Intermediate dimensions

We introduce a continuum of dimensions which are `intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $|U| \leq |V|^\theta$ for all sets $U, V$ used in a particular cover, where $\theta \in [0,1]$ is a parameter. Thus, when $\theta=1$ only covers using sets of the same size are allowable, and we recover the box dimensions, and when $\theta=0$ there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function of $\theta$), including proving that it is continuous on $(0,1]$ but not necessarily continuous at $0$, as well as establishing appropriate analogues of the mass distribution principle, Frostman's lemma, and the dimension formulae for products. We also compute, or estimate, the intermediate dimensions of some familiar sets, including sequences formed by negative powers of integers, and Bedford-McMullen carpets.


Intermediate dimensions: definitions and background
We work with subsets of R n throughout, although much of what we establish also holds in more general metric spaces. We denote the diameter of a set F by |F |, and when we refer to a cover {U i } of a set F we mean that F ⊆ i U i where {U i } is a finite or countable collection of sets.
Recall that Hausdorff dimension dim H may be defined without introducing Hausdorff measures, but using Hausdorff content. For F ⊆ R n , dim H F = inf s ≥ 0 : for all ε > 0 there exists a cover {U i } of F such that |U i | s ≤ ε , see [2,Chapter 2]. Expressed in this way, Hausdorff and box dimensions may be regarded as extreme cases of the same definition, one with no restriction on the size of covering sets, and the other requiring them all to have equal diameters. With this in mind, one might regard them as the extremes of a continuum of dimensions with increasing restrictions on the relative sizes of covering sets. This is the main idea of this paper, which we formalise by considering restricted coverings where the diameters of the smallest and largest covering sets lie in a geometric range δ 1/θ ≤ |U i | ≤ δ for some 0 ≤ θ ≤ 1. Similarly, we define the upper θ-intermediate dimension of F by dim θ F = inf s ≥ 0 : for all ε > 0 there exists δ 0 > 0 such that for all 0 < δ ≤ δ 0 , there is a cover {U i } of F such that δ 1/θ ≤ |U i | ≤ δ and |U i | s ≤ ε .
With these definitions, where dim B is the upper box dimension. Moreover, it follows immediately that, for a bounded set F and θ ∈ [0, 1], It is also immediate that dim θ F and dim θ F are increasing in θ, though as we shall see they need not be strictly increasing. Furthermore, dim θ is finitely stable, that is dim θ (F 1 ∪ F 2 ) = max{dim θ F 1 , dim θ F 2 }, and, for θ ∈ (0, 1], both dim θ F and dim θ F are unchanged on replacing F by its closure. In many situations, even if dim H F < dim B F , we still have dim B F = dim B F and dim θ F = dim θ F for all θ ∈ [0, 1]. In this case we refer to the box dimension dim B F = dim B F = dim B F and the θ-intermediate dimension dim θ F = dim θ F = dim θ F . This paper is devoted to understanding θ-intermediate dimensions. The hope is that dim θ F will interpolate between the Hausdorff and box dimensions in a meaningful way, a rich and robust theory will be discovered, and interesting further questions unearthed. We first derive useful properties of intermediate dimensions, including that dim 0 F and dim 0 F are continuous on (0, 1] but not necessarily at 0, as well as proving versions of the mass distribution principle, Frostman's lemma and product formulae. We then examine a range of examples illustrating different types of behaviour including sequences formed by negative powers of integers, and self-affine Bedford-McMullen carpets. A related approach to 'dimension interpolation' was recently considered in [5] where a new dimension function was introduced to interpolate between the box dimension and the Assouad dimension. In this case the dimension function was called the Assouad spectrum, denoted by dim θ A F (θ ∈ (0, 1)). Proposition 2.1. Let F be a non-empty bounded subset of R n and let 0 ≤ θ < φ ≤ 1.
