /θ , δ ] fordifferent θ . Continuity of the intermediate dimensions at θ = 0 has many powerful consequences, ∗ University of St Andrews, UK; email: [email protected] a r X i v : . [ m a t h . M G ] N ov n particular for the box dimensions of projections of the set (see [2]) and images of the setunder stochastic processes such as fractional Brownian motion (see [1]). However, it is clearfrom [6, Proposition 2.4] and Proposition 3.10 that there are many compact sets for which theintermediate dimensions are discontinuous at θ = 0, or even constant at the value of the boxdimension, in which case they give very little information at all; one simple example of such a setis { } ∪ { n : n ∈ N , n ≥ } , see [6, Example 1]. In this paper, we introduce the Φ-intermediatedimensions, by restricting the sizes of the covering sets to lie in a wider class of intervals of theform [Φ( δ ) , δ ] for a more general function Φ. These dimensions give even more refined geometricinformation than the intermediate dimensions about the sizes of balls needed to cover efficiently setsfor which the intermediate dimensions are discontinuous at θ = 0. Generally speaking, sets withdifferent Hausdorff and lower box dimensions are inhomogeneous in space, while sets with differentlower and upper box dimensions are inhomogeneous in scale. The Φ-intermediate dimensions of aset are affected by the inhomogeneity of the set in both space and scale.While many results for the Φ-intermediate dimensions are similar to results for the intermediatedimensions, others, such as Theorem 4.1 (the H¨older distortion estimates), and Theorem 6.1 on‘recovering the interpolation’ for sets whose intermediate dimensions are discontinuous at θ = 0,are different, and the Φ-intermediate dimensions give rise to a rich and workable theory in theirown right. In forthcoming work, we will investigate sharpness for some bounds in this paper andgive further examples and applications of the Φ-intermediate dimensions, including using some ofthe results in Section 4 to give information about whether certain sets are bi-Lipschitz equivalentor about the possible exponents of H¨older maps between them. One possible question whichhas not yet been explored is whether capacity theoretic methods and dimension profiles can beused to obtain results about the Φ-intermediate dimensions of projections of sets, as was donefor the intermediate dimensions in [2], or about the Φ-intermediate dimensions of images of setsunder stochastic processes such as fractional Brownian motion, as was done for the intermediatedimensions in [1].The intermediate dimensions are an example of dimension interpolation (discussed in [9]), anarea that was introduced relatively recently but which has already gathered significant interest. Theidea is to consider two different notions of dimension and to find a geometrically meaningful familyof dimensions which lie between these dimensions and share some of the characteristics of both.This family should satisfy the basic properties expected of any reasonable notion of dimension, sharesome of the properties of the two individual dimensions, and provide more geometric informationabout subsets than either of the two dimensions does individually. The hope is that, as well as beinginteresting to study for its own sake, dimension interpolation will help improve understanding ofthe two dimensions individually by illuminating why the two dimensions can give different valuesfor certain sets. Another example of dimension interpolation is the Assouad spectrum, introducedby Fraser and Yu in [12], which interpolates between the upper box and quasi-Assouad dimensions,giving information about the ‘thickest’ part of the set by fixing relative scales. A more general classof dimensions, variously called the φ -Assouad dimensions, Φ -dimensions or generalised Assouadspectra , were also introduced in [12], greatly developed by Garc´ıa, Hare and Mendivil in [15],and further studied in [14, 27], with a version for measures studied in [16]. They are defined byfixing the relative scales in more general ways than for the Assouad spectrum, and lie between theupper box and Assouad dimensions, giving more refined geometric information about sets whosequasi-Assouad dimension is less that the Assouad dimension. Philosophically, the Φ-intermediatedimensions are to the intermediate dimensions what the φ -Assouad dimensions are to the Assouadspectrum, and the condition that the intermediate dimensions of a set are discontinuous at θ = 0can be seen as parallel to the condition that the quasi-Assouad dimension of a set is less than itsAssouad dimension. Indeed, the φ -Assouad dimensions were part of our original motivation forconsidering the Φ-intermediate dimensions, and many of the results in this paper have analoguesfor the φ -Assouad dimensions in [15]. 2 ummary and discussion of main results In Section 2, we introduce the notation and the types of metric spaces that we work with. We definethe notions of dimension that we will need, and state some basic properties of these dimensions.In Section 3, we give relationships between the different notions of dimension, Propositions 3.1and 3.11. In Theorem 3.3 and Proposition 3.4 we prove quantitative continuity-like properties forthe Φ-intermediate dimensions, which intuitively say that if two functions Φ and Φ are ‘close’ toeach other then the dimensions do not differ too much for subsets whose Assouad dimension is nottoo large. From this we deduce a condition (Proposition 3.4.2 (ii)) for the Φ- and Φ -intermediatedimensions to coincide for all subsets with finite Assouad dimension, and in Proposition 3.11 weobtain bounds comparing the θ -intermediate and Φ-intermediate dimensions. For the special casethat is the intermediate dimensions, the continuity-like results give an improvement (Proposi-tion 3.8) of the continuity-like results of Falconer, Fraser and Kempton [6, Proposition 2.1] andFalconer [5, (2.6)]. This also allows us to obtain a sharp general lower bound (Proposition 3.10)for the intermediate dimensions of a set in terms of its box and Assouad dimensions, which im-proves the general lower bounds [6, Proposition 2.4] and [5, (2.6)]. Notably, our bound is strictlypositive for all θ ∈ (0 ,
1] if the box dimension of the set is positive; there is a ‘mutual depen-dency’ (Proposition 3.7) between the box and intermediate dimensions (in Euclidean space, thismutual dependency also follows from Falconer’s general lower bound [5, (2.6)]). This is similar tothe mutual dependency between the upper box dimension, Assouad spectrum and quasi-Assouaddimension, which follows from work in [11, 12, 13]. This shows that in order to check that theupper/lower box dimension of a set is 0, it suffices to check the a priori weaker condition that the(respectively) upper/lower θ -intermediate dimension of the set is 0 at any (small) θ ∈ (0 , θ close to 1 that is known at the time of writing.In Section 4 we prove H¨older distortion estimates for the Φ-intermediate dimensions in The-orem 4.1 which, interestingly, are different from the standard dim f ( F ) ≤ α − dim F bound forimages under α -H¨older maps which holds for the Hausdorff, box and θ -intermediate dimensions.The estimates imply bi-Lipschitz stability (Corollary 4.1.1 2.), which is an important propertythat most notions of dimension satisfy. This means that the Φ-intermediate dimensions provideyet another invariant for the classification of subsets up to bi-Lipschitz image.In Section 5 we prove a mass distribution principle (Lemma 5.1) and a converse, a Frostmantype lemma (Lemma 5.2) for the Φ-intermediate dimensions, similar to [6, Propositions 2.2 and2.3 respectively] for the intermediate dimensions. These two lemmas combine to give Theorem 5.3,a useful characterisation of the Φ-intermediate dimensions in terms of measures. We use themass distribution principle Lemma 5.1 to prove in Proposition 5.4 that the lower versions ofthe intermediate and Φ-intermediate dimensions are not finitely stable, in contrast to the upperversions of the dimensions which are finitely stable (and so in this sense better behaved). Wealso use Theorem 5.3 to prove Theorem 5.5 on the dimensions of product sets, giving new bounds(some of which are not stated for the intermediate dimensions in [6, Proposition 2.5]) in termsof the dimensions of the marginals, one of which we improve further in the case of self-products.Theorem 5.5 shows that (dim Φ , dim B ) and (dim θ , dim B ) are examples of ‘dimension pairs’ satisfyingthe standard inequalities (5.10) for dimensions of product sets that many pairs of dimensionssatisfy, although our upper bound for dim Φ ( E × F ) is different to what would be expected. In[6], the Frostman type lemma for the intermediate dimensions is proved in Euclidean space, butto prove Lemma 5.2 in more general metric spaces we use an analogue of the dyadic cubes ingeneral doubling metric spaces given in [18], so Lemma 5.2 and its applications to product sets areexamples of results where the extension to more general metric spaces is non-trivial.Proposition 5.4 also gives an example of a set to which the main result of Section 6 can beapplied. Theorem 6.1 shows that for any compact subset of an appropriate space there is afamily of functions Φ which fully interpolate between the Hausdorff and box dimensions. Thus3he Φ-intermediate dimensions can be used to ‘recover the interpolation’ between these dimensionswhen the intermediate dimensions of a set are discontinuous at θ = 0, and so provide morerefined geometric information than the intermediate dimensions alone. Moreover, there exists asingle family of Φ which interpolate for both the upper and lower versions of the dimensions, andwhose dimensions vary monotonically for all sets, but in Proposition 6.2 we show that there doesnot generally exist a family for which the dimensions interpolate for a given set and also varycontinuously for all other sets. Throughout, log will denote the natural logarithm, supp will denote the support of a measure, N = { , , , . . . } . We say that a sequence δ n of realnumbers is decreasing if δ n +1 ≤ δ n for all n ∈ N and strictly decreasing if δ n +1 < δ n for all n ∈ N .In the definitions in this section, we will use the convention that inf ∅ = inf {∞} = ∞ .Let ( X, d ) be a metric space. We will usually denote subsets of X by E or F . A subset F ⊆ X issaid to be bounded if sup { d ( x, y ) : x, y ∈ F } < ∞ . A cover of F will be a set of bounded subsets of X whose union contains F , and the diameter of F is | F | := sup e,f ∈ F d ( e, f ). For x ∈ X and r > B ( x, r ) = B X ( x, r ) := { y ∈ X : d ( x, y ) < r } is an open ball in X centred at x , noting that this setmay have diameter less than 2 r . For x ∈ F and r >
0, we write B F ( x, r ) := { y ∈ F : d ( x, y ) < r } for the open ball in the subset F , centred at x . For δ >
0, let N δ ( F ) be the smallest number of openballs in X with diameter less than or equal to δ with which it is possible to cover F . The subset F is totally bounded if for all δ > N δ ( F ) < ∞ . The subsets F which we consider will usuallybe non-empty and totally bounded. For C ∈ [0 , ∞ ), we say that a map f : ( X, d X ) → ( Y, d Y )of metric spaces is C -Lipschitz if d Y ( f ( x ) , f ( x )) ≤ Cd X ( x , x ) for all x , x ∈ X , Lipschitz if there exists C ∈ [0 , ∞ ) such that it is C -Lipschitz, and bi-Lipschitz if there exists C ∈ [1 , ∞ )such that C − d X ( x , x ) ≤ d Y ( f ( x ) , f ( x )) ≤ Cd X ( x , x ) for all x , x ∈ X . For α ∈ (0 ,
1] and C ∈ [0 , ∞ ), we say that f is α -H¨older or ( C, α )-H¨older if d Y ( f ( x ) , f ( x )) ≤ C d X ( x , x ) α forall x , x ∈ X , and H¨older if there exist α ∈ (0 ,
1] and C ∈ [0 , ∞ ) such that it is ( C, α )-H¨older.We will often require the metric spaces we work with to satisfy certain properties.
Definition 2.1.
For c ∈ (0 ,
1) we say a metric space X is c -uniformly perfect if for all x ∈ X and R ∈ R such that 0 < R < | X | we have B ( x, R ) \ B ( x, cR ) (cid:54) = ∅ . The space X is uniformly perfect if there exists c ∈ (0 ,
1) such that X is c -uniformly perfect.Intuitively, a metric space is uniformly perfect if it does not have islands which are very sepa-rated from the rest of the space. Definition 2.2.
A metric space is said to be doubling if there exists a constant M ∈ N (called the doubling constant ) such that for every x ∈ X and r >
0, there exists x , . . . , x M ∈ X such that B ( x, r ) ⊆ (cid:83) Mi =1 B ( x i , r ).The following notions of dimension give information about the ‘thickest’ part of subsets of ametric space. Definition 2.3.
The
Assouad dimension of a non-empty subset F of a metric space is defined bydim A F = inf { α : there exists C > x ∈ F and0 < r < R, we have N r ( B ( x, R ) ∩ F ) ≤ C ( R/r ) α } . Fraser and Yu [12] define the φ -Assouad dimension for continuous, increasing φ : [0 , → [0 , < φ ( R ) ≤ R for all R ∈ [0 , φ A F = inf { α : there exists C > x ∈ F and0 < r ≤ φ ( R ) ≤ R ≤ , we have N r ( B ( x, R ) ∩ F ) ≤ C ( R/r ) α } . R R ) instead of φ ( R ), and require that the functions satisfy slightly different condi-tions, but throughout we use the notation in Definition 2.3. The case φ ( R ) = R /θ for θ ∈ (0 , upper Assouad spectrum dim θ A . This can be expressed in terms of the Assouad spec-trum dim θ A , which is defined by fixing the scales r = φ ( R ) = R /θ and also introduced andstudied in [12], by dim θ A F = sup θ (cid:48) ∈ (0 ,θ ] dim θ (cid:48) A F for F ⊆ R d , see [11]. If F is a totally boundedset then dim B F ≤ dim φ A F ≤ dim A F and dim B F ≤ dim θ A F ≤ dim θ A F ≤ dim qA F ≤ dim A F ,and dim θ A F → dim B F as θ → + , see [12]. Also dim θ A F → dim qA F as θ → − where dim qA isthe quasi-Assouad dimension, see [11]. For a given set, the Assouad spectrum and upper Assouadspectrum are continuous in θ ∈ (0 , θ , butthe Assouad spectrum is not generally monotone [12].There are natural duals to the notions of dimension in Definition 2.3, which give informationabout the ‘thinnest’ part of subsets. The following is dual to the Assouad dimension. Definition 2.4.
The lower dimension of a subset F , with more than one point, of a metric spaceis defined bydim L F = sup { α : there exists C > x ∈ F and0 < r < R ≤ | F | , we have N r ( B ( x, R ) ∩ F ) ≥ C ( R/r ) α } . The Assouad and lower dimensions are studied in [7], where in Section 13.1.1 it is shown thata metric space is doubling if and only if dim A X < ∞ , and that a metric space with more than onepoint is uniformly perfect if and only if 0 < dim L X . In order to ensure that all the dimensionsare finite (see Proposition 3.1), in much of this paper we will work with an ambient metric space X that has more than one point and is uniformly perfect, so 0 < dim L X , to ensure that thebasic inequalities between the dimensions in Proposition 3.1 will hold. Such a metric space cannothave any isolated points, and so must be infinite. If we additionally assume that X is doubling, orequivalently dim A X < ∞ , then we will see in Proposition 3.1 that all the dimensions we work withwill be finite. Many of the results require F to have finite Assouad dimension, and this will alwaysbe the case if the underlying metric space X has finite Assouad dimension. Any Ahlfors regularmetric space with more than one point is clearly uniformly perfect and doubling; an importantexample of such a space is R n with the Euclidean metric, and an example of such a space whichis not bi-Lipschitz equivalent to any subset of R n for any n ∈ N is the Heisenberg group with itsusual Carnot-Carath´eodory metric, see [21, 23, 25].Recall the following definition. Definition 2.5.
The upper box dimension of a non-empty, totally bounded subset F of a metricspace is dim B F := lim sup δ → + log( N δ ( F )) − log δ and the lower box dimension is dim B F := lim inf δ → + log( N δ ( F )) − log δ . For non-empty, bounded subsets F of R n with the Euclidean metric, we can give an alternativedefinition of lower box dimension.dim B F = inf { s ≥ (cid:15) > { U , U , . . . } of F such that | U i | = | U j | for all i, j, and (cid:88) i | U i | s ≤ (cid:15) } , (2.1)see [4, Chapter 2], and we can define the Hausdorff dimension without using Hausdorff measure by5im H F = inf { s ≥ (cid:15) > { U , U , . . . } of F such that (cid:88) i | U i | s ≤ (cid:15) } , (2.2)see [4, Section 3.2].For the purposes of this paper, we make the following definition. Definition 2.6.
Let Y ∈ (0 , ∞ ). We say that a real-valued function Φ whose domain of definitioncontains a subinterval of R with left endpoint 0 and positive length is Y -admissible if 0 < Φ( δ ) ≤ δ for all δ ∈ (0 , Y ), and Φ( δ ) /δ → δ → + , and Φ( δ ) is monotonic on (0 , Y ). We say that Φ is admissible if there exists Y ∈ (0 , ∞ ) such that Φ is Y -admissible.Throughout, most of the functions Φ we consider will be admissible. The different conditionson Φ are explored in Section 3.2. We now give the main definition. Definition 2.7.
Let Φ be an admissible function as in Definition 2.6, and let F be a non-empty,totally bounded subset of a uniformly perfect metric space X with more than one point. We definethe upper Φ -intermediate dimension of F bydim Φ F = inf { s ≥ (cid:15) > δ ∈ (0 ,
1] such that for all δ ∈ (0 , δ ) there exists a cover { U , U , . . . } of F such that Φ( δ ) ≤ | U i | ≤ δ for all i, and (cid:88) i | U i | s ≤ (cid:15) } . Similarly, we define the lower Φ -intermediate dimension of F bydim Φ F = inf { s ≥ (cid:15) > δ ∈ (0 ,
1] there exists δ ∈ (0 , δ )and a cover { U , U , . . . } of F such that Φ( δ ) ≤ | U i | ≤ δ for all i, and (cid:88) i | U i | s ≤ (cid:15) } . If these two quantities coincide, we call the common value the Φ -intermediate dimension of F ,denoted dim Φ F .In the above definition, the cover { U i } of F is a priori countable, but note that since it satisfies0 < Φ( δ ) ≤ | U i | for all i , and (cid:80) i | U i | s ≤ (cid:15) , it must be finite. Note that if F were not totallybounded then the Φ-intermediate dimensions of F would be infinite according to Definition 2.7.If θ ∈ (0 ,
1) and Φ( δ ) = δ /θ for all δ ∈ [0 , Φ F = dim θ F and dim Φ F = dim θ F are the lower and upper intermediatedimensions at θ respectively, and dim F = dim F = dim H F , dim F = dim B F , dim F = dim B F .If dim θ F = dim θ F then the common value is called the intermediate dimension of F at θ , denoteddim θ F . These are introduced in [6], where it is shown that for any given non-empty, bounded F ⊂ R n , dim θ F and dim θ F are monotonically increasing in θ ∈ [0 ,
1] and continuous in θ ∈ (0 , θ = 0.The Φ-intermediate dimensions satisfy the following basic properties; the proofs are straight-forward applications of the definitions, and are left for the reader. Proposition 2.1.
Let E and F be non-empty, totally bounded subsets of a uniformly perfect metricspace with more than one point, and let Φ be an admissible function. Then(i) Both dim Φ and dim Φ are increasing for sets , that is if E ⊆ F then dim Φ E ≤ dim Φ F and dim Φ E ≤ dim Φ F (ii) Both dim Φ and dim Φ are stable under closure , that is dim Φ F = dim Φ F and dim Φ F =dim Φ F , where F is the metric closure of F .
6n example which illustrates several points is the following.
Example 2.1.
The set F := Q ∩ [0 , ⊂ R is countable, so satisfies dim H F = 0, but for any ad-missible Φ, directly from Definition 2.7 we have dim Φ F = dim Φ F = 1. This example demonstratesthat: • The dimensions dim Φ and dim Φ are different from dim H . • There are non-empty, bounded subsets of R , such as F , for which there does not exist afamily of admissible functions for which the Φ-intermediate dimensions interpolate betweenthe Hausdorff and box dimensions of the set. • The dimensions dim Φ and dim Φ are not countably stable.The Hausdorff dimension is countably stable but not stable under closure, whereas the (upperand lower) box, intermediate and Φ-intermediate dimensions are stable under closure but notcountably stable. In this sense, the intermediate and Φ-intermediate dimensions are more similarto the box dimensions than the Hausdorff dimension. Φ -intermediate dimensions We now examine general bounds and continuity-like properties for the Φ-intermediate dimensions.As with intermediate dimensions, we have the following inequalities between the different notionsof dimension.
Proposition 3.1.
For any non-empty, totally bounded subset F of a uniformly perfect metricspace X with more than one point, and any admissible Φ , we have dim H F ≤ dim Φ F ≤ dim Φ F ≤ dim B F ≤ dim A F ≤ dim A X ;dim Φ F ≤ dim B F ≤ dim B F. Proof.