In particular, θ → dim θ F and θ → dim θ F are continuous for θ ∈ (0, 1]. Proof. We will only prove (2.1) since (2.2) is similar. The left-hand inequality of (2.1) is just the monotonicity of dim θ F . The right-hand inequality is trivially satisfied when dim θ F = n, so we assume that 0 ≤ dim θ F < n. Suppose that 0 ≤ θ < φ ≤ 1 and that 0 ≤ dim θ F < s < n. Then, given ε > 0, we may find arbitrarily small δ > 0 and countable or finite covers {U i } i∈I of F such that i∈I |U i | s < ε and δ ≤ |U i | ≤ δ θ for all i ∈ I. Let For each i ∈ I 1 we may split U i into subsets of small coordinate cubes to get sets Taking sums with respect to this cover: (2.3). This holds for some cover for arbitrarily small ε and all s > dim θ F , giving dim φ F ≤ n + θ(dim θ F − n)/φ, which rearranges to give (2.1).
Finally, note that (2.2) follows by exactly the same argument noting that the assumption dim θ F < s gives rise to δ 0 > 0 such that for all δ ∈ (0, δ 0 ) we can find covers {U i } i∈I of F satisfying (2.3).

A mass distribution principle for dim θ and dim θ
The mass distribution principle is a powerful tool in fractal geometry and provides a useful mechanism for estimating the Hausdorff dimension from below by considering measures supported on the set, see [2, page 67]. We present natural analogues for dim θ and dim θ . Proposition 2.2. Let F be a Borel subset of R n and let 0 ≤ θ ≤ 1 and s ≥ 0. Suppose that there are numbers a, c, δ 0 > 0 such that for all 0 < δ ≤ δ 0 we can find a Borel measure µ δ supported by F with µ δ (F ) ≥ a, and with Then dim θ F ≥ s. Moreover, if measures µ δ with the above properties can be found only for a sequence of δ → 0, then the conclusion is weakened to dim θ F ≥ s.
so that i |U i | s ≥ a/c > 0 for every admissible cover and therefore dim θ F ≥ s. The weaker conclusion regarding the upper intermediate dimension is obtained similarly.
Note the main difference between Proposition 2.2 and the usual mass distribution principle is that a family of measures {µ δ } is used instead of a single measure. Since each measure µ δ is only required to describe a range of scales, in practice one can often use finite sums of point masses. Whilst the measures µ δ may vary, it is essential that they all assign mass at least a > 0 to F .

A Frostman type lemma for for dim θ
Frostman's lemma is another powerful tool in fractal geometry, which asserts the existence of measures of the type considered by the mass distribution principle, see [2, page 77]  Proposition 2.3. Let F be a compact subset of R n , let 0 < θ ≤ 1, and suppose 0 < s < dim θ F . There exists a constant c > 0 such that for all δ ∈ (0, 1) we can find a Borel probability measure µ δ supported on F such that for all x ∈ R n and δ 1/θ ≤ r ≤ δ, µ δ (B(x, r)) ≤ cr s .
Moreover, µ δ can be taken to be a finite collection of atoms.
Proof. This proof follows the proof of the classical version of Frostman's lemma given in [7, pages 112-114]. For m ≥ 0 let D m denote the familiar partition of [0, 1] n consisting of 2 nm pairwise disjoint half-open dyadic cubes of sidelength 2 −m , that is cubes of the form [a 1 , a 1 + 2 −m ) × · · · × [a n , a n + 2 −m ). By translating and rescaling we may assume without loss of generality that F ⊆ [0, 1] n and that F is not contained in any Q ∈ D 1 . It follows from the definition of dim θ F that there exists ε > 0 such that for all δ ∈ (0, 1) and for all covers (2.5) Given δ ∈ (0, 1), let m ≥ 0 be the unique integer satisfying 2 −m−1 < δ 1/θ ≤ 2 −m and let µ m be a measure defined on F as follows: for each Q ∈ D m such that Q ∩ F = ∅, then choose an arbitrary point x Q ∈ Q ∩ F and let where δ x Q is a point mass at x Q . Modify µ m to form a measure µ m−1 , supported on the same finite set, defined by where ν| E denotes the restriction of ν to E. The purpose of this modification is to reduce the mass of cubes which carry too much measure. This is done since we are ultimately trying to construct a measure which we can estimate uniformly from above. Continuing inductively, We terminate this process when we define µ m−l where l is the largest integer satisfying 2 −(m−l) n 1/2 ≤ δ. (We may assume that l ≥ 0 by choosing δ sufficiently small to begin with.) In particular, cubes Q ∈ D m−l satisfy |Q| = 2 −(m−l) n 1/2 ≤ δ. By construction we have for all k = 0, . . . , l and Q ∈ D m−k . Moreover, for all x ∈ F , there is at least one k ∈ {0, . . . , l} and Q ∈ D m−k with x ∈ Q such that the inequality in (2.6) is an equality. This is because all cubes at level m satisfy the equality for µ m and if a cube Q satisfies the equality for µ m−k , then either Q or its parent cube satisfies the equality for µ m−k−1 .