We first prove dim Φ F ≤ dim B F . Since X is uniformly perfect, there exists c ∈ (0 ,
1) suchthat X is c -uniformly perfect. Since Φ is admissible, there exists Y ∈ (0 , ∞ ) such that Φ is Y -admissible, and Φ( δ ) /δ → δ → + , so there exists ∆ ∈ (0 , Y ) such that for all δ ∈ (0 , ∆)we have Φ( δ ) /δ < c/
2. Let s > dim B F and (cid:15) >
0. Let t ∈ (dim B F, s ), so since t > dim B F , byDefinition 2.5 there exists δ ∈ (0 , min { (cid:15) s − t , ∆ , | X | , } ) such that for all δ ∈ (0 , δ ) there exists acover of F by δ − t or fewer sets { U i } , each having diameter at most δ . For each i , if U i ∩ F = ∅ ,we may remove U i from the cover, so we may assume without loss of generality that for each i wehave U i ∩ F (cid:54) = ∅ . For each i , if | U i | ≥ δ/ U i in the cover unchanged. If | U i | < δ/
2, thensince U i (cid:54) = ∅ there exists x i ∈ U i , and since | U | < δ/ < δ ≤ | X | and X is c -uniformly perfect,there exists y i ∈ B ( x i , δ/ \ B ( x i , cδ/ y i to U i , and call the resulting cover { V i } .By the triangle inequality and since δ < ∆, for all i we haveΦ( δ ) ≤ cδ/ ≤ | V i | ≤ δ/ δ/ δ. Moreover, (cid:88) i | V i | s ≤ δ − t δ s = δ s − t < δ s − t < (cid:15) since 0 < δ < δ < (cid:15) s − t . Thus by Definition 2.7 we have dim Φ F ≤ s . Letting s → (dim B F ) + givesdim Φ F ≤ dim B F , as required.The proof that dim Φ F ≤ dim B F is similar. Let s (cid:48) > dim B F and (cid:15) (cid:48) >
0. Let t (cid:48) ∈ (dim B F, s (cid:48) ), sosince t (cid:48) > dim B F , by Definition 2.5 for all δ (cid:48) ∈ (0 , min { ( (cid:15) (cid:48) ) s (cid:48)− t (cid:48) , ∆ , | X | , } ) there exists δ (cid:48) ∈ (0 , δ (cid:48) )7nd a cover of F by ( δ (cid:48) ) − t (cid:48) or fewer set, each having diameter at most δ (cid:48) . As above, we can use thiscover to form a cover { V (cid:48) j } which satisfies Φ( δ (cid:48) ) ≤ | V (cid:48) j | ≤ δ (cid:48) for all j and (cid:80) j | V (cid:48) j | s < (cid:15) (cid:48) . Thereforedim Φ F ≤ s (cid:48) , and letting s (cid:48) → (dim B F ) + gives dim Φ F ≤ dim B F , as required.The inequality dim H F ≤ dim Φ F holds by Definitions 2.2 and 2.7; when the class of admissiblecovers is more restricted, the infimum will be greater. The inequalities dim Φ F ≤ dim Φ F anddim B F ≤ dim B F also follow directly from the definitions. The inequality dim B F ≤ dim A F holdsby fixing R = | F | in Definition 2.3. The inequality dim A F ≤ dim A X follows from Definition 2.3since F ⊆ X .We assume that the ambient metric space X is uniformly perfect with more than one point,and that Φ( δ ) /δ → δ → + , to ensure that Proposition 3.1 will hold and to avoid cases likethe two-point metric space, which would have infinite intermediate and Φ-intermediate dimensionsaccording to Definition 2.7. There is no general relationship between the lower box dimension andthe upper intermediate or upper Φ-intermediate dimensions. It follows from Proposition 3.1 thatif F ⊂ R n is non-empty and bounded then dim Φ F ≤ dim Φ F ≤ n , and if in addition F is openwith respect to the Euclidean metric then dim Φ F = dim Φ F = n , as one would expect.We will need the following sufficient condition for the Φ-intermediate dimension always toequal the box dimension. The analogous result for the φ -Assouad dimensions is Proposition 3.12(i), which gives a sufficient condition on φ for the φ -Assouad dimension to coincide with the upperbox dimension for all sets. Proposition 3.2.
Let Φ be an admissible function such that log δ log Φ( δ ) → as δ → + . Then forany non-empty, totally bounded subset F of a uniformly perfect metric space with more than onepoint, we have dim Φ F = dim B F and dim Φ F = dim B F .Proof. We prove that dim Φ F = dim B F ; the proof of dim Φ F = dim B F is similar. By Propo-sition 3.1 we have dim Φ F ≤ dim B F . Assume for contradiction that dim Φ F < dim B F , and let s, t ∈ R be such that dim Φ F < s < t < dim B F . Since dim Φ F < s there exists δ > δ ∈ (0 , δ ) there exists a cover { U i } of F such that for all i we haveΦ( δ ) ≤ | U i | ≤ δ (3.1)and (cid:80) i | U i | s ≤ (cid:15) . Therefore N δ ( F ) δ t ≤ (cid:88) i δ t = (cid:88) i δ t | U i | s | U i | t − s | U i | t ≤ (cid:88) i δ t | U i | s δ t − s (Φ( δ )) t by (3.1)= (cid:18) δ t − s ) /t Φ( δ ) (cid:19) t (cid:88) i | U i | s ≤ (cid:18) δ t − s ) /t Φ( δ ) (cid:19) t . This converges to 0 as δ → + , since log δ log Φ( δ ) → < t − s ) /t . This contradicts t < dim B F andcompletes the proof.Note that the function Φ : (0 , / → (0 , /
5) defined by Φ( δ ) := δ − log δ is admissible andsatisfies log δ log Φ( δ ) → δ → + , so by Proposition 3.2, for any non-empty, totally bounded subset F of a uniformly perfect metric space with more than one point, we have dim Φ F = dim B F anddim Φ F = dim B F .We now consider continuity-like results for the Φ-intermediate dimensions. The main suchresult for is Theorem 3.3, which roughly implies that if two admissible functions Φ and Φ are in a8uantitative sense ‘close’ to each other then the Φ and Φ -intermediate dimensions of sets whoseAssouad dimension is not too large do not differ greatly. This is similar in spirit to the quantitativecontinuity result [6, Proposition 2.1] for the intermediate dimensions. Garc´ıa, Hare and Mendivil[15, Proposition 2.10 (ii)] prove a continuity result for the φ -Assouad dimensions. Theorem 3.3.
Let Φ , Φ be admissible functions, let X be a uniformly perfect metric space withmore than one point, and let F be a non-empty, totally bounded subset such that dim A F < ∞ .(i) Suppose that < dim Φ F < dim A F , and let η ∈ [0 , dim A F − dim Φ F ) . Then(1) If there exists ∆ > such that for all δ ∈ (0 , ∆) we have Φ ( δ α ) ≤ (Φ( δ )) dimΦ F dimΦ F + η where α := dim A F − dim Φ F dim A F − dim Φ F − η (3.2) then dim Φ F ≤ dim Φ F + η .(2) If for all δ > there exists δ ∈ (0 , δ ) such that (3.2) holds, then dim Φ F ≤ dim Φ F + η .(ii) Now suppose that < dim Φ F < dim A F , and let η ∈ [0 , dim A F − dim Φ F ) . Then if thereexists ∆ > such that for all δ ∈ (0 , ∆) we have Φ (cid:18) δ dimA F − dimΦ F dimA F − dimΦ F − η (cid:19) ≤ (Φ( δ )) dimΦ F dimΦ F + η (3.3) then dim Φ F ≤ dim Φ F + η .Proof. The idea of the proof is to consider a cover of F with diameters in [Φ( δ ) , δ ], and ‘fatten’the smallest sets in the cover to size Φ ( δ α ) and break up the largest sets in the cover to size δ α ;the ‘cost’ of each of these actions in terms of how much the dimension can increase is the same,namely η .The result follows directly from Definition 2.7 if η = 0, since if Φ ( δ ) ≤ Φ( δ ) then the infimumin the definition of the Φ-intermediate dimension will be greater, so henceforth assume that η > α > δ α < δ , and dim Φ F dim Φ F + η < δ )) dimΦ F dimΦ F + η > Φ( δ ). Since X is uniformlyperfect, there exists c ∈ (0 , /
2) such that X is c -uniformly perfect.(i) (1) Suppose that dim Φ F < dim A F , and let η ∈ (0 , dim A F − dim Φ F ). Assume there exists∆ > δ ∈ (0 , ∆) (3.2) is satisfied. Without loss of generality, by reducing ∆if necessary, we may assume that ∆ < min { , | X |} , that Φ and Φ are ∆-admissible, and thatfor all δ ∈ (0 , ∆) we have δ/ Φ ( δ ) ≥ /c . Let s ∈ (dim Φ F, dim A F − η ) and let (cid:15) >
0. Let a ∈ (dim A F, ∞ ) be close enough to dim A F that a − s − α ( a − s − η ) > . (3.4)Since a > dim A F , there exists C ∈ (0 , ∞ ) such that for all x ∈ F and 0 < r < R we have N r ( B ( x, R ) ∩ F ) ≤ C ( R/r ) a . Since min (cid:110) s, ( s + η )dim Φ F dim Φ F + η (cid:111) > dim Φ F , there exists δ ∈ (0 , ∆] suchthat for all δ ∈ (0 , δ ) there exists a cover { U i } i ∈ I of F such that Φ( δ ) ≤ | U i | ≤ δ for all i ∈ I , and (cid:88) i ∈ I | U i | min (cid:26) s, ( s + η )dimΦ F dimΦ F + η (cid:27) ≤ ((1 + 1 /c ) s + η + 1 + 2 a C ) − (cid:15). (3.5)Write I as a disjoint union I = I ∪ I ∪ I where I := { i ∈ I : Φ( δ ) ≤ | U i | < Φ ( δ α ) } I := { i ∈ I : Φ ( δ α ) ≤ | U i | ≤ δ α } I := { i ∈ I : δ α < | U i | ≤ δ } i ∈ I then leave U i in the cover unchanged. Nowsuppose that i ∈ I . Note that U i is non-empty, so choose any p i ∈ U i . Since X is c -uniformlyperfect, there exists q i ∈ X such that Φ ( δ α ) ≤ d ( p i , q i ) ≤ Φ ( δ α ) /c . By the triangle inequalityand the definition of I we haveΦ ( δ α ) ≤ d ( p i , q i ) ≤ | U i ∪ { q i }| ≤ | U i | + d ( p i , q i ) ≤ | U i | + Φ ( δ α ) /c< (1 + 1 /c )Φ ( δ α ) ≤ δ α . Moreover by (3.2), | U i ∪ { q i }| ≤ | U i | + Φ ( δ α ) /c ≤ | U i | + (Φ( δ )) dimΦ F dimΦ F + η c< (1 + 1 /c ) | U i | dimΦ F dimΦ F + η . (3.6)Now let j ∈ I . By the definition of C there exist x j, , . . . , x j, (cid:98) C (2 | U j | /δ α ) a (cid:99) ∈ F such that U j ∩ F ⊆ (cid:98) a C | U j | a δ − aα (cid:99) (cid:91) k =1 B ( x j,k , δ α / , noting that each of these balls has diameter at most δ α and at least cδ α / ≥ Φ ( δ α ). Then { U i ∪ { q i }} i ∈ I ∪ { U i } i ∈ I ∪ { B ( x j,k , δ α / } j ∈ I ,k ∈{ ,..., (cid:98) a C | U j | a δ − aα (cid:99)} covers F , and each set in this new cover has diameter in the interval [Φ ( δ α ) , δ α ]. Moreover, (cid:88) i ∈ I | U i ∪ { q i }| s + η + (cid:88) i ∈ I | U i | s + η + (cid:88) j ∈ I (cid:98) a C | U j | a δ − aα (cid:99) (cid:88) k =1 | B ( x j,k , δ α / | s + η ≤ (cid:88) i ∈ I (cid:18) (1 + 1 /c ) | U i | dimΦ F dimΦ F + η (cid:19) s + η + (cid:88) i ∈ I | U i | s + η + (cid:88) j ∈ I a C | U j | a δ − aα ( δ α ) s + η ≤ (1 + 1 /c ) s + η (cid:88) i ∈ I | U i | ( s + η )dimΦ F dimΦ F + η + (cid:88) i ∈ I | U i | s + 2 a C (cid:88) j ∈ I | U j | s | U j | a − s δ α ( − a + s + η ) ≤ (1 + 1 /c ) s + η (cid:88) i ∈ I | U i | ( s + η )dimΦ F dimΦ F + η + (cid:88) i ∈ I | U i | s + 2 a Cδ a − s − α ( a − s − η ) (cid:88) j ∈ I | U j | s ≤ ((1 + 1 /c ) s + η + 1 + 2 a C ) (cid:88) i ∈ I | U i | min (cid:26) s, ( s + η )dimΦ F dimΦ F + η (cid:27) (by (3.4)) ≤ (cid:15) (by (3.5)) . Therefore dim Φ F ≤ s + η . Letting s → (dim Φ F ) + gives dim Φ F ≤ dim Φ F + η , as required.(i) (2) If we only have that for all δ > δ ∈ (0 , δ ) such that (3.2) holds, then bythe same argument as in the proof of (i) (1), for all s > dim Φ F and (cid:15) > { V i } of F such that Φ ( δ α ) ≤ | V i | ≤ δ α and (cid:80) i | V i | s + η ≤ (cid:15) . Therefore dim Φ F ≤ s + η , and letting s → (dim Φ F ) + gives dim Φ F ≤ dim Φ F + η , as required.(ii) Under the assumptions of (ii), let s > dim Φ F and (cid:15) >
0, and choose a ∈ (dim A F, ∞ ) tosatisfy (3.4). Then for all δ ∈ (0 , ∆) there exists δ ∈ (0 , δ ) such that there exists a cover { U i } of F such that Φ( δ ) ≤ | U i | ≤ δ for all i , and such that (3.5) holds. Then as in the proof of (i) (1)(replacing dim with dim throughout), there exists a cover { V i } of F such that Φ ( δ α ) ≤ | V i | ≤ δ α for all i , and (cid:80) i | V i | s + η ≤ (cid:15) . Therefore dim Φ F ≤ s + η , and letting s → (dim Φ F ) + givesdim Φ F ≤ dim Φ F + η , as required. 10y Proposition 3.1 we have dim Φ F ≤ dim Φ F ≤ dim A F for any admissible Φ , so there isnothing more to be said in the case dim Φ F = dim A F . The only remaining nontrivial case iscovered by the following proposition. Proposition 3.4.
Let Φ , Φ be admissible functions, let X be a uniformly perfect metric space withmore than one point, and let F be a non-empty, totally bounded subset such that < dim A F < ∞ .Let η ∈ (0 , dim A F ) .(i) Suppose dim Φ F = 0 . Then(1) If there exist b, ∆ > such that for all δ ∈ (0 , ∆) we have Φ ( δ α ) ≤ (Φ( δ )) b where α = α ( η ) := dim A F dim A F − η (3.7) then dim Φ F ≤ η . In particular, if for all α (cid:48) > there exists ∆ > such that for all δ ∈ (0 , ∆) we have Φ ( δ α (cid:48) ) ≤ (Φ( δ )) b (or indeed if there exists ∆ > such that for all δ ∈ (0 , ∆) we have Φ ( δ ) ≤ (Φ( δ )) b ) then (3.7) is satisfied for all η > , so dim Φ F = 0 .(2) If for all δ > there exists δ ∈ (0 , δ ) such that (3.7) holds, then dim Φ F ≤ η . Inparticular, if for all α (cid:48) > and δ > there exists δ ∈ (0 , δ ) such that Φ ( δ α (cid:48) ) ≤ (Φ( δ )) b then dim Φ F = 0 .(ii) If dim Φ F = 0 and (3.7) holds for all δ ∈ (0 , ∆) , then dim Φ F ≤ η . In particular, if forall α (cid:48) > there exists ∆ > such that for all δ ∈ (0 , ∆) we have Φ ( δ α (cid:48) ) ≤ (Φ( δ )) b then dim Φ F = 0 .Proof. We sketch the proof of (i) (1) as it is similar to the proof of Theorem 3.3; the otherparts follow in a similar way as in Theorem 3.3. Suppose that the assumptions of (i) (1) hold.Let t ∈ (0 ,
1) (depending on η, b ) be small enough that t/ ( t + η ) < min { b, } . Let s, (cid:15) > a ∈ (dim A F, ∞ ) be close enough to dim A F that a − s − α ( a − s − η ) >
0. Sincemin (cid:110) s, ( s + η ) tt + η (cid:111) > dim Φ F = 0, there exists δ ∈ (0 , ∆] such that for all δ ∈ (0 , δ ) there exists acover { U i } i ∈ I of F such that Φ( δ ) ≤ | U i | ≤ δ for all i ∈ I , and (cid:88) i ∈ I | U i | min { s, ( s + η ) tt + η } ≤ ((1 + 1 /c ) s + η + 1 + 2 a C ) − (cid:15). As in the proof of Theorem 3.3 (i) (1), we can form a new cover of F with the diameter of eachcovering set lying in the interval [Φ ( δ α ) , δ α ], noting that instead of (3.6) we have | U i ∪ { q i }| ≤ | U i | + Φ ( δ α ) /c ≤ | U i | + (Φ( δ )) b /C ≤ | U i | + (Φ( δ )) tt + η c< (1 + 1 /c ) | U i | tt + η . Then as above, (cid:88) i ∈ I | U i ∪ { q i }| s + η + (cid:88) i ∈ I | U i | s + η + (cid:88) j ∈ I (cid:98) a C | U j | a δ − aα (cid:99) (cid:88) k =1 | B ( x j,k , δ α / | s + η ≤ (1 + 1 /c ) s + η (cid:88) i ∈ I | U i | ( s + η ) tt + η + (cid:88) i ∈ I | U i | s + 2 a Cδ a − s − α ( a − s − η ) (cid:88) j ∈ I | U j | s ≤ ((1 + 1 /c ) s + η + 1 + 2 a C ) (cid:88) i ∈ I | U i | min { s, ( s + η ) tt + η }≤ (cid:15), so dim Φ F ≤ s + η , and letting s → + gives dim Φ F ≤ η , as required.11orollary 3.4.1 says that if the underlying metric space is doubling, then if Φ and Φ are‘close’ in a way that depends only on X , then the difference between the Φ- and Φ -intermediatedimensions of subsets will be small, independently of the particular subset. A typical examplewhere Corollary 3.4.1 could be applied is X = R n with the Euclidean metric, where dim A X = n . Corollary 3.4.1.
Let Φ , Φ be admissible functions, let X be a uniformly perfect, doubling metricspace with more than one point, and let F be a non-empty, bounded subset. If there exists ∆ > such that for all δ ∈ (0 , ∆) we have Φ (cid:18) δ dimA X dimA X − η (cid:19) ≤ (Φ( δ )) dimA X dimA X + η (3.8) then(i) if dim Φ F < dim A F and η ∈ [0 , dim A F − dim Φ F ) then dim Φ F ≤ dim Φ F + η (ii) dim Φ F < dim A F and η (cid:48) ∈ [0 , dim A F − dim Φ F ) then dim Φ F ≤ dim Φ F + η (cid:48) .If for all δ > there exists δ ∈ (0 , δ ) such that (3.8) holds, and if dim Φ F < dim A F and η ∈ [0 , dim A F − dim Φ F ) , then dim Φ F ≤ dim Φ F + η .Proof. By Proposition 3.1,1 ≤ dim A X dim A X − η ≤ dim A F − dim Φ F dim A F − dim Φ F − η ≤ dim A F − dim Φ F dim A F − dim Φ F − η and 1 ≥ dim A X dim A X + η ≥ dim Φ F dim Φ F + η ≥ dim Φ F dim Φ F + η , so the result follows from Theorem 3.3 in the cases dim Φ F > Φ F >
0, and fromProposition 3.4 in the cases dim Φ F = 0 and dim Φ F = 0.We now define a relation ≡ on the set of admissible function by setting Φ ≡ Φ if for any non-empty, totally bounded subset F of any uniformly perfect metric space with more than one point,such that dim A F < ∞ , we have dim Φ F = dim Φ F and dim Φ F = dim Φ F ; it is straightforwardto see that this is an equivalence relation. We define a relation (cid:22) on the equivalence classes of ≡ bysetting [Φ ] (cid:22) [Φ ] if for all such F we have dim Φ F ≤ dim Φ F and dim Φ F ≤ dim Φ F ; it is easy tosee that (cid:22) is well-defined and a non-strict partial order, and we abuse notation by writing Φ (cid:22) Φ to mean [Φ ] (cid:22) [Φ ]. If none of Φ ≡ Φ , Φ (cid:22) Φ or Φ (cid:22) Φ hold then we say that Φ and Φ are incomparable , and that [Φ ] and [Φ ] are also incomparable . Consider the topology on the setof equivalence classes generated by the basis of open sets { N Φ ,α : Φ an admissible function, α ∈ (1 , ∞ ) } where N Φ ,α := { C : there exists Φ ∈ C and ∆ ∈ (1 , ∞ ) such that for all δ ∈ (0 , ∆)we have (Φ( δ α )) α ≤ Φ ( δ ) ≤ (Φ( δ /α )) /α } . Then for any given non-empty, totally bounded subset F of a uniformly perfect metric space withmore than one point, such that dim A F < ∞ , the maps [Φ] (cid:55)→ dim Φ F and [Φ] (cid:55)→ dim Φ F arewell-defined and, by Theorem 3.3 and Proposition 3.4, continuous with respect to this topology,which further justifies us calling these ‘continuity-like’ results.Corollary 3.4.2 (ii) gives a condition for the dimensions to coincide for all sets; Garc´ıa, Hareand Mendivil [15, Proposition 2.10 (i)] also give a condition for the φ -Assouad dimensions. Corollary 3.4.2.