For each x ∈ F , choosing the largest such Q yields a finite collection of cubes Q 1 , . . . , Q t which cover F and satisfy δ 1/θ ≤ |Q i | ≤ δ for i = 1, . . . , t. Therefore, using (2.5), which is clearly a probability measure supported on a finite collection of points. Moreover, for all x ∈ R n and δ 1/θ ≤ r ≤ δ, B(x, r) is certainly contained in at most c n cubes in D m−k where k is chosen to be the largest integer satisfying 0 ≤ k ≤ l and 2 −(m−k+1) < r, and c n is a constant depending only on n. Therefore, using (2.6), which completes the proof, setting c = c n ε −1 n s/2 2 s .

General bounds
Here we consider general bounds which rely on the Assouad dimension and which have interesting consequences for continuity. Namely, they provide numerous examples where the intermediate dimensions are discontinuous at θ = 0 and also provide another proof that the intermediate dimensions are continuous at θ = 1. The Assouad dimension of F ⊆ R n is defined by where N r (A) denotes the smallest number of sets of diameter at most r required to cover a set A. Proposition 2.4. Given any non-empty bounded F ⊆ R n and θ ∈ (0, 1), Proof. We will prove the lower bound for dim θ F , the proof for dim θ F is similar. Fix θ ∈ (0, 1) and assume that dim B F > 0, otherwise the result is trivial. Let and δ ∈ (0, 1) be given. By the definition of lower box dimension, there exists a uniform constant C 0 , depending only on F and b, such that there is a δ-separated set of points in F of cardinality at least C 0 δ −b . Let µ δ be a uniformly distributed probability measure on these points, i.e. a sum of C 0 δ −b point masses each with mass C −1 0 δ b . We use our mass distribution principle with this measure to prove the proposition.
Let U ⊆ R n be a Borel set with |U | = δ γ for some γ ∈ [θ, 1]. By the definition of Assouad dimension there exists a uniform constant C 1 , depending only on F and d, such that U intersects at most C 1 (δ γ /δ) d points in the support of µ δ . Therefore This proposition implies that for bounded sets with dim H F < dim B F = dim A F , the intermediate dimensions dim θ F and dim θ F are necessarily discontinuous at θ = 0. In fact the intermediate dimensions are constant on (0, 1] in this case. On the other hand, this gives an alternative demonstration that dim θ F and dim θ F are always continuous at θ = 1. Moreover, the proposition provides a quantitative lower bound near θ = 1. In Section 3.2 we will use Proposition 2.4 to construct examples exhibiting a range of behaviours.

Product formulae
A well-studied problem in dimension theory is how dimensions of product sets behave. The following product formulae for intermediate dimensions may be of interest in their own right, but in Section 3.2 they will be used to construct examples.
Proof. Fix θ ∈ (0, 1) throughout, noting that the cases when θ = 0, 1 are well-known, see [2,Chapter 7]. We begin by demonstrating the left-hand inequality. We may assume that dim θ E, dim θ F > 0 as otherwise the conclusion follows by monotonicity. Moreover, since E, F are bounded we may assume they are compact since all the dimensions considered are unchanged under taking closure. Let 0 < s < dim θ E and 0 < t < dim θ F . It follows from Proposition 2.3 that there exist constants C s , C t > 0 such that for all δ ∈ (0, 1) there exist Borel probability measures µ δ supported on E and ν δ supported on F such that for all x ∈ R n and δ 1/θ ≤ r ≤ δ, µ δ (B(x, r)) ≤ C s r s and ν δ (B(x, r)) ≤ C t r t .
Consider the product measure µ δ × ν δ which is supported on E × F . For z ∈ R n × R n and δ 1/θ ≤ r ≤ δ, (µ δ × ν δ )(B(z, r)) ≤ C s C t r s+t and Proposition 2.2 yields dim θ (E × F ) ≥ s + t; letting s → dim θ E and t → dim θ F gives the desired inequality. The middle inequality is trivial and so it remains to prove the right-hand inequality. Let s > dim θ E and d > dim A F , and let ε > 0. By the definition of dim θ E there exists a constant δ 0 ∈ (0, 1) such that for all 0 < δ < δ 0 we may find a cover of E by sets {U i } i∈U such that δ 1/θ ≤ |U i | ≤ δ for all i and i∈U |U i | s ≤ ε.
Since d > dim A F ≥ dim B F , there is a constant C > 0 depending only on F and d such that we may cover of F by fewer than Cδ −d balls {V j } j∈V where |V j | = δ for all j. We now build a cover of E × F . Using the definition of the Assouad dimension, there exists another constant C > 0 depending only on F and d such that we may cover each set U i × V j by fewer than C δ |U i | d sets of diameter |U i |. This can be achieved by first covering V j by sets {W k } k of diameter |U i |. Then {U i × W k } k is a cover by sets of diameter √ 2|U i |, each of which can be covered by a constant number of sets of diameter |U i |. Let {W (i, j) k } k∈K denote the obtained cover of U i × V j and note that i,j,k {W (i, j) k } is a cover of E × F satisfying

Convergent sequences
Let p > 0 and Since F p is countable, dim H F p = 0. It is well-known that dim B F p = 1/(p + 1), see [2,Chapter 2]. We obtain the intermediate dimensions of F p .
For the lower bound we put suitable measures on F p and apply Proposition 2.2. Fix s = θ/(p + θ). Let 0 < δ < 1 and again let M = δ −(s+θ(1−s))/(p+1) . Define µ δ as the sum of point masses on the points 1/k p (1 ≤ k < ∞) with by the choice of s. To see that (2.4) is satisfied, note that if 2 ≤ k ≤ M then, by a mean value theorem estimate, thus the gap between any two points of F p carrying mass is at least p/M p+1 . Let U be such that δ ≤ |U | ≤ δ θ . Then U intersects at most 1 + |U |/(p/M p+1 ) = 1 + |U |M p+1 /p of the points of F p which have mass δ s . Hence From Proposition 2.2, dim θ F p ≥ s = θ/(p + θ).

Simple examples exhibiting different phenomena
We use the convergent sequences F p from the previous section, the product formulae from Proposition 2.5, and that the upper intermediate dimensions are finitely stable to construct examples displaying a range of different features which are illustrated in Figure  1.
The natural question which began this investigation is 'does dim θ vary continuously between the Hausdorff and lower box dimension?'. This indeed happens for the convergent sequences considered in the previous section, but turns out to be false in general. The first example in this direction is provided by another convergent sequence.   This sequence converges slower than any of the polynomial sequences F p and it is wellknown and easy to prove that dim B F log = dim A F log = 1. It follows from Proposition 2.4 that dim θ F log = dim θ F log = 1, θ ∈ (0, 1].
Since dim 0 F log = dim H F log = 0 there is a discontinuity at θ = 0.
Example 2: Continuous at 0, part constant, part strictly increasing. In the opposite direction, it is possible that dim θ F = dim H F < dim B F for some θ > 0. Indeed, let F = F 1 ∪ E where F 1 = {0, 1/1, 1/2, 1/3, . . . } as before, and let E ⊂ R be any compact set with dim H E = dim B E = 1/3 (for example an appropriately chosen self-similar set). Then it is straightforward to deduce that It is also possible for dim θ F to approach a value strictly in between dim H F and dim B F as θ → 0. This is the subject of the next two examples.
Example 3: Discontinuous at 0, part constant, part strictly increasing. For an example that is constant on an interval adjacent to 0, let F = E ∪ F 1 where this time E ⊂ R is any closed countable set with dim B E = dim A E = 1/4. It is immediate that with dim H F = 0 and dim B F = 1/2.