Let Φ , Φ be admissible functions. i) (1) If for all α ∈ (1 , ∞ ) there exists ∆ > such that for all δ ∈ (0 , ∆) we have Φ ( δ α ) ≤ (Φ( δ )) /α (3.9) (noting that this will be the case if, for example, there exists C ∈ (0 , ∞ ) such that lim sup δ → + Φ ( Cδ )Φ ( δ ) < ∞ ), then Φ (cid:22) Φ .(2) If we only assume that for all α ∈ (1 , ∞ ) and δ > there exists δ ∈ (0 , δ ) such that (3.9) holds (noting that this will be the case if, for example, there exists C ∈ (0 , ∞ ) suchthat lim inf δ → + Φ ( Cδ )Φ ( δ ) < ∞ ), then for any non-empty, totally bounded subset F of auniformly perfect metric space with more than one point, such that dim A F < ∞ , wehave dim Φ F ≤ dim Φ F .(ii) If for all α ∈ (1 , ∞ ) there exists ∆ > such that for all δ ∈ (0 , ∆) we have (Φ( δ α )) α ≤ Φ ( δ ) ≤ (Φ( δ /α )) /α (3.10) (noting that this will be the case if, for example, there exist C , C ∈ (0 , ∞ ) such that lim sup δ → + Φ ( C δ )Φ ( δ ) < ∞ and lim sup δ → + Φ ( C δ )Φ ( δ ) < ∞ ), then Φ ≡ Φ .Proof. (i) (1) Let F be a non-empty, totally bounded subset of a uniformly perfect metric spacewith more than one point, assume that dim A F < ∞ , and assume that (3.9) holds for all δ ∈ (0 , ∆).Then if dim Φ F = dim A F then dim Φ F ≤ dim A F = dim Φ F by Proposition 3.1. If dim Φ F = 0then dim Φ F = 0 = dim Φ F by Proposition 3.4 (i) (1). If 0 < dim Φ F < dim A F then for all η > α = min (cid:40) dim A F − dim Φ F dim A F − dim Φ F − η , dim Φ F + η dim Φ F (cid:41) , we have that dim Φ F ≤ dim Φ F + η by Proposition 3.3 (i) (1), and letting η → + gives dim Φ F ≤ dim Φ F . Similarly, by Proposition 3.1 and Proposition 3.4 (ii) and Proposition 3.3 (ii), we havethat in all cases dim Φ F ≤ dim Φ F , so Φ (cid:22) Φ.(i) (2) If we only assume that for all δ > δ ∈ (0 , δ ) such that (3.9) holds, thensimilarly we have dim Φ F ≤ dim Φ F in all cases by Proposition 3.1 and Proposition 3.4 (i) (2) andProposition 3.3 (i) (2).(ii) Under the assumptions of (ii) left-hand inequality of (3.10) and (i) give that Φ (cid:22) Φ . Byreplacing δ by δ α in the right-hand inequality of (3.10), we see that by (i), Φ (cid:22) Φ. ThereforeΦ ≡ Φ , as required. Φ In this section we use Corollary 3.4.2 to explore the conditions that can be imposed on the func-tion Φ, and show that nothing is really lost by only considering functions that satisfy the niceproperties of being strictly increasing, invertible and continuous. First we make some remarksabout admissible functions. Only the behaviour of Φ close to 0 is relevant. Note that since X isuniformly perfect, there exists c ∈ (0 ,
1) such that X is c -uniformly perfect, and then the proof ofProposition 3.1 holds when the condition in Definition 2.6 that Φ( δ ) /δ → δ → + is relaxed tothe condition that there exists ∆ ∈ (0 , Y ) such that for all δ ∈ (0 , ∆) we have Φ( δ ) /δ < c/
2. Theproof of Proposition 3.2, however, shows that nothing is gained by considering these more generalΦ.
Proposition 3.5.
For any admissible function Φ there exists an admissible function Φ : (0 , → (0 , that is a strictly increasing, C ∞ diffeomorphism, such that Φ ≡ Φ . roof. Since Φ is admissible, there exists N ∈ N such that Φ is defined, positive and increasing on(0 , − N ], and such that Φ(2 − N ) <
1. We define a strictly increasing function Φ : (0 , → (0 , to be linear on [2 − N ,
1] with Φ (2 − N ) := Φ(2 − N ) and Φ (1) = 1 (noting thatΦ is strictly increasing on this interval), and defining Φ inductively on (0 , − N ) as follows.Suppose we have defined Φ on [2 − n ,
1] for some n ≥ N . If Φ(2 − n − ) < Φ (2 − n ) then defineΦ (2 − n − ) := Φ(2 − n − ) and define Φ to be linear on [2 − n − , − n ] (noting that Φ is strictlyincreasing on this interval), which completes the inductive definition. If, on the other hand,Φ(2 − n − ) = Φ (2 − n ), then let m be the smallest integer such that m > n and Φ(2 − m ) < Φ(2 − n )(which exists since Φ( δ ) → δ → + ), define Φ (2 − m ) := max { Φ (2 − n ) / , Φ(2 − m ) } , and defineΦ to be linear on [2 − m , − n ], noting that Φ is strictly increasing on this interval. Then byconstruction Φ is strictly increasing on (0 ,
1] and satisfies Φ ( δ/ ≤ Φ( δ ) and 2Φ (2 δ ) ≥ Φ( δ ) forall δ ∈ (0 , − N − ). Each of the countably many points of non-differentiability of Φ can be locallysmoothened to give an admissible function Φ : (0 , → (0 ,
1) that is C ∞ on (0 , ( δ ) / ≤ Φ ( δ ) ≤ ( δ ) for all δ ∈ (0 , − N ). ThenΦ ( δ ) /δ ≤ ( δ ) /δ ≤ δ ) /δ = 8Φ(4 δ ) / (4 δ ) −−−−→ δ → + , since Φ( δ ) /δ → δ → + , so Φ is admissible. Moreover,Φ( δ/ / ≤ Φ ( δ ) / ≤ Φ ( δ ) ≤ ( δ ) ≤ δ )for all δ ∈ (0 , − N − ), so by Corollary 3.4.2 (ii) we have Φ ≡ Φ. By the smooth inverse functiontheorem, Φ has a C ∞ inverse Φ − : (0 , → (0 , is a C ∞ diffeomorphism, as required.If desired, one could choose a Φ in the same equivalence class that is defined on (0 , ∞ ) insteadof on (0 , Proposition 3.6.
Let ∆ > and let Φ : (0 , ∆) → [0 , ∞ ) be any function (not necessarily mono-tonic) such that Φ( δ ) /δ → as δ → + , and define the Φ -intermediate dimensions as in Defini-tion 2.7, for non-empty, totally bounded subsets of a uniformly perfect metric space X with morethan one point. Let F be any such subset.(i) If Φ : (0 , ∆) → R is defined by Φ ( δ ) := sup { Φ( δ (cid:48) ) : δ (cid:48) ∈ [0 , δ ] } then dim Φ F = dim Φ F .(ii) (1) If for all δ > there exists δ ∈ (0 , δ ) such that Φ( δ ) = 0 then dim Φ F = dim H F .(2) Suppose there exists ∆ ∈ (0 , such that Φ( δ ) > for all δ ∈ (0 , ∆ ] but for all δ ∈ (0 , ∆ ] there exists δ ∈ (0 , δ ) such that inf { Φ( δ ) : δ ∈ [ δ , δ ] } = 0 . Then if F is compact then dim Φ F = dim H F . In particular, since dim Φ is stable under closure, bythe Heine-Borel theorem, if F is any non-empty, bounded subset of X = R n with theEuclidean metric then dim Φ F = dim H F .(3) If there exists ∆ ∈ (0 , ∆] such that Φ : (0 , ∆ ) → R defined by Φ ( δ ) := inf { Φ( δ (cid:48) ) : δ (cid:48) ∈ [ δ, ∆ ] } is positive for all δ ∈ (0 , ∆ ) then dim Φ F = dim Φ F .Proof. Since X is uniformly perfect, there exists c ∈ (0 , /
2) such that X is c -uniformly perfect.Since Φ( δ ) /δ → δ → + , we may assume without loss of generality (reducing ∆ if necessary)that ∆ < min { , | X |} and that for all δ ∈ (0 , ∆) we have Φ( δ ) ≤ (1 + 2 /c ) − δ . In the proofs of thedifferent sections of the proposition, the same symbols may take different values.14i) First note that for all δ ∈ (0 , ∆),Φ ( δ ) /δ = sup { Φ( δ (cid:48) ) /δ : δ (cid:48) ∈ (0 , δ ] } ≤ sup { Φ( δ (cid:48) ) /δ (cid:48) : δ (cid:48) ∈ (0 , δ ] } −−−−→ δ → + , and Φ ( δ ) is monotonic, so Φ is admissible. Also, Φ( δ ) ≤ Φ ( δ ) for all δ ∈ (0 , ∆) so directly fromDefinition 2.7 we have dim Φ F ≤ dim Φ F , so it remains to prove the reverse inequality.Note that for all δ ∈ (0 , ∆) we have Φ ( δ ) ≤ sup { (1 + 1 /c ) − δ (cid:48) : δ (cid:48) ∈ [0 , δ ] } ≤ sup { δ (cid:48) : δ (cid:48) ∈ [0 , δ ] } = δ . Let s > dim Φ F and (cid:15) >
0. Then there exists δ > δ ∈ (0 , min { δ , ∆ } )there exists a cover { U i } of F such that Φ( δ ) ≤ | U i | ≤ δ for all i , and (cid:88) i | U i | s ≤ − s (1 + 1 /c ) − s (cid:15). (3.11)Then if δ (cid:48) ∈ (0 , δ ), by the definition of Φ there exists δ ∈ (0 , δ (cid:48) ] such that Φ( δ ) ≥ Φ ( δ (cid:48) ) / { U i } be the cover corresponding to δ as above. For each i , if | U i | ≥ Φ ( δ (cid:48) ) then leave | U i | in the cover unchanged, noting that Φ ( δ (cid:48) ) ≤ | U i | ≤ δ ≤ δ (cid:48) . If Φ ( δ (cid:48) ) > | U i | , on the otherhand, note that | U i | is non-empty, so there exists p i ∈ U i , and there exists q i ∈ X such thatΦ ( δ (cid:48) ) ≤ d ( p i , q i ) ≤ Φ ( δ (cid:48) ) /c . Replace U i in the cover by U i ∪ { q i } , and denote the new cover of F by { V i } i . Then by the triangle inequality and since δ < ∆,Φ ( δ (cid:48) ) ≤ d ( p i , q i ) ≤ | U i ∪ { q i }| ≤ | U i | + Φ ( δ (cid:48) ) /c < (1 + 1 /c )Φ ( δ (cid:48) ) ≤ δ (cid:48) , and | U i ∪ { q i }| ≤ (1 + 1 /c )Φ ( δ (cid:48) ) ≤ /c )Φ( δ ) ≤ /c ) | U i | . Therefore (cid:88) i | V i | s ≤ (cid:88) i (2(1 + 1 /c ) | U i | ) s = 2 s (1 + 1 /c ) s (cid:88) i | U i | s ≤ (cid:15) by (3.11), so dim Φ F ≤ s , and letting s → (dim Φ F ) + gives dim Φ F ≤ dim Φ F , so dim Φ F ≤ dim Φ F as required.(ii) (1) Follows directly from (2.2) and Definition 2.7.(ii) (2) Assume that F is compact. It follows directly from (2.2) and Definition 2.7 thatdim H F ≤ dim Φ F , so it remains to show the reverse inequality.Let s > dim H F , (cid:15) > δ ∈ (0 , δ ∈ (0 , δ ) such that inf { Φ( δ ) : δ ∈ [ δ , δ ] } = 0. Since s > dim H F , by (2.2) there exists a finite or countable cover { U i } of F suchthat (cid:80) i | U i | s ≤ min { δ s , (cid:15) } . This means that each | U i | ≤ δ . Since F is compact, there is a finitecover { V i } , so min i {| V i |} >
0, and each | V i | ≤ δ . Since inf { Φ( δ ) : δ ∈ [ δ , δ ] } = 0, there exists δ ∈ [ δ , δ ] such that Φ( δ ) ∈ (0 , min i {| V i |} ). Then for each i we have 0 ≤ Φ( δ ) ≤ min i {| V i |} ) ≤| V i | ≤ δ ≤ δ and (cid:80) i | V i | s ≤ (cid:80) i | U i | s ≤ (cid:15) . As (cid:15) and δ were arbitrary, we have dim Φ F ≤ s .Letting s → (dim H F ) + gives dim Φ F ≤ dim H F , hence dim Φ F = dim H F , as required.(ii) (3) Note that Φ ( δ ) is monotonic, and Φ ( δ ) ≤ Φ( δ ) for all δ ∈ (0 , ∆ ) so Φ ( δ ) /δ ≤ Φ( δ ) /δ → δ → + , so Φ is admissible and dim Φ F ≤ dim Φ F , and it remains to prove thereverse inequality. Let s > dim Φ F and (cid:15) >
0. Let δ > δ ∈ (0 , Φ (min { ∆ , δ } ) / s > dim Φ F , there exists δ ∈ (0 , δ ) and a cover { U i } of F such that Φ ( δ ) ≤ | U i | ≤ δ for all i , and (cid:88) i | U i | s ≤ − s (1 + 1 /c ) − s (cid:15). (3.12)By the definition of Φ , there exists δ ∈ [ δ, ∆ ] such that Φ( δ ) < ( δ ). But since Φ ( δ ) ≤ Φ( δ ) ≤ δ ≤ δ < Φ (min { ∆ , δ } ) /
2, it must be the case that δ < min { ∆ , δ } . If | U i | ≥ Φ( δ )then leave U i in the cover unchanged, noting that Φ( δ ) ≤ | U i | ≤ δ ≤ δ . If, on the other hand, | U i | < Φ( δ ), then note that since U i (cid:54) = ∅ , there exists p i ∈ U i , and there exists q i ∈ X such thatΦ( δ ) ≤ d ( p i , q i ) ≤ Φ( δ ) /c ; replace U i in the cover with U i ∪ { q i } and call the new cover { V i } .Now, Φ( δ ) ≤ d ( p i , q i ) ≤ | U i ∪ { q i }| ≤ | U i | + d ( p i , q i ) < (1 + 1 /c )Φ( δ ) < δ , δ < ∆. Also, | U i ∪ { q i }| ≤ (1 + 1 /c )Φ( δ ) ≤ /c )Φ ( δ ) ≤ /c ) | U i | . Therefore (cid:88) i | V i | s ≤ (cid:88) i (2(1 + 1 /c ) | U i | ) s = 2 s (1 + 1 /c ) s (cid:88) i | U i | s ≤ (cid:15), by (3.12). Therefore dim Φ F ≤ s , and letting s → (dim Φ F ) + gives dim Φ F ≤ dim Φ F , as required. We now explore the consequences of general results proved for the Φ-intermediate dimensions inSection 3.1 for the special case of the intermediate dimensions. From Proposition 3.4 we obtainthe following mutual dependency result for the box and intermediate dimensions.
Proposition 3.7.
Let F be a non-empty, totally bounded subset of a uniformly perfect metricspace with more than one point, such that dim A F < ∞ . Then(i) If dim B F > then for all θ ∈ (0 , we have dim θ F > .(ii) If dim B F > then for all θ ∈ (0 , we have dim θ F > .Proof. (i) We prove this by contraposition; let θ ∈ (0 ,
1) and suppose that dim θ F = 0. If dim A F =0 then dim B F = 0 by Proposition 3.1. If, on the other hand, 0 < dim A F < ∞ , then lettingΦ( δ ) = δ /θ and Φ ( δ ) = δ/ ( − log δ ) for all δ ∈ (0 , / ( δ ) ≤ δ = (Φ( δ )) θ , so0 = dim Φ F = dim B F , where the first equality follows from Proposition 3.4 (i) (1) and the lastequality follows from Proposition 3.2.(ii) follows from Proposition 3.4 (ii) in a similar way to how (i) follows from Proposition 3.4 (i)(1).For subsets of Euclidean space, Proposition 3.7 also follows from Falconer’s general lower bound[5, (2.6)]. Proposition 3.7 means that in order to check that the upper/lower box dimension of aset is 0, it suffices to check the a priori weaker condition that the (respectively) upper/lower θ -intermediate dimension of the set is 0 at any (small) θ ∈ (0 , B F = 0 if and only if dim θ A F = 0 for all (equivalently,any) θ ∈ (0 ,
1) if and only if dim qA F = 0, which follows from work in [11, 12, 13] (the analogueof the quasi-Assouad dimension of F in the context of the intermediate dimensions would be thequantity lim θ → + (dim θ F ) or lim θ → + (dim θ F )).Proposition 3.8 is a quantitative continuity result for the intermediate dimensions. Falconer,Fraser and Kempton’s [6, Proposition 2.1] and Falconer’s [5, (2.1)] are two very different continuityresults for subsets of Euclidean space, and Proposition 3.8 improves upon each of these. The proofof [6, Proposition 2.1] involves breaking up the largest sets in the cover, similar to the argumentconcerning I in the proof of Theorem 3.3, while [5, (2.1)] is proved by ‘fattening’ the smallest setsin the cover, much like the argument concerning I in the proof of Theorem 3.3. The novelty inthe proof of Theorem 3.3 (from which Proposition 3.8 follows) is to do both of these things at thesame time in such a way that the ‘cost’ of each of these (in terms of how much the dimension canincrease) is the same, which is why Proposition 3.8 improves both previous continuity results. Proposition 3.8.
Let F be a non-empty, totally bounded subset of a uniformly perfect metricspace with more than one point, such that < dim A F < ∞ . Let < θ ≤ φ ≤ . Then ( i ) dim θ F ≤ dim φ F ≤ dim θ F + ( φ − θ )dim θ F (dim A F − dim θ F )( φ − θ )dim θ F + θ dim A F ;16 ii ) dim θ F ≤ dim φ F ≤ dim θ F + ( φ − θ )dim θ F (dim A F − dim θ F )( φ − θ )dim θ F + θ dim A F .
Furthermore, it follows that the functions θ (cid:48) (cid:55)→ dim θ (cid:48) F and θ (cid:48) (cid:55)→ dim θ (cid:48) F are continuous for θ (cid:48) ∈ (0 , ; indeed they are both dim A F θ -Lipschitz on [ θ, . Therefore by Lipschitz continuity and Rademacher’s theorem, or alternatively by monotonic-ity and Lebesgue’s theorem, the functions θ (cid:48) (cid:55)→ dim θ (cid:48) F and θ (cid:48) (cid:55)→ dim θ (cid:48) F are differentiable atLebesgue-almost every θ (cid:48) ∈ (0 , X is doubling, then we can take the Lipschitz constant for both of these functions on [ θ,
1] to bedim A X/ (4 θ ), independent of F (for example if X = R n with the Euclidean metric then we cantake both Lipschitz constants to be n/ (4 θ )). Proof.
Note first that the expressions in (i) and (ii) are well-defined because 0 < θ ≤ φ anddim A F > θ F ≤ dim φ F is monotonicity of the intermediate dimensions: if φ = 1 then dim φ F = dim B F so thisfollows from Propositions 3.1. If φ < δ /θ ≤ δ /φ for all δ ∈ (0 , φ -intermediate dimension will be greater, and so dim θ F ≤ dim φ F (this alsofollows as a very special case of Corollary 3.4.2 (i) (1)).We now prove the other inequality. If θ = φ then the inequality reduces to dim φ F ≤ dim θ F ,which is trivial. If dim θ F = dim A F then the inequality is simply dim φ F ≤ dim A F , which followsfrom Proposition 3.1. If dim θ F = 0 then by Propositions 3.1 and 3.7 we have dim φ F ≤ dim B F = 0,which is the desired inequality. Therefore we may assume henceforth that 0 < θ < φ ≤ < dim θ F < dim A F . Define Φ( δ ) := δ /θ for δ ∈ (0 , φ < ( δ ) := δ /φ for δ ∈ (0 , φ = 1 then define Φ ( δ ) := δ/ ( − log δ ) for δ ∈ (0 , / φ = 1, we have dim Φ F = dim θ F and dim Φ F = dim φ F . Define η := ( φ − θ )dim θ F (dim A F − dim θ F )( φ − θ )dim θ F + θ dim A F , so 0 < η < dim A F − dim θ F . Define α := dim A F − dim θ F dim A F − dim θ F − η . By a direct algebraic manipulation, dim θ Fθ (dim θ F + η ) = α/φ, so Φ ( δ α ) ≤ δ α/φ = δ dim θFθ (dim θF + η ) = (Φ( δ )) dim θF dim θF + η . Therefore by Theorem 3.3, dim Φ F ≤ dim Φ F + η , which is the required inequality.To deduce Lipschitz continuity on [ θ, < θ ≤ θ (cid:48) ≤ φ ≤ φ F − dim θ (cid:48) F ≤ ( φ − θ (cid:48) )dim θ (cid:48) F (dim A F − dim θ (cid:48) F )( φ − θ (cid:48) )dim θ (cid:48) F + θ (cid:48) dim A F ≤ dim θ (cid:48) F (dim A F − dim θ (cid:48) F ) θ (cid:48) dim A F ( φ − θ (cid:48) ) ≤ ((dim A F ) / θ (cid:48) dim A F ( φ − θ (cid:48) ) ≤ dim A F θ ( φ − θ (cid:48) ) , as required. Lipschitz continuity of θ (cid:48) (cid:55)→ dim θ (cid:48) F for θ (cid:48) ∈ [ θ,
1] follows similarly.17alconer noted that his continuity result [5, (2.1)] shows that dim θ Fθ and dim θ Fθ are monotoni-cally decreasing in θ ∈ (0 ,
1] and so the graphs of θ → dim θ F and θ → dim θ F are starshaped withrespect to the origin. Proposition 3.9 shows that in fact, under mild conditions, the graphs are strictly starshaped with respect to the origin. Proposition 3.9.
Let F be a non-empty, totally bounded subset of a uniformly perfect metricspace with more than one point, such that < dim A F < ∞ . Then(i) If dim B F > then dim θ Fθ is strictly decreasing in θ ∈ (0 , .(ii) If dim B F > then dim θ Fθ is strictly decreasing in θ ∈ (0 , .Proof. We prove (i); the proof of (ii) is similar. Assume that dim B F > < θ < φ ≤
1. By Proposition 3.7 (i), dim θ F >
0, so by Proposition 3.8 (i) and a direct algebraic manipulation,dim φ Fφ ≤ φ (cid:18) dim θ F + ( φ − θ )dim θ F (dim A F − dim θ F )( φ − θ )dim θ F + θ dim A F (cid:19) < dim θ Fθ , as required.If we know the value of dim θ F or dim θ F for one value of θ ∈ (0 ,
1] then Proposition 3.8 givesan upper bound for dim φ F or dim φ F respectively, for all φ ∈ [ θ, φ F ≥ φ dim A F dim θ Fθ dim A F − ( θ − φ )dim θ F (3.13)for all φ ∈ (0 , θ ], and the same bound holds replacing dim with dim throughout. Of particularinterest is the lower bound for the intermediate dimensions in terms of the box dimension, becausethe box dimension of many sets is known independently; Proposition 3.10 follows immediately byletting θ = 1 in (3.13) and then renaming φ by θ . Proposition 3.10.
Let F be a non-empty, totally bounded subset of a uniformly perfect metricspace with more than one point, such that < dim A F < ∞ . Then for all θ ∈ (0 , ,(i) dim θ F ≥ θ dim A F dim B F dim A F − (1 − θ )dim B F (ii) dim θ F ≥ θ dim A F dim B F dim A F − (1 − θ )dim B F We make several remarks about the bound in (i), f ( θ ) := θ dim A F dim B F dim A F − (1 − θ )dim B F ;the same remarks hold for the bound in (ii), with dim replaced by dim throughout. • If dim B F = dim A F then for all θ ∈ (0 ,
1] we have dim θ F = dim B F = dim A F . Forsubsets of Euclidean space this follows from the general lower bound [6, Proposition 2.4], butProposition 3.10 extends this to more general metric spaces. • The function f ( θ ) → dim B F as θ → − and, if dim B F < dim A F , then f ( θ ) → θ → + . • If dim B F > f ( θ ) > θ ∈ (0 , • The function f ( θ ) is continuous for θ ∈ (0 ,
1] and real analytic for θ ∈ (0 , f (cid:48) ( θ ) = dim A F dim B F (dim A F − dim B F )(dim A F − (1 − θ )dim B F ) − ≥ , so f ( θ ) is increasing in θ ∈ (0 , f (cid:48)(cid:48) ( θ ) = − A F − (1 − θ )dim B F ) − dim A F (dim B F ) (dim A F − dim B F ) ≤ , f ( θ ) is concave for θ ∈ (0 , < dim B F < dim A F then for all θ ∈ (0 , f (cid:48) ( θ ) isbounded below by a positive constant depending only on F , and f (cid:48) ( θ ) → dim A F dim B F (dim A F − dim B F ) − as θ → + f (cid:48) ( θ ) → dim B dim A F (dim A F − dim B ) as θ → − ;also, f (cid:48)(cid:48) ( θ ) is bounded above by a negative constant depending only on F , so f ( θ ) is strictlyincreasing and strictly concave for θ ∈ (0 , θ (cid:55)→ dim θ F and θ (cid:55)→ dim θ F for many natural sets F . • The general lower bound [6, Proposition 2.4] for subsets of Euclidean space is proved using adifferent method (a mass distribution argument). If dim B F = dim A F then Proposition 3.10(i) is the same as [6, Proposition 2.4]; if dim B F < dim A F then Proposition 3.10 (i) improves[6, Proposition 2.4] for all θ ∈ (0 , B F > θ F ≥ θ dim B F [5, (2.6)] for subsets of Euclidean space forall θ ∈ (0 , f ( θ ) at θ = 1 is lim θ → − f (cid:48) ( θ ) = dim B F − (dim B F ) dim A F which,if 0 < dim B F < dim A F , is less than both dim A F − dim B (the gradient of the bound [6,Proposition 2.4] at θ = 1) and dim B F (the gradient of the bound [5, (2.6)]). • For fixed θ ∈ (0 ,
1) and a fixed value of dim B F , as the value of dim A F decreases to dim B F ,the value of f ( θ ) increases to dim B F . This is also the case with the lower bound [6, Proposi-tion 2.4]. There is an intuitive geometric interpretation of this interplay between the Assouadand intermediate dimensions, as both of these dimensions capture some sort of spatial inho-mogeneity of sets in different ways. Indeed, this result suggests that if the scaling behaviourat even the thickest parts of the set (the Assouad dimension) is not much worse than the scal-ing behaviour of the whole set (the box dimension), then that means that the set has enoughspatial homogeneity that even when covers with sizes in the interval [ δ /θ , δ ] are allowed (the θ -intermediate dimension), the exponent cannot drop too much below the exponent when allthe covers have approximately the same size (the box dimension) unless θ is very small. • A well-studied class of fractals are the self-affine carpets of Bedford and McMullen; for thedefinition and a survey of the dimension theory of such carpets, see [8]. If F is any Bedford-McMullen carpet with non-uniform fibres then 0 < dim H F < dim B F = dim B F < dim A F ;the intermediate dimensions have been studied in [6, Section 4] and [19], but a precise formularemains elusive. Proposition 3.10 provides the first lower bound that is both strictly positivefor all θ ∈ (0 ,
1] and strictly concave for θ ∈ [0 , θ close to 1 that is known at the time of writing. Indeed, using the notation in [8, 19], let F be the carpet with parameters m = M = 2, n = 100, N = 1, N = 100, N = 101, which hastwo columns and many rows with a very different number of contractions in each column,so that the difference between the Hausdorff and box dimensions is quite large. Then usingthe formulae in [8, Theorem 2.1], we have dim H F ≈ . B F ≈ .
852 and dim A F = 2.Therefore the gradient of the lower bound in Proposition 3.10 at θ = 1 is dim B F − (dim B F ) dim A F ≈ . B F as θ → − , while the gradient ofKolossv´ary’s lower bound [19, Theorem 1.4] at θ = 1 is log( N/M ) − log N M log n ≈ .
352 by [19,Remark 4.2]. We have already seen that for sets such as F with 0 < dim B F < dim A F ,Proposition 3.10 improves both general lower bounds [6, Proposition 2.4] and [5, (2.6)], andindeed in this case the gradient at θ = 1 of the former is dim A F − dim B F ≈ . B F ≈ . R with the usual metric. For p ∈ (0 , ∞ ), let F p := { } ∪ { n − p : n ∈ N } .It is well known that dim A F p = 1 and dim B F p = dim B F p = p +1 . Falconer, Fraser and Kempton[6, Proposition 3.1] showed that dim θ F p = dim θ F p = θp + θ for all θ ∈ [0 , < θ ≤ φ ≤ θ F p + ( φ − θ ) dim θ F p (dim A F p − dim θ F p )( φ − θ ) dim θ F p + θ dim A F p = θp + θ + ( φ − θ ) θp + θ (cid:16) − θp + θ (cid:17) ( φ − θ ) θp + θ + θ = φp + φ = dim φ F p , so the upper bound of Proposition 3.8 is attained, which shows that this bound is sharp. Thismeans that Theorem 3.3 can also be sharp, at least if Φ and Φ are chosen appropriately, suchas Φ( δ ) = δ /θ and Φ ( δ ) = δ /φ for 0 < θ < φ <
1. The continuity result [6, Proposition 2.1],on the other hand, is not sharp: this bound can be improved by replacing the dimension n of theambient Euclidean space by dim A F in the formula (using a similar proof), but when F = F p , wehave dim A F = 1 = n and yet Proposition 3.8 still improves [6, Proposition 2.1] for these sets (theimprovement comes from ‘fattening’ the smallest sets in the cover in the proof of Theorem 3.3).If 0 < θ < ddφ (dim φ F θ ) (cid:12)(cid:12)(cid:12)(cid:12) φ = θ = ddφ (cid:18) φθ + φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) φ = θ = (cid:18) θ ( θ + φ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) φ = θ = 14 θ = dim A F θ θ , so the Lipschitz constant θ = dim A F θ θ for φ (cid:55)→ dim φ F θ , φ ∈ [ θ,
1] given in Proposition 3.8 cannotbe improved in this case, and so is sharp in general.For all p ∈ (0 , ∞ ), θ dim A F dim B F dim A F − (1 − θ ) dim B F = θ p +1 − (1 − θ ) p +1 = θp + θ = dim θ F p , so the lower bound of Proposition 3.10 is attained in this case, so this bound is sharp in general.The fact that this bound is attained means that of all sets with box dimension p +1 and Assouaddimension 1, F p has the smallest possible intermediate dimensions. Intuitively, this suggests thatof all sets whose box and Assouad dimensions take these values, F p is (in some sense) as inhomo-geneous in space as possible. This example also suggests that Proposition 3.10 will be useful forcomputing the intermediate dimensions of certain sets with known box and Assouad dimensions;if Falconer, Fraser and Kempton [6, Proposition 3.1] had had this bound at their disposal, theywould not have needed to use the mass distribution argument specific to the F p sets when provingthe lower bound dim θ F p ≥ θp + θ . A consequence of Corollary 3.4.2 is the following relationships between the Φ-intermediate andintermediate dimensions.
Proposition 3.11.
Let Φ be any admissible function, and let θ := lim inf δ → + log δ log Φ( δ ) ; θ := lim sup δ → + log δ log Φ( δ ) , (3.14) noting that ≤ θ ≤ θ ≤ . Then for any non-empty, totally bounded subset F of a uniformlyperfect metric space with more than one point, such that dim A F < ∞ , we have the followingbounds.Upper bounds for dim Φ F : i) If < θ then dim Φ F ≤ dim θ F (compare Proposition 3.12 (iii))(ii) If θ = θ then for any θ ∈ (0 , we have dim Φ F ≤ dim θ F (compare Proposition 3.12(iv))Lower bounds for dim Φ F :(iii) dim θ F ≤ dim Φ F (iv) dim θ F ≤ dim Φ F (compare Proposition 3.12 (ii))Upper bounds for dim Φ F :(v) If < θ then dim Φ F ≤ dim θ F (vi) If < θ then dim Φ F ≤ dim θ F (compare Proposition 3.12 (iii))(vii) If θ then for any θ ∈ (0 , we have dim Φ F ≤ dim θ F (viii) If θ = θ then for any θ ∈ (0 , we have dim Φ F ≤ dim θ F (compare Proposition 3.12(iv))Lower bound for dim Φ F :(ix) dim θ F ≤ dim Φ F (compare Proposition 3.12 (ii) for the φ -Assouad dimensions) For clarity, we state some consequences of gathering together some of the inequalities. • If 0 = θ = θ = lim δ → + log δ log Φ( δ ) then for any θ ∈ (0 ,
1] we have dim Φ F ≤ dim θ F anddim Φ F ≤ dim θ F . • If 0 = θ < θ then dim θ F ≤ dim Φ F ≤ dim θ F (so if dim θ F exists then dim Φ F = dim θ F ),and for any θ ∈ (0 ,
1] we have dim Φ F ≤ min { dim θ F, dim θ F } . • If 0 < θ ≤ θ then dim θ F ≤ dim Φ F ≤ min { dim θ F, dim θ F } (so if dim θ F exists thendim Φ F = dim θ F ), and max { dim θ F, dim θ F } ≤ dim Φ F ≤ dim θ F (so if dim θ F existsthen dim Φ F = dim θ F ). • If 0 < θ = θ = lim δ → + log δ log Φ( δ ) then dim Φ F = dim θ F and dim Φ F = dim θ (compare [15,Corollary 2.11] for the φ -Assouad dimensions). Proof. (of Proposition 3.11). The proof is an application of the continuity of the intermediatedimensions for θ ∈ (0 ,
1] (Proposition 3.8) and Corollary 3.4.2.(i) Suppose 0 < θ . If θ = 1 then the result is dim Φ F ≤ dim B F = dim θ F which follows fromProposition 3.1. If θ ∈ (0 ,
1) then letting η ∈ (0 , − θ ), by the definition of θ we havelim sup δ → + Φ( δ ) δ / ( θ + η ) = 0 < ∞ so Corollary 3.4.2 (i) (1) gives dim Φ F ≤ dim θ + η F , and the intermediate dimensions are continuousat θ > η → + .(ii) Follows from Corollary 3.4.2 (i) (1) since if 0 = θ then for all θ ∈ (0 ,
1] we havelim sup δ → + Φ( θ ) δ /θ = 0 < ∞ . (iii) If 0 = θ then the result follows from Proposition 3.1, so suppose 0 < θ and let η ∈ (0 , θ ).Then lim inf δ → + δ / ( θ − η ) Φ( δ ) = 0 < ∞
21o by Corollary 3.4.2 (i) (2), we have dim θ − η F ≤ dim Φ F , and by continuity of intermediatedimensions, the result follows by letting η → + .(iv) If 0 = θ then the result follows from Proposition 3.1, so suppose 0 < θ and let η ∈ (0 , θ ).Then lim sup δ → + δ / ( θ − η ) Φ( δ ) = 0 < ∞ so dim θ − η F ≤ dim Φ F by Corollary 3.4.2 (i) (1), and by continuity of intermediate dimensions,the result follows by letting η → + .(v) Suppose 0 < θ . If θ = 1 then by Proposition 3.1, dim Φ F ≤ dim Φ F ≤ dim B F = dim θ F is the result. If θ ∈ (0 ,
1) then letting η ∈ (0 , − θ ) we havelim inf δ → + Φ( δ ) δ / ( θ + η ) = 0 < ∞ so Corollary 3.4.2 (i) (2) gives dim Φ F ≤ dim θ + η F , and by continuity of intermediate dimensions,the result follows by letting η → + .(vi) Suppose 0 < θ . If θ = 1 then by Proposition 3.1, dim Φ F ≤ dim B F = dim θ F is theresult, so suppose θ ∈ (0 , η ∈ (0 , − θ ). Thenlim sup δ → + Φ( δ ) δ / ( θ + η ) = 0 < ∞ , so by Corollary 3.4.2 (i) (1) we have dim Φ F ≤ dim θ + η F , and the by continuity of intermediatedimensions, the result follows by letting η → + .(vii) Follows from Corollary 3.4.2 (i) (2) since if 0 = θ then for all θ ∈ (0 ,
1] we havelim inf δ → + Φ( θ ) δ /θ = 0 < ∞ . (viii) Follows from Corollary 3.4.2 (i) (1) since if 0 = θ then for all θ ∈ (0 ,
1] we havelim sup δ → + Φ( δ ) δ /θ = 0 < ∞ . (ix) If 0 = θ then by Proposition 3.1 we have dim H F = dim θ F ≤ dim Φ F , which is the result,so suppose 0 < θ and let η ∈ (0 , θ ). By the definition of θ ,lim sup δ → + δ / ( θ − η ) Φ( δ ) = 0 < ∞ so by Proposition 3.4.2 (i) (1), we have dim θ − η F ≤ dim Φ F , and the lower intermediate dimensionsare continuous at θ since θ >
0, so the result follows by letting η → + .For sets whose upper intermediate dimensions are continuous at θ = 0, we usually will notstudy the Φ-intermediate dimensions, because much information about the general Φ-intermediatedimensions of such sets can be obtained directly from results about their intermediate dimen-sions and these inequalities. We state results for the φ -Assouad dimensions which are similar toProposition 3.11, and follow straightforwardly from work in [11, 12, 15], in particular the remarksafter Definition 2.3. When comparing the intermediate dimensions to the Assouad spectrum, re-spectively, we consider the box dimensions to correspond to the upper box dimension (as thesedimensions are always attained as the limit of the respective spectra) and the Hausdorff dimensionto correspond to the Assouad dimension (as these dimensions are not generally attained as thelimit of the respective spectra). 22 roposition 3.12. Let φ : [0 , → [0 , satisfy < φ ( R ) ≤ R for all R ∈ [0 , . Define θ := lim inf R → + log R log Φ( R ) ; θ := lim sup R → + log R log Φ( R ) . Then using notation as in Definition 2.3, we have that for all totally bounded subsets F of a metricspace,(i) If θ = θ = 0 then dim φ A F = dim B F (compare Proposition 3.2)(ii) If θ ∈ (0 , then dim φ A F ≤ dim θ A F (compare Proposition 3.11 (ix) and (iv))(iii) If θ ∈ (0 , then dim φ A F ≥ dim θ A F ≥ dim θ A F (compare Proposition 3.11 (vi) and (i))(iv) If θ = θ = 1 then dim φ A F ≥ dim qA F (compare Proposition 3.11 (ii) and (ii))(v) If inf R ∈ (0 , φ ( R ) /R > (and so in particular θ = θ = 1 ) then dim φ A F = dim A F , see [15,Proposition 2.9] (no analogue for the Φ -intermediate dimensions, as is clear from the proofof Proposition 5.4 (ii), for example) We now investigate how these dimensions behave under H¨older, Lipschitz and bi-Lipschitz maps.The familiar upper bound dim f ( F ) ≤ α − dim F for the image of a ‘reasonable’ set F underan α -H¨older map f , which holds for the Hausdorff, box and intermediate dimensions (see Corol-lary 4.1.2), is different to the bound that can be obtained (see Theorem 4.1) for the Φ-intermediatedimensions, which demonstrates that the Φ-intermediate dimensions can behave in a way whichis genuinely different to the intermediate dimensions and can be interesting to study in their ownright. In Corollary 4.1.1 2. we prove that for subsets of uniformly perfect metric spaces with morethan one point which are non-empty, totally bounded and have finite Assouad dimension, dim Φ and dim Φ are stable under bi-Lipschitz maps, which is an important property that most notionsof dimension satisfy. This shows that the Φ-intermediate dimensions provide further invariants forthe classification of such subsets up to bi-Lipschitz image. Bi-Lipschitz stability has already beenproven for the Hausdorff and box dimensions in [4, Propositions 2.5 and 3.3] and, for subsets of R n , for the intermediate dimensions in [9, Lemma 3.1]. Fraser [10] uses the Assouad spectrumto give bounds on the possible H¨older exponents of maps from an interval to a natural class ofspirals that are better than bounds that have been obtained using any other notion of dimension.In forthcoming work, we will give similar applications of some of the results in this section todetermine whether certain spaces are bi-Lipschitz equivalent or, if not, to give information aboutthe possible H¨older exponents of maps between them, and we will also investigate sharpness forsome of the estimates in this section. Theorem 4.1.