Example 4: Discontinuous at 0, strictly increasing. Finally, for an example where dim θ F is smooth, strictly increasing but not continuous at θ = 0, let Here dim H F = 0 and dim B F = 3/2 and Proposition 2.5 gives

Bedford-McMullen carpets
A well-known class of fractals where the Hausdorff and box dimensions differ are the selfaffine carpets; this is a consequence of the alignment of the component rectangles in the iterated construction. The first studies of planar self-affine carpets were by Bedford [1] and McMullen [8] independently, see also [9] and these Bedford-McMullen carpets have been widely studied and generalised, see for example [3] and references therein.
There exists a unique non-empty compact set F ⊆ [0, 1] 2 satisfying that is F is the attractor of the iterated function system S (p,q) (p,q)∈D . We call such a set F a Bedford-McMullen self-affine carpet. It is sometimes convenient to denote pairs in D by = (p , q ). We model our carpet F via the symbolic space D N , which consists of all infinite words over D and is equipped with the product topology. We write i ≡ (i 1 , i 2 , . . .) for elements of D N and (i 1 , . . . , i k ) for words of length k in D k , where i j ∈ D. Then the canonical projection τ : D N → [0, 1] 2 is defined by This allows us to switch between symbolic and geometric notation since Bedford [1] and McMullen [8] showed that where m 0 is the number of p such that there is a q with (p, q) ∈ D, that is the number of columns of the array containing at least one rectangle. They also showed that where equality of the sums follows from the definitions of n p and a We assume that the non-zero n p are not all equal, otherwise dim H F = dim B F ; in particular this implies that a := max ∈D a ≥ 2.
We denote the kth-level iterated rectangles by We also write R k (i) ≡ R k (i 1 , i 2 , . . .) for this rectangle when we wish to indicate the kthlevel iterated rectangle containing the point τ (i) = τ (i 1 , i 2 , . . .). We will associate a probability vector {b } ∈D with D and let µ = k∈N i k ∈D b i k δ i k be the natural Borel product probability measure on D N , where δ is the Dirac measure on D concentrated at . Then the measure noting that ∈D b = 1, to get a measure µ on F ; thus the measures of the iterated rectangles are µ R k (i 1 , . . . , i k ) = b i 1 · · · b i k = m −kd (a i 1 · · · a i k ) (log n m−1) . (4.6) Approximate squares are well-known tools in the study of self-affine carpets. Given k ∈ N let l(k) = k log n m so that k log n m ≤ l(k) ≤ k log n m + 1 (4.7) For such k and i = (i 1 , i 2 , . . .) ∈ D N we define the approximate square containing τ (i) as the union of m −k × n −k rectangles: . . . , i k ) : p i j = p i j for j = 1, . . . , k and q i j = q i j for j = 1, . . . , l(k) , recalling that = (p , q ). This approximate square has sides m −k ×n −l(k) where n −1 m −k ≤ n −l(k) ≤ m −k . Note that, by virtue of self-affinity and since the sequence (p i 1 , · · · p i k ) is the same for all level-k rectangles R k (i 1 , . . . , i k ) in the same approximate square, the a i l(k)+1 · · · a i k level-k rectangles that comprise the approximate square Q k (i) all have equal µ-measure. Thus, writing L = log n m, We now obtain an upper bound for dim θ F which implies continuity at θ = 0 and so on [0, 1]. Recall that a = max ∈D a ≥ 2. Proof. For i = (i 1 , i 2 , . . .), rewriting (4.9) gives µ Q k (i) = m −kd (a i 1 · · · a i k ) 1/k (a i 1 · · · a i l(k) ) 1/l(k) Lk a i 1 · · · a i l(k) (kL/l(k))−1 . (4.11) We consider the two bracketed terms on the right-hand side of (4.11) in turn. We show that the first term cannot be too small for too many consecutive k and that the second term is bounded below by 1.