Let Φ and Φ be admissible functions and let ( X, d X ) and ( Y, d Y ) be uniformlyperfect metric spaces with more than one point. Let F be a non-empty, totally bounded subset of X . Let f : F → Y be an α -H¨older map for some α ∈ (0 , , and assume that dim A f ( F ) < ∞ . Let γ ∈ [1 , /α ] .(i) Assume that dim Φ F < α dim A f ( F ) .(1) Assume there exists δ (cid:48) > such that for all δ ∈ (0 , δ (cid:48) ) we have Φ ( δ ) ≤ (Φ( δ / ( αγ ) )) α . (4.1) Then dim Φ f ( F ) ≤ dim Φ F + α ( γ −
1) dim A f ( F ) αγ . he particular case γ = 1 states that if Φ ( δ ) ≤ (Φ( δ /α )) α for all δ ∈ (0 , δ (cid:48) ) then dim Φ f ( F ) ≤ α − dim Φ F . The particular case γ = 1 /α , Φ = Φ gives that dim Φ f ( F ) ≤ dim Φ F + (1 − α ) dim A f ( F ) . (2) If we only assume that for all δ (cid:48) > there exists δ ∈ (0 , δ (cid:48) ) such that (4.1) holds, thenwe can conclude only that dim Φ f ( F ) ≤ dim Φ F + α ( γ −
1) dim A f ( F ) αγ . The particular case γ = 1 states that if for all δ (cid:48) > there exists δ ∈ (0 , δ (cid:48) ) such that Φ ( δ ) ≤ (Φ( δ /α )) α then dim Φ f ( F ) ≤ α − dim Φ F .(ii) Assume that dim Φ F < α dim A f ( F ) and that there exists δ (cid:48) > such that for all δ ∈ (0 , δ (cid:48) ) we have Φ ( δ ) ≤ (Φ( δ / ( αγ ) )) α . Then dim Φ f ( F ) ≤ dim Φ F + α ( γ −
1) dim A f ( F ) αγ . The particular case γ = 1 states that if Φ ( δ ) ≤ (Φ( δ /α )) α for all δ ∈ (0 , δ (cid:48) ) then dim Φ f ( F ) ≤ α − dim Φ F . The particular case γ = 1 /α , Φ = Φ gives that if dim Φ f ( F ) ≤ dim Φ F + (1 − α ) dim A f ( F ) . Note that if, contrary to the assumption of Theorem 4.1 (i), we have dim Φ F ≥ α dim A f ( F )then by Proposition 3.1 dim Φ f ( F ) ≤ dim A f ( F ) ≤ α − dim Φ F, and if moreover dim Φ f ( F ) ≥ α dim A f ( F ) then alsodim Φ f ( F ) ≤ dim A f ( F ) ≤ α − dim Φ F. Proof. (i) (1) Assume that dim A f ( F ) < ∞ and dim Φ F < α dim A f ( F ) < ∞ . The idea of theproof is to consider a cover of F with diameters in [Φ( δ ) , δ ], consider cover of f ( F ) formed by theimages under f of this cover, and ‘fatten’ the smallest sets in the new cover to size Φ ( δ αγ ) andbreak up the largest sets in the new cover to size δ αγ .Since f is α -H¨older, there exists C ∈ [0 , ∞ ) such that f is ( C, α )-H¨older. Fix any (cid:15) >
0. Let t > dim Φ F + α ( γ −
1) dim A f ( F ) αγ , so there exist s > dim Φ F and a > dim A f ( F ) such that t > s + α ( γ − aαγ and s < αa . Therefore if we define g ( η ) := ηs + αa ( γ − η ) αγ then g ( η ) is linear and decreasing in η , so for all η ∈ [1 , γ ] we have t > g (1) ≥ g ( η ) ≥ g ( γ ) = s/α .Without loss of generality we may assume that the constant C ≥
1. Since Φ is admissible,there exists Y ∈ (0 ,
1) such that Φ and Φ are Y -admissible, and we may assume that δ (cid:48) < Y .Since a > dim A f ( F ), there exists M ∈ N such that for all y ∈ f ( F ) and 0 < r < R , we have N r ( B ( y, R ) ∩ f ( F )) ≤ M ( R/r ) a . Since Y is uniformly perfect, there exists c ∈ (0 ,
1) such that Y is c -uniformly perfect. Since Φ is admissible, Φ( δ ) /δ → δ → + , so there exists ∆ > δ ∈ (0 , ∆) we have Φ( δ ) /δ < c/ ( δ ) /δ < c/
2. Since s > dim Φ F , there exists δ ∈ (0 , min { ∆ /α , ( c | Y | ) /α , ( δ (cid:48) ) /α , } ) such that for all δ ∈ (0 , δ ) there exists a cover { U i } of F such that Φ( δ ) ≤ | U i | ≤ δ for all i , and (cid:88) i | U i | s ≤ (( C + c − ) s/α + M (2 C ) a + γg (1) ) − (cid:15)/ . (4.2)We may assume without loss of generality that for all i we have U i ∩ F (cid:54) = ∅ . Now, { f ( U i ) } covers f ( F ), and | f ( U i ) | ≤ C | U i | α for all i . Now we have two cases.Case 1: Suppose i is such that | f ( U i ) | ≤ δ αγ /
2. Then f ( U i ) (cid:54) = ∅ so there exists some y i ∈ f ( U i ).Since Y is c -uniformly perfect and Φ ( δ αγ ) /c ≤ δ αγ /c ≤ δ α /c < δ α /c < | Y | there exists y (cid:48) i ∈ Y such that Φ ( δ αγ ) ≤ d Y ( y i , y (cid:48) i ) ≤ Φ ( δ αγ ) /c , noting that by the assumption on Φ , this meansthat d Y ( y i , y (cid:48) i ) ≤ (Φ( δ )) α /c . Let V i := f ( U i ) ∪ { y (cid:48) i } , so f ( U i ) ⊆ V i . By the triangle inequality andthe fact that δ αγ ≤ δ αγ < ∆, we haveΦ ( δ αγ ) ≤ d Y ( y i , y (cid:48) i ) ≤ | V i | ≤ | f ( U i ) | + d Y ( y i , y (cid:48) i ) ≤ | f ( U i ) | + Φ ( δ αγ ) /c ≤ δ αγ / δ αγ / δ αγ . (4.3)Moreover, by the assumption about Φ , | V i | ≤ | f ( U i ) | + Φ ( δ αγ ) /c ≤ C | U i | α + (Φ( δ )) α /c ≤ ( C + c − ) | U i | α . (4.4)Case 2: Now suppose that i is such that δ αγ / < | f ( U i ) | ≤ Cδ α . Then (2 C ) − /α δ γ < | U i | ≤ δ sothere exists β i ∈ [1 , γ ] such that (2 C ) − /α δ β i < | U i | ≤ δ β i . Then δ αγ / < | f ( U i ) | ≤ Cδ αβ i ≤ Cδ α .By the definition of M , there exists a collection of M (2 C ) a δ α ( β i − γ ) a ≤ M (2 C ) a | U i | αa (1 − γ/β i ) orfewer balls which cover f ( U i ) ∩ f ( F ) and such that each of these balls has diameter at most δ αγ / Y is c -uniformly perfect, for each of these balls, we can add a point in Y whose distancefrom the centre of the ball is between Φ ( δ αγ ) and Φ ( δ αγ ) /c , and since Φ ( δ αγ ) /c ≤ δ αγ /
2, eachof the new sets, which we call { W i,j } j , will satisfyΦ ( δ αγ ) ≤ | W i,j | ≤ δ αγ / δ αγ / δ αγ (4.5)by the triangle inequality. Moreover, | W i,j | ≤ δ αγ = (2 C ) γ/β i ((2 C ) − /α δ β i ) αγ/β i ≤ (2 C ) γ/β i | U i | αγ/β i . (4.6)Recall that for all η ∈ [1 , γ ] we have t > g (1) ≥ g ( η ) ≥ g ( γ ) = s/α , so in particular t > g ( β i ) forall i . Therefore, using (4.4) and (4.6) and the definition of g we have (cid:88) k | V k | t + (cid:88) i,j | W i,j | t < (cid:88) k | V k | s/α + (cid:88) i,j | W i,j | g ( β i ) ≤ (cid:88) k (( C + c − ) | U k | α ) s/α + (cid:88) i M (2 C ) a | U i | αa (1 − γ/β i ) ((2 C ) γ/β i | U i | αγ/β i ) g ( β i ) ≤ ( C + c − ) s/α (cid:88) k | U k | s + M (2 C ) a + γg ( β i ) /β i (cid:88) i | U i | s ≤ ( C + c − ) s/α (cid:88) k | U k | s + M (2 C ) a + γg (1) (cid:88) i | U i | s ≤ (cid:15) { V k } k ∪ { W i,j } i,j covers f ( F ), and noting (4.3)and (4.5), we have dim Φ f ( F ) ≤ t . Letting t → (cid:32) dim Φ F + α ( γ −
1) dim A f ( F ) αγ (cid:33) + gives dim Φ f ( F ) ≤ dim Φ F + α ( γ −
1) dim A f ( F ) αγ , as required.(i) (2) If we only have that for all δ (cid:48) > δ ∈ (0 , δ (cid:48) ) such that (4.1) holds, then as inthe proof of (i) (1), for all t > dim Φ F + α ( γ −
1) dim A f ( F ) αγ we can construct a cover { Z i } of f ( F ) suchthat Φ ( δ ) ≤ | Z i | ≤ δ for all i , and (cid:80) i | Z i | t ≤ (cid:15) . Therefore dim Φ f ( F ) ≤ t , and letting t → (cid:32) dim Φ F + α ( γ −
1) dim A f ( F ) αγ (cid:33) + gives dim Φ f ( F ) ≤ dim Φ F + α ( γ −
1) dim A f ( F ) αγ , as required.(ii) If (4.1) holds for all δ ∈ (0 , δ (cid:48) ), then for all t > dim Φ F + α ( γ −
1) dim A f ( F ) αγ there exists s > dim Φ F and a > dim A f ( F ) such that t > s + α ( γ − aαγ and s < αa . Then for all δ ∈ (0 , min { ∆ /α , ( c | Y | ) /α , ( δ (cid:48) ) /α , } ) there exist δ ∈ (0 , δ ) and a cover { U i } of F such that Φ( δ ) ≤| U i | ≤ δ for all i , and (cid:88) i | U i | s ≤ (( C + c − ) s/α + M (2 C ) a + γg (1) ) − (cid:15)/ . Then as in the proof of (i) (1), there exists a cover | Z i | of f ( F ) such that Φ ( δ αγ ) ≤ | Z i | ≤ δ αγ for all i , and (cid:80) i | Z i | t ≤ (cid:15) . Therefore dim Φ f ( F ) ≤ t , and letting t → (cid:16) dim Φ F + α ( γ −
1) dim A f ( F ) αγ (cid:17) + gives dim Φ f ( F ) ≤ dim Φ F + α ( γ −
1) dim A f ( F ) αγ , as required.Corollary 4.1.1 shows that in particular the intermediate and Φ-intermediate dimensions cannotincrease under Lipschitz maps. By contrast, the Assouad spectrum can increase under Lipschitzmaps, see [7, Theorem 3.4.12]. Corollary 4.1.1.
Let Φ be an admissible function and let ( X, d X ) and ( Y, d Y ) be uniformly perfectmetric spaces with more than one point. Let F be a non-empty, totally bounded subset of X , let f : F → Y be a Lipschitz map, and assume that dim A f ( F ) < ∞ . Then1. (i) dim Φ f ( F ) ≤ dim Φ F (ii) dim Φ f ( F ) ≤ dim Φ F .2. If moreover f is bi-Lipschitz then dim Φ f ( F ) = dim Φ F and dim Φ f ( F ) = dim Φ F . (Note thatin this case the assumption dim A f ( F ) < ∞ is equivalent to dim A F < ∞ since the Assouaddimension is stable under bi-Lipschitz maps, see [7, Lemma 2.4.2] for a proof in R n , forexample.)Proof.
1. (i) If dim Φ F ≥ dim A f ( F ) then dim Φ f ( F ) ≤ dim A f ( F ) ≤ dim Φ F by Proposition 3.1;if dim Φ F < dim A f ( F ) then the case α = γ = 1, Φ = Φ, of Theorem 4.1 (i) (1) givesdim Φ f ( F ) ≤ dim Φ F .(ii) If dim Φ F ≥ dim A f ( F ) then dim Φ f ( F ) ≤ dim A f ( F ) ≤ dim Φ F ; if dim Φ F < dim A f ( F )then the case α = γ = 1 of Theorem 4.1 (ii) gives dim Φ f ( F ) ≤ dim Φ F .26. If f is bi-Lipschitz then there exists a Lipschitz map g : f ( F ) → F that is inverse to f .Applying 1. (i) to f and g gives dim Φ f ( F ) = dim Φ F . Applying 1. (ii) to f and g givesdim Φ f ( F ) = dim Φ F , as required.The particular form Φ( δ ) = δ /θ for the intermediate dimensions gives a better H¨older distortionestimate for the intermediate dimensions. For subsets of Euclidean space, Corollary 4.1.2 was notedin [5, Section 2.1 5.], and it also follows from the m = n case of the stronger result [1, Theorem 3.1](using [2, Lemma 3.3 and Theorem 4.1]) which is proven using capacity theoretic methods anddimension profiles, but we include it nonetheless because our proof works for more general metricspaces. Corollary 4.1.2.
Let Φ be an admissible function and let ( X, d X ) and ( Y, d Y ) be uniformly perfectmetric spaces with more than one point. Let F be a non-empty, totally bounded subset of X , let f : F → Y be an α -H¨older map for some α ∈ (0 , and assume that dim A f ( F ) < ∞ . Then for all θ ∈ [0 , we have(i) dim θ f ( F ) ≤ α − dim θ F (ii) dim θ f ( F ) ≤ α − dim θ F Proof.
These estimates hold for the Hausdorff and lower and upper box dimensions (left for thereader, similar to [4, Exercise 2.2 and Proposition 3.3]), so assume that θ ∈ (0 ,
1) and let Φ( δ ) =Φ ( δ ) = δ /θ .(i) If dim θ F ≥ α dim A f ( F ) then dim θ f ( F ) ≤ dim A f ( F ) ≤ α − dim θ F . If dim θ F < α dim A f ( F )then since Φ ( δ ) = Φ( δ ) = δ /θ = (( δ /α ) /θ ) α = Φ( δ /α ) α , the case γ = 1 of Theorem 4.1 (i) (1) gives that dim θ f ( F ) ≤ α − dim θ F , as required.(ii) follows from Theorem 4.1 (ii) in a similar way to how (i) follows from Theorem 4.1 (i) (1). In this section we prove a mass distribution principle for the Φ-intermediate dimensions and aconverse result (a Frostman type lemma), before proving some applications regarding product setsand finite stability.
The mass distribution principle for the intermediate dimensions [6, Proposition 2.2] is a useful toolto bound these dimensions from below by putting a measure on the set. The following naturalgeneralisation for the Φ-intermediate dimensions holds.
Lemma 5.1.
Let Φ be an admissible function, let F be a non-empty, totally bounded subset of auniformly perfect metric space X with more than one point, and let s ≥ , a, c, δ > be constants.(i) If there exists a positive decreasing sequence δ n → such that for each n ∈ N there existsa Borel measure µ n with support supp ( µ n ) ⊆ F with µ n ( supp ( µ n )) ≥ a , and such that for everyBorel subset U ⊆ X with Φ( δ n ) ≤ | U | ≤ δ n we have µ n ( U ) ≤ c | U | s , then dim Φ F ≥ s .(ii) If, moreover, for all δ ∈ (0 , δ ) there exists a Borel measure µ δ with support supp ( µ δ ) ⊆ F with µ δ ( supp ( µ δ )) ≥ a , and such that for every Borel subset U ⊆ X with Φ( δ ) ≤ | U | ≤ δ we have µ δ ( U ) ≤ c | U | s , then dim Φ F ≥ s .Proof. (i) If n ∈ N and { U i } is a cover of F such that Φ( δ n ) ≤ | U i | ≤ δ n for all i , then thetopological closures U i are Borel, satisfy Φ( δ n ) ≤ | U i | = | U i | ≤ δ n for all i , and cover supp( µ n ), so a ≤ µ n (supp( µ n )) = µ n (cid:32)(cid:91) i U i (cid:33) ≤ (cid:88) i µ n ( U i ) ≤ c (cid:88) i | U i | s = c (cid:88) i | U i | s . (5.1)27herefore (cid:80) i | U i | s ≥ a/c >
0, so dim Φ F ≥ s .(ii) If δ ∈ (0 , δ ) and { V i } is a cover of F such that Φ( δ ) ≤ | V i | ≤ δ for all i , then as in (5.1) wehave (cid:80) i | V i | s ≥ a/c >
0, so dim Φ F ≥ s . Another powerful tool in fractal geometry and geometric measure theory is Frostman’s lemma,dual to the mass distribution principle. We have the following analogue of Frostman’s lemma forthe Φ-intermediate dimensions, generalising [6, Proposition 2.3] for the intermediate dimensionsboth to more general functions Φ and more general metric spaces.
Lemma 5.2.
Let Φ be an admissible function and let F be a non-empty, totally bounded subset ofa uniformly perfect metric space ( X, d ) with more than one point, and assume that dim A F < ∞ .(i) If dim Φ F > then for all s ∈ (0 , dim Φ F ) there exists a constant c ∈ (0 , ∞ ) (depending on s ) such that for all δ > there exists δ ∈ (0 , δ ) and a Borel probability measure µ δ with finitesupport supp ( µ δ ) ⊆ F (so µ δ is the sum of finitely many atoms located at points in F ) such thatif x ∈ X and Φ( δ ) ≤ r (cid:48) ≤ δ then µ δ ( B ( x, r (cid:48) )) ≤ cr (cid:48) s . (ii) If dim Φ F > then for all s ∈ (0 , dim Φ F ) there exist constants c, δ ∈ (0 , ∞ ) such that forall δ (cid:48) ∈ (0 , δ ) there exists a Borel probability measure µ δ (cid:48) with finite support supp ( µ δ (cid:48) ) ⊆ F suchthat if x ∈ X and Φ( δ (cid:48) ) ≤ r ≤ δ (cid:48) then µ δ (cid:48) ( B ( x, r )) ≤ cr s .Proof. The idea of the proof is to put point masses on an analogue of dyadic cubes of size approx-imately Φ( δ ) so that the measure of sets with diameter approximately Φ( δ ) is controlled by theΦ-intermediate dimension of F , and then iteratively reduce the masses so that the mass of largercubes is not too large either.(ii) We first prove (ii). The proof is based on the proof of [6, Proposition 2.3] for the intermediatedimensions which is in turn based on [22, pages 112-114]. We note that in [6, Proposition 2.3] theassumption that the set F is closed is not necessary as it is not used in the proof. The maindifference with the proof of [6, Proposition 2.3] is that in R n we have the dyadic cubes to workwith, but here we use the fact that dim A F < ∞ , which implies that F is doubling, and use ananalogue of the dyadic cubes constructed in [18] for general doubling metric spaces. We apply [18,Theorem 2.2], using its notation, with the quasi-metric ρ simply being the metric d restricted to F , so the usual triangle inequality holds and A = 1. Let δ := 1 /
20 (in fact any δ ∈ (0 , / F is non-empty so there exists x F ∈ F , and note that F ⊆ B F ( x F , | F | ). Sincedim A F < ∞ , for each k ∈ N there exists D k ∈ N such that N δ k / ( B F ( x F , | F | )) ≤ D k . Bythe pigeonhole principle, all δ k -separated subsets of B F ( x F , | F | ) must have cardinality at most D k , so there exists a finite δ k -separated subset { z kα } α of B F ( x F , | F | ) = F , of maximum possiblecardinality. Then applying the theorem with c = C = 1, c = 1 / C = 2, we have that for each k ∈ N there exist subsets Q k := { Q kα } α of F such that:1. for all k ∈ N we have F = (cid:83) α Q kα with the union disjoint;2. B F ( z kα , (20) − k / ⊆ B F ( z kα , c (20) − k ) ⊆ Q kα ⊆ B F ( z kα , C (20) − k )= B F ( z kα , − k );3. if k, l ∈ N with k ≤ l then for any α, β , either Q kα ∩ Q lβ = ∅ or Q lβ ⊆ Q kα , and in the lattercase, also B F ( z lβ , − k ) ⊆ B F ( z kα , − k ), and we call Q kα the parent cube of Q lβ .28e say that Q kα is a dyadic cube with centre z kα .Now, Φ is admissible so there exists Y ∈ (0 ,
1) such that Φ is Y -admissible. Suppose dim Φ F > s ∈ (0 , dim Φ F ). Then there exists (cid:15) > δ ∈ (0 , Y ) such that for any δ (cid:48) ∈ (0 , δ ) andany cover { U i } of F satisfying Φ( δ (cid:48) ) ≤ | U i | ≤ δ (cid:48) for all i , we have (cid:88) i | U i | s ≥ (cid:15). (5.2)Since X is uniformly perfect, there exists c ∈ (0 ,
1) such that X is c -uniformly perfect. Since Φis admissible, there exists δ ∈ (0 , δ ) such that for all δ (cid:48) ∈ (0 , δ ) we have Φ( δ (cid:48) ) /δ (cid:48) < c / δ (cid:48) ∈ (0 , δ ) and let m = m ( δ (cid:48) ) be the largest natural number satisfying Φ( δ (cid:48) ) ≤ (20) − m . Definethe Borel measure µ m by µ m := (cid:88) α − ms M z kα where M z kα is a unit point mass at z kα . Let l be the largest integer such that 8(20 − ( m − l ) ) ≤ δ (cid:48) , notingthat since δ (cid:48) < δ we have Φ( δ (cid:48) ) /δ (cid:48) < /
320 and so l ≥
1. In particular, for all Q m − l ∈ Q m − l we have | Q m − l | ≤ δ (cid:48) /
2. In order to reduce the mass of cubes which carry too much measure,having defined µ m − k for some k ∈ { , , . . . , l − } , inductively define the Borel measure µ m − k − ,supported on the same finite set as µ m , by µ m − k − | Q m − k − := min (cid:26) , − ( m − k − s µ m − k ( Q m − k − ) (cid:27) µ m − k | Q m − k − for all Q m − k − ∈ Q m − k − . By construction, if k ∈ { , , . . . , l } and Q m − k ∈ Q m − k then µ m − l ( Q m − k ) ≤ − ( m − k ) s ≤ s c − s | Q m − k | s , (5.3)by condition 2. Moreover, each Q m ∈ Q m satisfies µ m ( Q m ) = 20 − ms , and if k ∈ { , , . . . , l − } and if Q m − k ∈ Q m − k satisfies µ m − k ( Q m − k ) = 20 − ( m − k ) s and Q m − k − ∈ Q m − k − is theparent cube of Q m − k then by the construction of µ m − k − , either µ m − k − ( Q m − k ) = 20 − ( m − k ) s or µ m − k − ( Q m − k − ) = 20 − ( m − k − s . Therefore for all y ∈ F there is at least one k ∈ { , , . . . , l } and Q y ∈ Q m − k with y ∈ Q y such that µ m − l ( Q y ) = 20 − ( m − k ) s ≥ − s | Q y | s (5.4)where the inequality is by condition 2. For each y ∈ F , choosing Q y satisfying (5.4) and Q y ∈ Q m − k for the largest possible k ∈ { , , . . . , l } yields a finite collection of cubes { Q i } which cover F . Foreach i , let z i be the centre of Q i , and by the uniformly perfect condition, since Φ( δ (cid:48) ) /c ≤ δ < | X | ,there exists p i ∈ X such that Φ( δ (cid:48) ) ≤ d ( p i , z i ) ≤ Φ( δ (cid:48) ) /c ≤ δ (cid:48) /
2. Letting U i := Q i ∪ { p i } , bythe triangle inequality and the definitions of m and l and condition 2 we have Φ( δ (cid:48) ) ≤ | U i | ≤ δ (cid:48) / δ (cid:48) / δ (cid:48) . Since { Q i } covers F , also { U i } covers F . Moreover, | U i | ≤ | Q i | + Φ( δ (cid:48) ) /c ≤ (1 + 1 /c ) | Q i | . Therefore by (5.2) and (5.4) we have µ m − l ( F ) = (cid:88) i µ m − l ( Q i ) ≥ (cid:88) i − s | Q i | s ≥ − s (1 + 1 /c ) − s (cid:88) i | U i | s ≥ − s (1 + 1 /c ) − s (cid:15). (5.5)Define µ δ (cid:48) := ( µ m − l ( F )) − µ m − l , which is clearly a Borel probability measure with finite supportsupp( µ δ (cid:48) ) ⊆ F . Now, since dim A F < ∞ there exists C ∈ N such that for all P ∈ F and d > N d ( B F ( P, d )) ≤ C . Let x ∈ X and r ∈ [Φ( δ (cid:48) ) , δ (cid:48) ]. Let j = j ( r ) be the largestinteger in { , , . . . , l } such that 20 − ( m − j +1) < r (such an integer exists by the definition of m ). If B X ( x, r ) ∩ F = ∅ then µ δ (cid:48) ( B X ( x, r )) = 0, so suppose that there exists some x ∈ B X ( x, r ) ∩ F ,so B X ( x, r ) ⊆ B F ( x , r ). Suppose B X ( x, r ) ∩ Q m − j (cid:54) = ∅ for some Q m − j ∈ Q m − j , with centre z m − j , say. Then there exists z ∈ B ( x, r ) ∩ Q m − j , and by the triangle inequality, condition 2 andthe definition of j , we have d ( x , z m − j ) ≤ d ( x , z ) + d ( z, z m − j ) ≤ r + 2(20) − ( m − j ) ≤ − ( m − j ) . z m − j ∈ B F ( x , − ( m − j ) ), and all the centres of the cubes in Q m − j which intersect B X ( x, r ) form a 20 − ( m − j ) -separated subset of B F ( x , − ( m − j ) ). But N − ( m − j ) / ( B F ( x , − ( m − j ) )) ≤ C, so by the pigeonhole principle there are most C such centres, and so at most C elements of Q m − j which intersect B X ( x, r ). Therefore by (5.3) and (5.5) and the definition of j , we have µ δ (cid:48) ( B X ( x, r )) = ( µ m − l ( F )) − µ m − l ( B X ( x, r )) ≤ C ( µ m − l ( F )) − − ( m − j ) s ≤ cr s , where c := C s (1 + 1 /c ) s (cid:15) − (20) s , as required.(i) The proof of (i) is similar. The special case of this result for the upper intermediate dimen-sions in R n follows from a similar argument to the proof of [6, Proposition 2.3] but is not notedthere. Suppose dim Φ F > s ∈ (0 , dim Φ F ). Then there exists (cid:15) > δ > δ ∈ (0 , min { δ , δ , Y } ), where δ and Y are as in the proof of (ii), and a cover { U i } of F satisfying Φ( δ ) ≤ | U i | ≤ δ for all i , and (cid:80) i | U i | s ≥ (cid:15) . Then as in the proof of (ii), thereexists a Borel probability measure µ δ with finite support supp( µ δ ) ⊆ F such that if x ∈ X andΦ( δ ) ≤ r (cid:48) ≤ δ then µ δ ( B ( x, r (cid:48) )) ≤ cr (cid:48) s , where c := C s (1 + 1 /c ) s (cid:15) − (20) s , as above.Putting Lemma 5.1 and Lemma 5.2 together, we obtain a useful characterisation of the Φ-intermediate dimensions. Theorem 5.3.
Let Φ be an admissible function and let F be a non-empty, totally bounded subsetof a uniformly perfect metric space X with more than one point, such that dim A F < ∞ . Then ( i ) dim Φ F = sup { s ≥ there exists a constant C ∈ (0 , ∞ ) such that for all δ > there exists δ ∈ (0 , δ ) and a Borel probabilitymeasure µ δ with support supp ( µ δ ) ⊆ F such that if U is a Borel subset of X and satisfies Φ( δ ) ≤ | U | ≤ δ then µ δ ( U ) ≤ C | U | s } ( ii ) dim Φ F = sup { s ≥ there exist C, δ ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ) there exists a Borel probability measure µ δ with support supp ( µ δ ) ⊆ F such that if U is a Borelsubset of X and satisfies Φ( δ ) ≤ | U | ≤ δ then µ δ ( U ) ≤ C | U | s } Proof.
We prove (ii) using Lemma 5.1 (ii) and Lemma 5.2 (ii); (i) follows from Lemma 5.1 (i) andLemma 5.2 (i) in a similar way. We denote by ‘sup’ the supremum on the right-hand side of theequation (ii). The function Φ is admissible, so there exists Y ∈ (0 , ∞ ) such that Φ is Y -admissible,and F is non-empty so there exists y ∈ F . If s = 0, then letting δ := Y , C := 1 and, for all δ ∈ (0 , δ ), letting µ δ be a unit point mass at y , we see that sup is well-defined and non-negative.Suppose that dim Φ F > s ∈ (0 , dim Φ F ). Then by the Frostman type Lemma 5.2 (ii), thereexist constants c, δ ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ) there exists a Borel probability measure µ δ with finite support supp( µ δ ) ⊆ F such that if x ∈ X and Φ( δ ) ≤ r ≤ δ then µ δ ( B ( x, r )) ≤ cr s . If U is a Borel subset of X and satisfies Φ( δ ) ≤ | U | ≤ δ , then if U ∩ F = ∅ then µ δ ( U ) = 0. Suppose thereexists some x ∈ U ∩ F . Then U ∩ supp( µ δ ) ⊆ B ( x, | U | ), so there exist x , . . . , x M ∈ B F ( x, | U | ),where M is the doubling constant of F , such that U ∩ supp( µ δ ) ⊆ B F ( x, | U | ) ⊆ (cid:83) Mi =1 B F ( x i , | U | ).Therefore µ δ ( U ) ≤ M (cid:88) i =1 µ δ ( B F ( x i , | U | )) = M (cid:88) i =1 µ δ ( B X ( x i , | U | )) ≤ C | U | s , C := M c . Thus s ≤ sup, and letting s → (dim Φ F ) − , we have dim Φ F ≤ sup.To prove the reverse inequality, suppose sup > t ∈ (0 , sup). Then there exists C, δ ∈ (0 , ∞ ) such that all δ ∈ (0 , δ ) there exists a Borel probability measure µ δ with supportsupp( µ δ ) ⊆ F such that if U is a Borel subset of X and satisfies Φ( δ ) ≤ | U | ≤ δ then µ δ ( U ) ≤ C | U | t . Therefore by the mass distribution principle Lemma 5.1 (ii) we have t ≤ dim Φ F . Letting t → (sup) − we have sup ≤ dim Φ F .Therefore if either sup or dim Φ F is positive then they are equal. But they are both non-negative,so they must always be equal, as required. We will see several applications of the mass distribution principle, and we give one here, whichillustrates an important difference between the upper and lower Φ-intermediate dimensions. Itwas stated in [5, Section 2.1 2.] that in Euclidean space, the upper intermediate dimensions arefinitely stable but the lower intermediate dimensions are not. In Proposition 5.4 we prove thatfor any admissible Φ, the upper Φ-intermediate dimension is finitely stable but the lower versionis not. This difference in finite stability supports the general principle that the upper version ofa dimension is often better behaved than the lower version, a principle which is also supportedby the fact that, for example, the modified upper box dimension can be defined using packingmeasure but the modified lower box dimension cannot (see [4]) and that the Assouad dimensionand Assouad spectrum are monotone for subsets but the lower dimension and lower spectrum arenot (see [7]).
Proposition 5.4. (i) The dimension dim Φ is finitely stable : if E and F are non-empty, totallybounded subsets of a uniformly perfect metric space with more than one point, then dim Φ ( E ∪ F ) =max { dim Φ E, dim Φ F } .(ii) Working in the metric space R with the usual metric, for any admissible Φ , there existcompact sets E, F ⊂ R such that dim Φ ( E ∪ F ) > max { dim Φ E, dim Φ F } , so dim Φ is not finitely stable.Proof. (i) First note that E ⊆ E ∪ F and F ⊆ E ∪ F so by Proposition 2.1 (i), we have dim Φ E ≤ dim Φ ( E ∪ F ) and dim Φ F ≤ dim Φ ( E ∪ F ), so max { dim Φ E, dim Φ F } ≤ dim Φ ( E ∪ F ). To prove thereverse inequality, note that if s > max { dim Φ E, dim Φ F } then there exists δ > δ ∈ (0 , δ ), if { U i } is a cover of E with Φ( δ ) ≤ | U i | ≤ δ for all i then (cid:80) i | U i | s ≤ (cid:15)/
2, and thereexists δ > δ ∈ (0 , δ ), if { V l } is a cover of F with Φ( δ ) ≤ | V l | ≤ δ for all l then (cid:80) l | V l | s ≤ (cid:15)/
2. Letting δ ∈ (0 , min { δ , δ } ) and letting { U i } and { V l } be the corresponding coversas above, then { W j } := { U i } ∪ { V l } covers E ∪ F , and (cid:88) j | W j | s ≤ (cid:88) i | U i | s + (cid:88) l | V l | s ≤ (cid:15)/ (cid:15)/ (cid:15). Therefore dim Φ ( E ∪ F ) ≤ s , and letting s → (max { dim Φ E, dim Φ F } ) + gives dim Φ ( E ∪ F ) ≤ max { dim Φ E, dim Φ F } , and sodim Φ ( E ∪ F ) = max { dim Φ E, dim Φ F } , as required.(ii) We take inspiration from [4, Exercises 2.8, 2.9]. The idea is to construct generalised Cantorsets E and F , each of which looks ‘large’ on most scales but ‘small’ on some sequence of scales.We do this in such a way that the sequences of scales where the two sets look small do not even31pproximately coincide, so for each small δ , either E looks large at every scale between δ and Φ( δ ),or F looks large at every scale between δ and Φ( δ ).The function Φ is admissible, so there exists Y ∈ (0 ,
1] such that Φ is Y -admissible, and wecan set Φ( δ ) := δ for δ ∈ [ Y,
1] without changing dim Φ or dim Φ for any set, so without loss ofgenerality we may assume that Y = 1. We inductively define the numbers k n ∈ { , , , . . . } and e kn , f kn >
0, for n = 0 , , , . . . , as follows. Let k := 0, e k = f k = 1. Having defined k n , e kn , f kn for some n = 0 , , , . . . , we have two cases depending on the parity of n . If n iseven, let k n +1 be the smallest integer such that k n +1 > k n and(1 / kn +1 − kn f kn < Φ((1 / kn +1 − kn (1 / kn +2 − kn +1 e kn ) , (5.6)and let e kn +1 := (1 / kn +1 − kn (1 / kn +1 − kn +1 e kn , and let f kn +1 := (1 / kn +1 − kn f kn .If, on the other hand, n is odd, then let k n +1 be the smallest integer such that k n +1 > k n and(1 / kn +1 − kn e kn < Φ((1 / kn +1 − kn (1 / kn +2 − kn +1 f kn ) , (5.7)and let f kn +1 := (1 / kn +1 − kn (1 / kn +1 − kn +1 f kn , and let e kn +1 := (1 / kn +1 − kn e kn .Now that we have defined these sequences, let E := [0 ,
1] and for j ∈ N , if 10 k n < j ≤ k n +1 for some even n ∈ { , , , . . . } then obtain E j by removing the middle 3/5 of each interval in E j − , otherwise obtain E j by removing the middle 1/3 of each interval in E j − . Let F := [2 , j ∈ N , if 10 k n < j ≤ k n +1 for some odd n ∈ { , , , . . . } then obtain F j from removingthe middle 3/5 of each interval in F j − , otherwise obtain F j by removing the middle 1/3 of eachinterval in F j − . Define E := (cid:84) ∞ j =1 E j and F := (cid:84) ∞ j =1 F j , noting that both are non-empty andcompact subsets of R . For all j ∈ N , let e j and f j be the lengths of each of the 2 j intervals in E j and F j respectively, noting that for each n ∈ N , the two different definitions that we have given for e kn and f kn agree by induction. By induction, e j and f j are sequences in (0 ,
1] which convergemonotonically to 0.We now find an upper bound for dim B E . Let n ∈ N be even. Then E kn +1 is made up of2 kn +1 intervals, each of length e kn +1 = (1 / kn +1 − kn e kn ≤ (1 / kn +1 − kn . Covering E with these intervals, we see that for all n ∈ N we havelog N e kn +1 F ( E ) − log( e kn +1 ) ≤ log 2 kn +1 log 5 kn +1 − kn = 10 k n +1 log 2(10 k n +1 − k n ) log 5= 10(10 k n ) log 29(10 k n ) log 5 = 10 log 29 log 5 . Therefore dim B E ≤
10 log 29 log 5 , and similarly using F kn +1 for n odd to cover F , we have thatdim B F ≤
10 log 29 log 5 . Therefore 10 log 29 log 5 ≥ max { dim B E, dim B F } . (5.8)To bound dim Φ ( E ∪ F ) from below, we use the mass distribution principle Lemma 5.1. Definethe sequence ( r n ) n ≥ by r n := (cid:40) e kn +2 = (1 / kn +1 − kn (1 / kn +2 − kn +1 e kn if n even, f kn +2 = (1 / kn +1 − kn (1 / kn +2 − kn +1 f kn if n odd . This sequence is strictly decreasing, because if n ≥ r n +1 = f kn +1+2 < f kn +1 < Φ( e kn +2 ) ≤ e kn +2 = r n , and similarly if n is odd then by (5.7) we have r n +1 < r n . Let δ ∈ (0 , r ). Let n δ ∈ N be theunique natural number such that r n δ ≤ δ < r n δ − . We have two cases depending on the parity of n δ . If n δ is even, then let µ δ be any Borel probability measure on F which gives mass 2 − knδ +1 to each of the 2 knδ +1 intervals in F knδ +1 . Let U be a Borel subset of R with Φ( δ ) ≤ | U | ≤ δ .32et j ∈ N be the unique natural number (depending on | U | ) such that f j ≤ | U | < f j − . By (5.6)we have f knδ +1 ≤ Φ( e knδ +2 ) = Φ( r n δ ) ≤ Φ( δ ) ≤ | U | < f j − , so j − < k nδ +1 , and we also have that f j ≤ | U | ≤ δ < r n δ − = f knδ − , so in fact10 k nδ − +2 < j ≤ k nδ +1 . Therefore by the construction of F , we have f j ≥ (cid:18) (cid:19) knδ − (cid:18) (cid:19) j − knδ − > (cid:18) (cid:19) j/ (cid:18) (cid:19) j/ . (5.9)Since | U | < f j − , U can intersect at most two of the 2 j − intervals in F j − , so U can intersect atmost 2(2 knδ +1 − j ) of the 2 knδ +1 intervals in F knδ +1 (of mass 2 − knδ +1 ). Therefore µ δ ( U ) ≤ knδ +1 − j )(2 − knδ +1 )= 2(2 − j )= 2 (cid:32)(cid:18) (cid:19) j/ (cid:18) (cid:19) j/ (cid:33) ≤ f j by (5.9) ≤ | U | If, on the other hand, n δ is odd, then letting µ δ be a Borel probability measure on E whichgives mass 2 − knδ +1 to each of the 2 knδ +1 intervals in E knδ +1 we similarly have that if Φ( δ ) ≤| U | ≤ δ then µ δ ( U ) ≤ | U | . Therefore by the mass distribution principle Lemma 5.1 (ii) andProposition 3.1 and (5.8), we have thatdim Φ ( E ∪ F ) ≥ >
10 log 29 log 5 ≥ max { dim B E, dim B F } ≥ max { dim Φ E, dim Φ F } , as required.The same calculation as the lower bound for dim Φ ( E ∪ F ) in the above proof at the sequence ofscales δ n := f kn +2 , together with the the mass distribution principle Lemma 5.1 (i), shows that ≤ dim Φ F . Also from the above proof, dim Φ F ≤ dim B F ≤
10 log 29 log 5 , so we havedim H F ≤ dim Φ F ≤
10 log 29 log 5 < ≤ dim Φ F ≤ dim B F, and similarly for E . It is straightforward to see that there are many admissible functions Φ suchthat log δ log Φ( δ ) → δ → + , for example Φ( δ ) = e − δ − . . If F is the set constructed in the proofof Proposition 5.4 corresponding to such a Φ, then by Proposition 3.11 (ii), for any θ ∈ (0 ,
1] wehave dim H F < ≤ dim Φ F ≤ dim θ F, so dim θ F is discontinuous at θ = 0. Then let Φ be an admissible function such that for allsufficiently large n we have Φ ( f kn +1 ) ≤ Φ ( f kn +2+1 ). Then for all sufficiently small δ , thereexists an odd integer n ( δ ) such that Φ( δ ) ≤ f kn ( δ )+1 ≤ δ , and by considering the natural cover of F kn ( δ )+1 with 2 kn ( δ )+1 intervals, each of length f kn ( δ )+1 , we see that dim Φ F ≤
10 log 29 log 5 < .In fact, we will see in Theorem 6.1 (i) that for this F , as well as more generally for any othernon-empty compact subset F of a uniformly perfect metric space with more than one point, forall s ∈ [dim H F, dim B F ] there exists an admissible function Φ s such that dim Φ s F = s , even whendim θ F is discontinuous at θ = 0. 33ote also that it follows from Propositions 5.4 and 3.1 and the fact that the Hausdorff di-mension is countably stable that for any admissible functions Φ and Φ , the three notions ofdimension dim H , dim Φ and dim Φ are pairwise-distinct (in particular for any θ , θ ∈ (0 , H ,dim θ , dim θ are pairwise-distinct), even when we only consider non-empty, compact subsets of R with the usual metric. This is quite different from the φ -Assouad dimensions, where there arecertain φ for which the φ -Assouad dimension coincide with the upper box dimension for all to-tally bounded sets, and other φ for which the φ -Assouad dimensions coincide with the Assouaddimension for all such sets, see Proposition 3.12 (i) and (v). It is a well-studied problem to bound the dimensions of product sets in terms of the dimensionsof the marginals. Very often, dimensions come in pairs (dim, Dim), where dim F ≤ dim F for all‘reasonable’ sets F , and which satisfy the boundsdim E + dim F ≤ dim( E × F ) ≤ dim E + dim F ≤ dim( E × F ) ≤ dim E + dim F (5.10)for all ‘reasonable’ sets E and F and ‘reasonable’ metrics on the product space. Examples of suchpairs are (Hausdorff, packing) [17], (lower box, upper box) [24], (lower, Assouad) and (modifiedlower, Assouad) [7, Corollary 10.1.2] and, for any fixed θ ∈ (0 , θ , Assouadspectrum at θ ) and (modified lower spectrum at θ , Assouad spectrum at θ ) [12, Proposition 4.4].In Theorem 5.5 (ii), we show that for any admissible function Φ, (lower Φ-intermediate, upperbox) (and hence (lower θ -intermediate, upper box) for any θ ∈ (0 , Φ ( E × F ) is dim Φ E + dim B F (analogously to [6, Proposition 2.5] for theintermediate dimensions), rather than the expected dim Φ E + dim Φ F . Theorem 5.5 generalises[6, Proposition 2.5] on the intermediate dimensions of product sets to more general functions Φand more general metric spaces, and also gives an improved lower bound for dim Φ ( E × F ) and animproved upper bound for dim Φ ( E × F ). It is common that bounds can be improved further inthe case of self-products, as in [12, Proposition 4.5] for the Assouad-type spectra, and indeed weimprove the lower bound for dim Φ ( E × F ) further for self-products in (iii). Theorem 5.5.