For k > 1 with l(k) = k log n m as usual, define We claim that for all K ≥ L/(1 − L) and all i = (i 1 , i 2 , . . .) ∈ D N , there exists k with Suppose this is false for some (i 1 , i 2 , . . .) and K ≥ 1/L(L − 1), so for all K ≤ k ≤ K/θ, f k (i) < λ := a log L/ log(L/2θ) , that is (a i 1 a i 2 · · · a i k ) 1/k < λ(a i 1 a i 2 · · · a i l ) 1/l(k) . (4.14) Define a sequence of integers k r (r = 0, 1, 2, . . .) inductively by k 0 = K and for r ≥ 1 taking k r to be the least integer such that k r log n m = k r−1 . Then k r ≤ k r−1 /L + 1, and a simple induction shows that Fix N to be the greatest integer such that k N ≤ K/θ. Then From (4.14) Combining with (4.15) we obtain λ > a −1/N ≥ a log L/ log(L/2θ) which contradicts the definition of λ. Thus the claim (4.13) is established. For the second bracket on the right-hand side of (4.11) note that 0 ≤ (kL/l(k)) − 1 = k log n m − k log n m l(k) Geometrically this means that for K ≥ L/(1 − L) every point z ∈ F belongs to at least one approximate square, Q k(z) say, with K ≤ k(z) ≤ K/θ and with µ(Q k(z) ) ≥ m −k(d+ε(θ)) . Since the approximate squares form a nested hierarchy we may choose a subset {Q k(zn) } N n=1 ⊂ {Q k(z) : z ∈ F } that is disjoint (except possibly at boundaries of approximate squares) and which cover F . Thus where |Q k | denotes the diameter of the approximate square Q k , noting that |Q k | ≤ 2 1/2 m −k . It follows that dim θ F ≤ d + ε(θ) as claimed.
The following lemma brings together some basic estimates that we will need to obtain a lower bound for the intermediate dimensions of F . Lemma 4.2. Let ε > 0. There exists K 0 ∈ N and a set E ⊂ F with µ(E) ≥ 1 2 such that for all i with τ (i) ∈ E and k ≥ K 0 , and µ(R k (i)) ≥ exp(−k(H(µ) + ε)), (4.18) where d = dim H F and H(µ) ∈ (0, log |D|) is the entropy of the measure µ.

Proof. McMullen [8, Lemmas 3,4(a)] shows that for
Thus by Egorov's theorem we may find a set E 1 ⊂ D N with µ( E 1 ) ≥ 3 4 , and K 1 ∈ N such that (4.17) holds for all i ∈ E 1 and k ≥ K 1 .
We now obtain a lower bound for dim θ F , showing in particular that dim θ F > dim H F for all θ > 0.
Proof. Fix θ ∈ (0, 1), let E ⊂ F and K 0 be given by Lemma 4.2, and let K ≥ K 0 . We define a measure ν K which assigns equal mass to all level-K rectangles, and then subdivides this mass among sub-rectangles using the same weights as for the measure µ, given by (4.5). This gives a measure to which we can apply the mass distribution principle, Lemma 2.2. Thus for k ≥ K, writing b = a (log n m−1) /m d as in (4.5), The formula for ν K differs from that of µ only in the mass it assigns according to the first K letters, and so the ν K -measure of an approximate square of side length m −k can be expressed in relation to the µ-measure of such a square. Thus for τ (i) ∈ E and k ≥ K, the approximate square Q k (i) that contains the point i has ν K -measure using Lemma 4.2. (Alternatively (4.21) may be verified directly using (4.6), (4.9) and (4.20).) Then for all i with τ (i) ∈ E and integers k ∈ [K, K/θ], where µ(E) ≥ 1 2 . To use our mass distribution principle we need equation (4.23) to hold on a set of i of large ν K mass, whereas currently we have that it holds on a set E of large µ mass. Let E = τ {i : inequality (4.23) is satisfied for all k ∈ [K, K/θ]}.
Note that, in (4.19), with equality if and only if µ gives equal mass to all cylinders of the same length, which happens if and only if each column in our construction contains the same number of rectangles. This happens exactly when the box and Hausdorff dimension coincide. Thus our lower bounds give that dim θ F > dim H F whenever θ > 0, provided that the Hausdorff and box dimensions of F are different.
Since the measures that we have constructed to give lower bounds on dim θ F are rather crude, it is unlikely that our lower bound for dim θ F converges to dim B F as θ → 1. However, a lower bound near which does approach dim B F as θ → 1 is given by Proposition 2.4, noting that dim A F > dim B F = dim B F provided dim B F > dim H F , see [6,4].
Many questions on the intermediate dimensions of Bedford-McMullen carpets remain, most notably finding the exact forms of dim θ F and dim θ F . In that direction we would at least conjecture that these intermediate dimensions are equal and strictly monotonic.