Let ( X, d X ) and ( Y, d Y ) be uniformly perfect metric spaces with more than onepoint. Let d X × Y be a metric on X × Y such that there exist constants c , c ∈ (0 , ∞ ) such that c max( d X , d Y ) ≤ d X × Y ≤ c max( d X , d Y ) , (5.11) the same condition as [24, (2.4)], noting that d X × Y := max { d X , d Y } and d X × Y := ( d pX + d pY ) /p for p ∈ [1 , ∞ ) are examples of familiar metrics which satisfy this condition. Then ( X × Y, d X × Y ) is auniformly perfect metric space with more than one point. Moreover, if E ⊆ X and F ⊆ Y are non-empty, totally bounded subsets with finite Assouad dimension, then E × F is a non-empty, totallybounded subset of ( X × Y, d X × Y ) , and in fact dim A ( E × F ) < ∞ . Directly from Definition 2.7 wehave dim Φ ( E × F ) ≤ dim Φ ( E × F ) , and we also have ( i ) max { dim Φ E +dim Φ F, dim Φ E +dim Φ F } ≤ dim Φ ( E × F ) ≤ min { dim Φ E +dim B F, dim B E +dim Φ F } (ii) dim Φ E + dim Φ F ≤ dim Φ ( E × F ) ≤ min { dim Φ E + dim B F, dim B E + dim Φ F } In the case of self-products, one of the bounds can be improved further; we trivially have dim Φ ( F × F ) ≤ dim Φ ( F × F ) , and also(iii) Φ F ≤ dim Φ ( F × F ) ≤ dim Φ F + dim B F (iv) Φ F ≤ dim Φ ( F × F ) ≤ dim Φ F + dim B F Proof.
The idea of the proof of the upper bounds is to consider a cover of one of the sets E withdiameters in [Φ( δ ) , δ ], and, for each set U i in that cover, to form a cover of that other set F with all34he diameters approximately equal to | U i | , with the number of sets in this cover controlled by theupper box dimension of F . We can then cover the product set with approximate squares with sizesapproximately between Φ( δ ) and δ to get the result. The idea of the proof of the lower bounds isto use the Frostman type Lemma 5.2 to put measures on each of the marginal sets such that themeasure of sets with diameter in [Φ( δ ) , δ ] is controlled by the Φ-intermediate dimensions of thesets, and then apply the mass distribution principle Lemma 5.1 with the product measure on theproduct set.The sets X and Y each have more than one point, so X × Y has more than one point. We firstshow that X × Y is uniformly perfect. Since X is uniformly perfect, there exists c ∈ (0 ,
1) suchthat X is c -uniformly perfect. If x ∈ E , y ∈ F and 0 < δ < | X | then there exists p ∈ E such that cδ/c ≤ d X ( x, p ) ≤ δ/c . Then c cδ/c ≤ c d X ( x, p ) ≤ d X × Y (( x, y ) , ( p, y )) ≤ c d X ( x, p ) ≤ δ. Thus X × Y is uniformly perfect.We now show that dim A ( E × F ) < ∞ . Since dim A E < ∞ and dim A F < ∞ , there exists C ∈ N depending only on E and F such that for all x ∈ E , y ∈ F and δ > N δ/c ( B E ( x, δ/c ) ∩ X ) ≤ C and N δ/c ( B F ( y, δ/c ) ∩ Y ) ≤ C . Let x ∈ E , y ∈ F and δ >
0, so there exists x , . . . , x C ∈ E and y , . . . , y C ∈ F such that B E ( x, δ/c ) ⊆ C (cid:91) i =1 B E ( x i , δ/c ); B F ( y, δ/c ) ⊆ C (cid:91) i =1 B F ( y i , δ/c ) . Then if x (cid:48) ∈ E and y (cid:48) ∈ F are such that 2 δ ≥ d X × Y (( x, y ) , ( x (cid:48) , y (cid:48) )) then2 δ ≥ c max { d X ( x, x (cid:48) ) , d Y ( y, y (cid:48) ) } so there exist i, j such that d X ( x i , x (cid:48) ) < δ/c and d Y ( y j , y (cid:48) ) < δ/c . Then d X × Y (( x i , x (cid:48) ) , ( y j , y (cid:48) )) ≤ c max { d X ( x i , x (cid:48) ) , d Y ( y j , y (cid:48) ) } ≤ δ, so ( x (cid:48) , y (cid:48) ) ∈ B E × F (( x i , y j ) , δ ). Therefore B E × F (( x, y ) , δ ) ⊆ C (cid:91) i =1 C (cid:91) j =1 B E × F (( x i , y j ) , δ ) , so N δ ( B E × F (( x, y ) , δ )) ≤ C < ∞ . Therefore E × F is doubling so has finite Assouad dimension.The same symbols may take different values in parts (i), (ii), (iii).(i) We first prove the upper bound of (i), following the proof of the upper bound in [6, Propo-sition 2.5]. Let (cid:15) >
0. We have shown that X × Y is uniformly perfect, so there exists c p ∈ (0 , X × Y is c p -uniformly perfect, and without loss of generality we may assume that0 < c p < c < < c < ∞ . Since dim A ( E × F ) < ∞ there exists A ∈ N such that for all P ∈ E × F and r > N r ( B E × F ( P, c r )) ≤ A . Since Φ is admissible, there exists Y ∈ (0 ,
1) such that Φ is Y -admissible. Also Φ( δ ) /δ → δ → + , so there exists ∆ ∈ (0 , Y )such that for all δ ∈ (0 , ∆) we have Φ( δ ) /δ < c p /
2. Let s > dim Φ E and d > dim B F . By thedefinition of dim B E there exists δ ∈ (0 , ∆) such that for all r ∈ (0 , δ ) there is a cover of F by r − d or fewer sets, each having diameter at most r . By the definition of dim Φ E there exists δ ∈ (0 , δ )such that for all δ ∈ (0 , δ ) there exists a cover { U i } of E such that Φ( δ ) ≤ | U i | ≤ δ for all i and (cid:88) i | U i | s ≤ A − ( c + c − p ) − ( s + d ) (cid:15). (5.12)For such a cover, for each i let { U i,j } j be a cover of F by | U i | − d or fewer sets, each having diameter | U i,j | ≤ | U i | . Then for all i, j we have | U i × U i,j | ≤ c max {| U i | , | U i,j |} = c | U i | ≤ c δ, (5.13)35o by the definition of A , U i × U i,j can be covered by A or fewer sets { U i,j,k } k , each having diameterat most min { δ/ , | U i × U i,j |} . We may assume that each of these sets is non-empty, so there existssome p i,j,k ∈ U i,j,k . By the uniformly perfect condition, there exists q i,j,k ∈ X × Y such thatΦ( δ ) ≤ d X × Y ( p i,j,k , q i,j,k ) ≤ Φ( δ ) /c p . Let V i,j,k := U i,j,k ∪ { q i,j,k } , so by the triangle inequalityΦ( δ ) ≤ d X × Y ( p i,j,k , q i,j,k ) ≤ | V i,j,k | ≤ | U i,j,k | + d X × Y ( p i,j,k , q i,j,k ) ≤ δ/ δ ) /c p ≤ δ, (5.14)since δ < δ < ∆. Also, by (5.13) we have | V i,j,k | ≤ | U i,j,k | + d X × Y ( p i,j,k , q i,j,k ) ≤ c | U i | + Φ( δ ) /c p ≤ ( c + c − p ) | U i | . (5.15)Therefore by (5.15) and (5.12) we have (cid:88) i,j,k | V i,j,k | s + d ≤ (cid:88) i A | U i | − d (( c + c − p ) | U i | ) s + d ≤ A ( c + c − p ) s + d (cid:88) i | U i | s ≤ (cid:15). Also E × F ⊆ (cid:91) i,j,k U i,j,k ⊆ (cid:91) i,j,k V i,j,k , and noting (5.14), we have that dim Φ ( E × F ) ≤ s + d . Letting s → (dim Φ E ) + and t → (dim B F ) + we have dim Φ ( E × F ) ≤ dim Φ E + dim B F . The bound dim Φ ( E × F ) ≤ dim B E + dim Φ F followssimilarly.The proof of the lower bound is somewhat similar to the proof of the lower bound in [6,Proposition 2.5]. If dim Φ F = 0 then since F is non-empty, there exists f ∈ F . By (5.11), thenatural embedding E (cid:44) −→ X × Y , x (cid:55)→ ( x, f ), is bi-Lipschitz onto its image, so by Corollary 4.1.12. and Proposition 2.1 (i),dim Φ E + dim Φ F = dim Φ E = dim Φ ( E × { f } ) ≤ dim Φ ( E × F ) . Similarly, if dim Φ E = 0 thendim Φ E + dim Φ F = dim Φ F ≤ dim Φ F ≤ dim Φ ( E × F ) . Now assume that dim Φ E > Φ F >
0. Let t ∈ (0 , dim Φ E ) and t ∈ (0 , dim Φ F ). ByLemma 5.2 (i) there exists c E ∈ (0 , ∞ ) such that for all δ > δ ∈ (0 , δ ) anda Borel probability measure µ δ with supp( µ δ ) ⊆ E such that if x ∈ X and Φ( δ ) ≤ r ≤ δ then µ δ ( B X ( x, r )) ≤ c E r t . By Lemma 5.2 (ii) there exist c F , δ ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ) there exists a Borel probability measure ν δ with supp( ν δ ) ⊆ F such that if y ∈ Y andΦ( δ ) ≤ r ≤ δ then ν δ ( B Y ( y, r )) ≤ c F r t . If δ >
0, then there exists δ ∈ (0 , min { δ , δ } ) andBorel probability measures µ δ and ν δ as above, and let µ δ × ν δ be the product measure, whichsatisfies supp( µ δ × ν δ ) ⊆ E × F . If U ⊆ X × Y is Borel and satisfies Φ( δ ) ≤ | U | ≤ δ then U isnon-empty, so there exists ( x, y ) ∈ U , and U ⊆ B X × Y (( x, y ) , | U | ) ⊆ B X ( x, | U | /c ) × B Y ( y, | U | /c ) . (5.16)By the definition of C , there exist x , . . . , x C ∈ E and y , . . . , y C ∈ F such that E ∩ B X ( x, | U | /c ) ⊆ C (cid:91) i =1 B X ( x i , | U | ); F ∩ B Y ( y, | U | /c ) ⊆ C (cid:91) i =1 B Y ( y i , | U | ) . E × F ) ∩ ( B X ( x, | U | /c ) × B Y ( y, | U | /c )) = ( E ∩ B X ( x, | U | /c )) × ( F ∩ B Y ( y, | U | /c )) ⊆ (cid:32) C (cid:91) i =1 B X ( x i , | U | ) (cid:33) × C (cid:91) j =1 B Y ( y j , | U | ) = C (cid:91) i =1 C (cid:91) j =1 ( B X ( x i , | U | ) × B Y ( y j , | U | )) , so by (5.16) and the definition of the product measure we have( µ δ × ν δ )( U ) ≤ ( µ δ × ν δ )( B X ( x, | U | /c ) × B Y ( y, | U | /c ))= ( µ δ × ν δ )( B X ( x, | U | /c ) × B Y ( y, | U | /c ) ∩ ( E × F )) ≤ ( µ δ × ν δ ) C (cid:91) i =1 C (cid:91) j =1 ( B X ( x i , | U | ) × B Y ( y j , | U | )) ≤ C (cid:88) i =1 C (cid:88) j =1 ( µ δ × ν δ )( B X ( x i , | U | ) × B Y ( y j , | U | ))= C (cid:88) i =1 C (cid:88) j =1 µ δ ( B X ( x i , | U | )) ν δ ( B Y ( y j , | U | )) ≤ C (cid:88) i =1 C (cid:88) j =1 c E | U | t c F | U | t = C c E c F | U | t + t . Therefore by the mass distribution principle Lemma 5.1 (i), dim Φ ( E × F ) ≥ t + t , and letting t → (dim Φ E ) − and t → (dim Φ F )) − , we have dim Φ ( E × F ) ≥ dim Φ E + dim Φ F , as required. Thebound dim Φ ( E × F ) ≥ dim Φ E + dim Φ F follows similarly.(ii) The proof of (ii) is similar to the proof of (i), so we sketch the proof. For the upper bound,let (cid:15) > s > dim Φ E and d > dim B F then there exists δ ∈ (0 ,
1) such that for all r ∈ (0 , δ ) there is a cover of F by ( r (cid:48) ) − d or fewer sets, each having diameter at most r (cid:48) , and forall δ > δ ∈ (0 , δ ) and a cover { U i } of E such that Φ( δ ) ≤ | U i | ≤ δ for all i and (cid:88) i | U i | s ≤ A − ( c + c − p ) − ( s + d ) (cid:15). Then as in the proof of (i) there exists a cover { V i,j,k } of E × F such that Φ( δ ) ≤ | V i,j,k | ≤ δ and (cid:80) i,j,k | V i,j,k | s + d ≤ (cid:15) , so dim Φ ( E × F ) ≤ s + d , and letting s → (dim Φ E ) + and t → (dim B F ) + we have dim Φ ( E × F ) ≤ dim Φ E + dim B F . The bound dim Φ ( E × F ) ≤ dim B E + dim Φ F followssimilarly.The proof of the lower bound is similar to the proof of the lower bound of (i), and is also similarto the proof of the lower bound in [6, Proposition 2.5]. If dim Φ F = 0 then by letting f ∈ F andconsidering the natural bi-Lipschitz map E → E × { f } we havedim Φ E + dim Φ F = dim Φ E = dim Φ ( E × { f } ) ≤ dim Φ ( E × F ) , and similarly if dim Φ E = 0 then dim Φ E + dim Φ F ≤ dim Φ ( E × F ). Suppose dim Φ E > Φ F >
0, and let t ∈ (0 , dim Φ E ) and t ∈ (0 , dim Φ F ). By Lemma 5.2 (ii) there exist c , δ ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ) there exist Borel probability measures µ δ and ν δ satisfying supp( µ δ ) ⊆ E and supp( ν δ ) ⊆ F such that if x ∈ X , y ∈ Y and Φ( δ ) ≤ r ≤ δ then µ δ ( B X ( x, r )) ≤ c r t and ν δ ( B Y ( y, r )) ≤ c r t . Now, supp( µ δ × ν δ ) ⊆ E × F , and ifΦ( δ ) ≤ | U | ≤ δ then as in the proof of the lower bound of (i), ( µ δ × ν δ )( U ) ≤ C c | U | t + t .37herefore by the mass distribution principle Lemma 5.1 (ii), dim Φ ( E × F ) ≥ t + t , and letting t → (dim Φ E ) − and t → (dim Φ F ) − , we have dim Φ ( E × F ) ≥ dim Φ E + dim Φ F , as required.(iii) The upper bound is just the upper bound of (i) with E = F ; the improved bound isthe lower bound. The lower bound is trivial if dim Φ F = 0, so assume that dim Φ F > t ∈ (0 , dim Φ F ). By Lemma 5.2 (i) there exists c F ∈ (0 , ∞ ) such that for all δ > δ ∈ (0 , δ ) and a Borel probability measure µ δ with supp( µ δ ) ⊆ F such that if x ∈ X andΦ( δ ) ≤ r ≤ δ then µ δ ( B X ( x, r )) ≤ c F r t . Then supp( µ δ × µ δ ) ⊆ F × F and as in the proof ofthe lower bound of (i), if Φ( δ ) ≤ | U | ≤ δ then ( µ δ × µ δ )( U ) ≤ C c F | U | t . Therefore by the massdistribution principle Proposition 4.1 (i), dim Φ ( F × F ) ≥ t , and letting t → (dim Φ F ) − , we havedim Φ ( F × F ) ≥ Φ F , as required.(iv) Follows immediately by setting E = F in (ii).In the particular case Φ : (0 , / → (0 , / δ ) := δ − log δ , for any non-empty, totally boundedsubset G of a uniformly perfect metric space we have dim Φ G = dim B G and dim Φ G = dim B G byProposition 3.2. Therefore from (i) and (ii) we recover the inequalities for the upper and lowerbox dimensions of product sets in [24, Theorem 2.4] (which is proven directly, without puttingmeasures on the sets). Note that bounds on the dimensions of products of more than two sets canbe obtained by applying Theorem 5.5 iteratively, for exampledim Φ ( E × F × G ) ≥ dim Φ ( E × F ) + dim Φ G ≥ dim Φ E + dim Φ F + dim Φ G (under the natural assumptions which allow Theorem 5.5 to be applied). Recall that [6, Proposition 2.4] and Proposition 3.10 show that there are many compact setswith intermediate dimensions discontinuous at θ = 0, and for these sets the intermediate di-mensions do not fully interpolate between the Hausdorff and box dimensions. Proposition 3.11suggests that when attempting to use the Φ-intermediate dimensions to ‘recover the interpolation’between the Hausdorff and box dimensions for such sets, any relevant admissible Φ will satisfylim inf δ → + log δ log Φ( δ ) = 0. Theorem 6.1 shows that for any compact set there is indeed a family ofΦ for which the Φ-intermediate dimensions interpolate all the way between the Hausdorff and boxdimensions of the set, thus giving finer geometric information about the set than the Hausdorff,box or intermediate dimensions when the intermediate dimensions of the set are discontinuous at θ = 0. Moreover, there exists a family of Φ which interpolates for both the upper and lower versionsof the dimensions, and forms a chain in the partial order introduced in Section 3.1. The remarksafter Proposition 5.4 give an example of a set with upper intermediate dimensions discontinuousat θ = 0 and give an indication of how one might construct admissible functions Φ which recoverthe interpolation for that particular set. Theorem 6.1.
Let X be a uniformly perfect metric space with more than one point, and let F bea non-empty, compact subset of X . Then for all s ∈ [dim H F, dim B F ] , there exists an admissiblefunction Φ s such that these functions satisfy(i) dim Φ s F = s ;(ii) dim Φ s F = min { s, dim B F } ;(iii) if dim H F ≤ s ≤ t ≤ dim B F then Φ s (cid:22) Φ t . As mentioned previously, the set Q ∩ [0 , ⊂ R shows that the assumption that F is compactcannot be removed in general. The key definition in the proof is (6.1), and compactness allowsus to take a finite subcover in Definition 2.2 of Hausdorff dimension, which ensures that Φ s ( δ ) iswell-defined and positive. 38 roof. Note first that since F is compact it is totally bounded, so all the dimensions are well-defined. Since X is uniformly perfect, there exists c ∈ (0 , /
2) such that X is c -uniformly perfect.Since 1 / ( − log δ ) → δ → + , there exists ∆ ∈ (0 , /
5) such that for all δ ∈ (0 , ∆) we have0 < δ − log δ < cδ/
3. If dim H F = dim B F then Φ dim H F ( δ ) = δ − log δ will satisfy all the conditionsby Proposition 3.2, so henceforth assume that dim H F < dim B F . The same symbols may takedifferent values in the proofs of parts (i), (ii), (iii).(i) Define Φ dim B F ( δ ) := δ − log δ , so (i) and (ii) are satisfied for s = dim B F by Proposition 3.2.For now, let s ∈ (dim H F, dim B F ). Since s > dim H F , for all δ ∈ (0 , ∆) there exists a finite orcountable cover { V i } i ≥ of F such that | V i | ≤ δ for all i and (cid:80) i | V i | s ≤ − − s . We may assumewithout loss of generality that for each i , V i is non-empty, so there exists p i ∈ V i . Now, for all i , V i ⊆ B ( p i , max { | V i | , − − i/s ) } ) so { B ( p i , max { | V i | , − − i/s } ) } i ≥ is an open cover for F , andeach set in this open cover has positive diameter since X is uniformly perfect. Since F is compact,there is a finite subset { U i } ⊆ { B ( p i , max { | V i | , − − i/s } ) } which also covers F . Now, (cid:88) i | U i | s ≤ (cid:88) i ≥ | B ( p i , max { | V i | , − − i/s } ) | s ≤ ∞ (cid:88) i =1 (2 − i/s ) s + (cid:88) i ≥ (4 | V i | ) s = 1 / s (cid:88) i | V i | s < . Since { U i } is a finite collection of sets with positive diameter, min i | U i | >
0. Therefore Φ s : (0 , ∆) → R is well-defined and positive if we defineΦ s ( δ ) := sup { x ∈ [0 , δ/ ( − log δ )] : there exists a finite cover { U i } of F such that x ≤ | U i | ≤ δ for all i and (cid:88) i | U i | s ≤ } . (6.1)By construction, Φ s ( δ ) /δ ≤ (cid:16) δ − log δ (cid:17) /δ → δ → + , and Φ s is increasing in δ , so Φ s isadmissible.We now show that dim Φ s F ≤ s . Given η, (cid:15) >
0, let δ := min { (cid:15) /η c s/η − s/η , ∆ } . By definition (6.1), for all δ ∈ (0 , δ ) there exists a finite cover { W i } of F satisfying Φ s ( δ ) / ≤| W i | ≤ δ and (cid:80) i | W i | s ≤
1, noting that each W i is non-empty. If | W i | ≥ Φ s ( δ ) then leave W i inthe cover unchanged. If | W i | < Φ s ( δ ) then pick any w i ∈ W i , and since X is c -uniformly perfect,there exists q i ∈ X such that Φ s ( δ ) ≤ d ( q i , w i ) ≤ Φ s ( δ ) /c . Replace W i in the cover by W i ∪ { q i } and call the new cover Y i . By the triangle inequality,Φ s ( δ ) ≤ d ( q i , w i ) ≤ | W i ∪ { q i }| ≤ | W i | + d ( q i , w i ) < Φ s ( δ ) + Φ s ( δ ) /c ≤ s ( δ ) /c ≤ δ/ ( − c log δ ) ≤ δ since δ < ∆. Also | W i ∪ { q i }| ≤ s ( δ ) /c ≤ (4 /c )Φ s ( δ ) / ≤ | W i | /c. Therefore (cid:88) i | Y i | s + η ≤ (cid:88) i | Y i | s δ η ≤ δ η (4 /c ) s (cid:88) i | W i | s ≤ (cid:15). As (cid:15) was arbitrary, we have dim Φ s F ≤ s + η . Letting η → + gives that dim Φ s F ≤ s .39o prove the reverse inequality, assume for a contradiction that dim Φ s F < s . Then thereexists δ ∈ (0 , ∆) such that for all δ ∈ (0 , δ ) there exists a cover { Z i } of F such that forall i we have Φ s ( δ ) ≤ | Z i | ≤ δ , and (cid:80) i | Z i | s ≤ − s c s , noting that each Z i is non-empty. Sincedim Φ s F < s < dim B F , by Proposition 3.2 there exists δ ∈ (0 , δ ) such that Φ s ( δ ) < δ / ( − log δ ),and let { Z i } be the cover corresponding to this δ , as above. Choose any z i ∈ Z i and let x i ∈ X be such that 2 | Z i | ≤ d ( z i , x i ) ≤ | Z i | /c . Then by the triangle inequality,2Φ s ( δ ) ≤ | Z i | ≤ d ( z i , x i ) ≤ | Z i ∪ { x i }| ≤ | Z i | + 2 | Z i | /c ≤ | Z i | /c ≤ (3 /c ) δ / ( − log δ ) < δ since δ < ∆. Moreover, { Z i ∪ { x i }} i covers F , and since | Z i ∪ { x i }| ≤ | Z i | /c we have (cid:88) i | Z i ∪ { x i }| s ≤ (cid:88) i (3 | Z i | /c ) s ≤ s c − s (cid:88) i | Z i | s ≤ . Therefore by the definition of Φ s ( δ ) we haveΦ s ( δ ) ≥ min { s ( δ ) , δ / ( − log δ ) } > Φ s ( δ ) , a contradiction. Hence for all s ∈ (dim H F, dim B F ) we have dim Φ s F ≥ s , and so dim Φ s F = s .Now consider the case s = dim H F . Let N ∈ N satisfy N > max (cid:26) B F − dim H F , (cid:27) . Define Φ s ( δ ) := min { Φ s +1 /N ( δ ) , . . . , Φ s +1 /n ( δ ) } for δ ∈ ( n +1 , n ] ( n ≥ N ). For all δ ∈ (0 , /N ] wehave Φ s ( δ ) ≤ Φ s +1 /N ( δ ) ≤ δ/ ( − log δ ) so Φ s ( δ ) /δ → δ → + . For all n ≥ N and δ ∈ (0 , ∆)we have Φ s +1 /n ( δ ) >
0, so if δ > s ( δ ) >
0. Moreover, if δ ≤ δ , say δ ∈ ( n +1 , n ] and δ ∈ ( m +1 , m ], n ≥ m ≥ N , thenΦ s ( δ ) = min { Φ s +1 /N ( δ ) , . . . , Φ s +1 /n ( δ ) }≤ min { Φ s +1 /N ( δ ) , . . . , Φ s +1 /m ( δ ) }≤ min { Φ s +1 /N ( δ ) , . . . , Φ s +1 /m ( δ ) } = Φ s ( δ )by the monotonicity of each Φ s +1 /i , so Φ s is monotonic. Therefore Φ s is admissible. For all n ≥ N ,for all δ ∈ (0 , /n ) we have Φ s ( δ ) ≤ Φ s +1 /n ( δ ), so by Proposition 3.1 and Corollary 3.4.2 (i) (1)we have s = dim H F ≤ dim Φ s F ≤ dim Φ s F ≤ dim Φ s +1 /n F = s + 1 n . Letting n → ∞ gives dim Φ s F = dim Φ s F = s , as required.(iii) For the family defined in (6.1), if dim H F < s ≤ t ≤ dim B F then for all δ ∈ (0 , ∆) wehave Φ s ( δ ) ≤ Φ t ( δ ) by construction. Moreover, there exists n ∈ N such that n > N (where N is as in the construction of Φ dim H F ) and dim H F + 1 /n < t , and for all δ ∈ (0 , /n ) we haveΦ dim H F ( δ ) ≤ Φ /n ( δ ) ≤ Φ t ( δ ). Therefore by Corollary 3.4.2 (i) (1), if dim H F ≤ s ≤ t ≤ dim B F then Φ s (cid:22) Φ t .(ii) By Proposition 3.1 and (i), for all s ∈ [dim H F, dim B F ] we havedim Φ s F ≤ min { dim Φ s F, dim B F } = min { s, dim B F } . It remains to prove the reverse inequality. By Propositions 3.1 and 3.2, if s = dim H F thenmin { s, dim B F } = s ≤ dim Φ s F , and if s = dim B F then dim Φ s F = dim B F = min { s, dim B F } .Now suppose s ∈ (dim H F, dim B F ] ∩ (dim H F, dim B F ). Assume for contradiction that dim Φ s F (1 + 2 /c ) − s , and − log δ ≥ /c ). Now, t > dim Φ s F so for all δ > δ ∈ (0 , min { ∆ , δ } ) and a cover { U i } such that Φ s ( δ ) ≤ | U i | ≤ δ for all i , and(1 + 2 /c ) − s ≥ (cid:88) i | U i | t ≥ (cid:88) i | U i | s . (6.2)But δ < ∆ so (1 + 2 /c ) − s < δ t − t (cid:48) ( − log δ ) t = δ − t (cid:48) (cid:18) δ − log δ (cid:19) t , so there exists i such that δ/ ( − log δ ) > | U i | ≥ Φ s ( δ ). If i is such that | U i | ≥ min { s ( δ ) , δ/ ( − log δ ) } then leave U i in the cover unchanged. If, however, i is such that | U i | < min { s ( δ ) , δ/ ( − log δ ) } then note that U i (cid:54) = ∅ , so choose any p ∈ U i . Then there exists q ∈ X such that 2Φ s ( δ ) ≤ d ( p, q ) ≤ s ( δ ) /c , and replace U i in the cover by U i ∪ { q } , and call the new cover { V i } i . In thecase | U i | < min { s ( δ ) , δ/ ( − log δ ) } , by the triangle inequality we have2Φ s ( δ ) ≤ d ( p, q ) ≤ | U i ∪ { q }| ≤ | U i | + d ( p, q ) ≤ | U i | + 2Φ s ( δ ) /c ≤ (1 + 2 /c ) | U i | < /c )Φ s ( δ ) ≤ /c ) δ/ ( − log δ ) ≤ δ. Then for each i we have min { δ/ ( − log δ ) , s ( δ ) } ≤ | V i | ≤ δ , and (cid:88) i | V i | s ≤ (cid:88) i ((1 + 2 /c ) | U i | ) s = (1 + 2 /c ) s (cid:88) i | U i | s ≤ , by (6.2). This means that Φ s ( δ ) ≥ min { s ( δ ) , δ/ ( − log δ ) } > Φ s ( δ ), a contradiction. Hencedim Φ s F ≥ s for all s ∈ (dim H F, dim B F ].Now suppose s ∈ (dim B F, dim B F ). By (iii) we have Φ dim B F (cid:22) Φ s , so by what we have justproved, we have min { s, dim B F } = dim B F ≤ dim Φ dimB F F ≤ dim Φ s F .Together, the cases show that for all s ∈ [dim H F, dim B F ] we have dim Φ s F ≥ min { s, dim B F } and hence dim Φ s F = min { s, dim B F } , as required.In the definition (6.1) of Φ s , any positive constant would work in place of the constant 1, sothere are many different Φ s that will work. The family of dimensions dim Φ s and dim Φ s may notvary continuously for all sets, as shown by the following proposition. Proposition 6.2.
There exist non-empty, compact subsets
F, F of R (with the usual metric) suchthat (i) and (ii) are satisfied:(i) if (Φ s ) s ∈ (dim H F, dim B F ) is any family of admissible functions such that dim Φ s F = s for all s ∈ (dim H F, dim B F ) then the function s (cid:55)→ dim Φ s F is not continuous on (dim H F, dim B F ) .(ii) if (Φ s ) s ∈ (dim H F, dim B F ) is any family of admissible functions such that dim Φ s F = s for all s ∈ (dim H F, dim B F ) then the function s (cid:55)→ dim Φ s F is not continuous on (dim H F, dim B F ) .Proof. Let F := { } ∪ { /n : n ∈ N } , so dim θ F = θ θ for all θ ∈ [0 ,
1] by [6, Proposition 3.1].Let F = E ∪ F for any compact countable set E ⊂ R with dim B E = dim A E = 1 /
4, so as in [6,Example 3] dim H F = 0 and dim θ F = max (cid:110) θ θ , / (cid:111) for all θ ∈ (0 , s ) s ∈ (dim H F, dim B F ) is any family of admissible functions such that dim Φ s F = s for all s ∈ (dim H F, dim B F ). Then for all s > / Φ s F = s > / / F so byProposition 3.11 (i), lim sup δ → + log Φ s ( δ )log δ > / Φ s F ≥ dim / F = 1 /
4. By Proposition 3.11 (iii) for all s < / log Φ s ( δ )log δ → δ → + , so since dim θ F = θ θ → θ →
0, by Proposition 3.11 (ii)dim Φ s F = 0. Therefore the function s (cid:55)→ dim Φ s F is discontinuous at s = 1 /
4, as required.(ii) follows from Proposition 3.11 in a similar way to (i).It is natural to ask the parallel questions about the extent to which one can recover the in-terpolation between the upper box and Assouad dimensions for sets for which the quasi-Assouaddimension is less than the Assouad dimension (a condition which can be considered parallel tothe condition that the intermediate dimensions are discontinuous at θ = 0). Garc´ıa, Hare andMendivil [15, Corollary 3.9] show that if 0 < α < β < F ⊂ [0 ,
1] suchthat dim qA F = α , dim A F = β , and such that for all s ∈ [ α, β ] there exists a function φ for whichdim φ A F = s . On pages 4 and 10 they suggest that “it may be difficult to find a one-parameterfamily of continuous dimension functions that interpolates precisely between the quasi-Assouadand Assouad dimensions,” and say that they do not know whether such a family exists. Acknowledgements
The author would like to thank Kenneth Falconer, Istv´an Kolossv´ary, Justin Tan, and especiallyJonathan Fraser, for helpful discussions and insightful comments. The author was financiallysupported by a Leverhulme Trust Research Project Grant (RPG-2019-034).
References [1] S. A. Burrell. Dimensions of fractional Brownian images. Preprint, 2020. arXiv: .[2] S. A. Burrell, K. J. Falconer, and J. M. Fraser. Projection theorems for intermediate dimen-sions.
J. Fractal Geom. (to appear). arXiv: . Preprint, 2019.[3] S. A. Burrell, K. J. Falconer, and J. M. Fraser. The fractal structure of elliptical polynomialspirals. Preprint, 2020. arXiv: .[4] K. J. Falconer.
Fractal Geometry: Mathematical Foundations and Applications . 3rd ed. Wiley,2014.[5] K. J. Falconer. Intermediate dimensions - a survey. Preprint, 2020. arXiv: .[6] K. J. Falconer, J. M. Fraser, and T. Kempton. Intermediate dimensions.
Math. Z. (2020),813–830.[7] J. M. Fraser.
Assouad Dimension and Fractal Geometry . Tracts in Mathematics Series. Cam-bridge University Press, 2020.[8] J. M. Fraser. Fractal geometry of Bedford-McMullen carpets. In:
Fall 2019 Jean-Morlet Chairprogramme . Springer Lecture Notes Series (to appear), Preprint, 2020. arXiv: .[9] J. M. Fraser. Interpolating between dimensions. In:
Fractal Geometry and Stochastics VI .Birkh¨auser, Progr. Probab. (to appear), Preprint, 2019. arXiv: .[10] J. M. Fraser. On H¨older solutions to the spiral winding problem. Preprint, 2019. arXiv: .[11] J. M. Fraser, K. E. Hare, K. G. Hare, S. Troscheit, and H. Yu. The Assouad spectrum andthe quasi-Assouad dimension: a tale of two spectra.
Ann. Acad. Sci. Fenn. Math. (2019),379–387. 4212] J. M. Fraser and H. Yu. New dimension spectra: Finer information on scaling and homo-geneity. Adv. Math. (2018), 273–328.[13] I. Garc´ıa and K. E. Hare. Properties of Quasi-Assouad dimension. Preprint, 2017. arXiv: .[14] I. Garc´ıa, K. E. Hare, and F. Mendivil. Almost sure Assouad-like Dimensions of Comple-mentary sets.
Math. Z. (to appear). arXiv: . Preprint, 2019.[15] I. Garc´ıa, K. E. Hare, and F. Mendivil. Intermediate Assouad-like Dimensions.
J. FractalGeom. (to appear). arXiv: . Preprint 2020.[16] K. E. Hare and K. G. Hare. Intermediate Assouad-like dimensions for measures. Preprint,2020. arXiv: .[17] J. D. Howroyd. On Hausdorff and packing dimension of product spaces.
Math. Proc. Cam-bridge Philos. Soc. (1996), 715–727.[18] T. Hyt¨onen and A. Kairema. Systems of dyadic cubes in a doubling metric space.
Colloq.Math. (2010).[19] I. Kolossv´ary. On the intermediate dimensions of Bedford-McMullen carpets. Preprint, 2020.arXiv: .[20] I. Kukavica and Y. Pei. An estimate on the parabolic fractal dimension of the singular setfor solutions of the Navier–Stokes system.
Nonlinearity (2012), 2775–2783.[21] E. Le Donne, S. Li, and T. Rajala. Ahlfors-regular distances on the Heisenberg group withoutbiLipschitz pieces. Proc. Lond. Math. Soc. (2015).[22] P. Mattila.
Geometry of Sets and Measure in Euclidean Spaces . Cambridge University Press,1995.[23] P. Pansu. M´etriques de Carnot-Carath´eodory et quasiisom´etries des espaces sym´etriques derang un.
Ann. Math. (1989), 1–60.[24] J. C. Robinson and N. Sharples. Strict Inequality in the Box-Counting Dimension ProductFormulas.
Real Anal. Exchange (2010), 95–119.[25] S. Semmes. On the nonexistence of bi-Lipschitz parameterizations and geometric problemsabout A ∞ -weights. Rev. Mat. Iberoamericana (1996), 337–410.[26] J. T. Tan. On the intermediate dimensions of concentric spheres and related sets. Preprint,2020. arXiv: .[27] S. Troscheit. Assouad spectrum thresholds for some random constructions. Canad. Math.Bull. (2019), 434–453. Amlan BanajiSchool of Mathematics and StatisticsUniversity of St AndrewsSt Andrews, KY16 9SS, UK
Email: [email protected]@st-andrews.ac.uk