Projection theorems for intermediate dimensions

Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, relying on this, we show that the intermediate dimensions of the projection of a set $E \subset \mathbb{R}^n$ onto almost all $m$-dimensional subspaces depend only on $m$ and $E$, that is, they are almost surely independent of the choice of subspace. Our approach is based on `intermediate dimension profiles' that are expressed in terms of capacities. We discuss several applications at the end of the paper, including a surprising result that relates the box dimensions of the projections of a set to the Hausdorff dimension of the set.


Introduction
Theorems on dimensions of projections of fractals in Euclidean space have a long history. In 1954 Marstrand [12] proved that the Hausdorff dimension of the orthogonal projections of a Borel set E ⊂ R 2 onto linear subspaces was almost-surely constant. More specifically, dim H π V E = min{dim H E, 1}, for almost all one-dimensional subspaces V , where π V denotes orthogonal projection onto V . Kaufman gave a potential-theoretic proof of Marstrand's results [11], and in 1975 Mattila extended them to Borel sets E ⊂ R n and almost all V in the Grassmannian G(n, m) [13]. These seminal results set in motion a sustained interest in the behaviour of dimension under projections, see [1,14] for basic expositions and [5,15,16] for recent surveys.
It is natural to seek projection results for the various other dimensions that occur throughout fractal geometry. For example, in 1997 Falconer and Howroyd showed that the upper and lower box-counting dimensions of the projections of a set are almost surely constant and given by what they termed a 'dimension profile' [6,10], reflecting how a set in R n appears when viewed from an m-dimensional perspective. The dimension profiles were, however, implicitly defined and somewhat awkward to work with, leading to a recent re-working of the theory using a potential-theoretic approach [2,3] where box-counting dimensions are defined in terms of capacities, which are then used to study projections. θ = 0) and box-counting dimensions (θ = 1). These dimensions are defined by restricting the diameters of sets used in admissible coverings of E to a range [r, r θ ] for small r. A general discussion of this and other forms of dimension interpolation may be found in the recent survey [8].
In this paper, potential-theoretic methods are used to study intermediate dimensions, first to give a definition of these dimensions in terms of capacities with respect to certain kernels and then to prove a Marstrand-type theorem to give the almost sure intermediate dimensions of projections of sets in terms of capacities, see Theorem 5. 1. Some examples and applications are given in the final section.

Intermediate dimensions
Intermediate dimensions were introduced by Falconer, Fraser and Kempton in [4] It is easy to see from (2.1) and (2.2) that dim θ E and dim θ E are the infima of s for which these lower and upper limits equal 0; that there are unique such values follows from the following lemma.
Lemma 2.1. Let θ ∈ (0, 1] and E ⊂ R n . For each 0 < r < 1, Taking infima over all such covers yields r s−t S t r,θ (E) ≤ S s r,θ (E) ≤ r θ(s−t) S t r,θ (E), from which (2.6) follows. These inequalities carry over on taking lower limits of the quotients so in particular lim inf r→0 log S s r,θ (E) − log r is strictly monotonic decreasing and continuous for s ∈ [0, n]. Since S 0 r,θ (E) is bounded below by the box-counting number of E at scale r θ , it follows that Also S n r,θ (E) is bounded above by the n-dimensional volume of a ball containing E so Continuity now gives a unique s ∈ [0, n] such that lim inf In Section 4 we will show how dim θ E and dim θ E can be represented in terms of capacities of E ⊂ R n with respect to certain kernels. Then in Section 5 we will show that by changing a parameter in the kernels we obtain the intermediate dimensions of the orthogonal projections of E onto almost all m-dimensional subspaces.

Capacities and Dimension Profiles
In this section we introduce a notion of dimension derived from capacities that is closely related to the intermediate dimensions and which is amenable to studying projections.
Throughout this section, let θ ∈ (0, 1] and m ∈ {1, . . . , n}. For 0 ≤ s ≤ m and 0 < r < 1, define the potential kernels When s = m this becomes and so corresponds to the kernel φ m r (x) used in [2,3] in the context of box-counting dimensions. As one would expect, this kernel is also recovered when θ = 1 where φ s,m r,θ is independent of s. Note that φ s,m r,θ (x) is continuous in x and monotonically decreasing in |x|. Letting M(E) denote the set of Borel probability measures supported on E, we say that the energy of µ ∈ M(E) with respect to φ s,m r,θ is and the potential of µ at x ∈ R n is φ s,m r,θ (x − y) dµ(y).
We define the capacity C s,m r,θ (E) of E to be the reciprocal of the minimum energy achieved by probability measures on E, that is Since φ s,m r,θ (x) is continuous in x and strictly positive and E is compact, C s,m r,θ (E) is positive and finite. For bounded, but not necessarily closed, sets we take the capacity of a set to be that of its closure.
The existence of equilibrium measures for kernels and the relationship between the minimal energy and the corresponding potentials is standard in classical potential theory. We state this in a convenient form; it is easily proved for continuous kernels, see, for example, [3, Lemma 2.1].
for all x ∈ E, with equality for µ-almost all x ∈ E.
As we will see, these capacities are closely related to the sums considered in Section 2. The following lemma, which parallels Lemma 2.1, enables us to define 'intermediate dimension profiles'.
Proof. By comparison of the kernels it is easily checked that, for Using the definition of capacity and that an equilibrium measure on E for the kernel φ s,m r,θ is a candidate for an equilibrium measure for φ t,m r,θ and vice-versa, we obtain . Taking logarithms and rearranging gives (3.3).

The inequalities (3.3) remain true on taking lower limits of the quotients so lim inf
Since the kernels are bounded above by 1, Proof. This follows immediately noting that the kernels φ t,m r,θ (x) are clearly decreasing in m.
The remainder of the paper concerns the relationship between the intermediate dimensions of a set E, defined in terms of the sums over restricted covers of E, and intermediate dimension profiles, defined in terms of the capacities. In particular, we will see that for with respect to the natural invariant measure on the Grassmannian.

Capacities and Intermediate Dimensions
The main result in this section characterises intermediate dimensions of sets E ⊂ R n in terms of dimension profiles which we have defined in terms of capacities C s,n r,θ (E) with respect to the kernels φ s,n r,θ .
This will follow immediately from the following proposition together with the definitions (2.4), (2.5), (3.4) and (3.5). We may assume throughout that E is compact since the intermediate dimensions are stable under taking closures, see [4].
Proposition 4.2. Let E ⊂ R n be compact, θ ∈ (0, 1], and 0 ≤ s ≤ n. Then there is a number r 0 > 0 such that for all 0 < r ≤ r 0 , (4.1) r s C s,n r,θ (E) ≤ S s r,θ (E) ≤ a n ⌈log 2 (|E|/r) + 1⌉r s C s,n r,θ (E) where the number a n depends only on n. Consequently Proof of Proposition 4.2 We prove the left hand inequality of (4.1) in Lemma 4.3 and the right hand inequality in Lemma 4.4. Then Proof. By Lemma 3.1 there exists an equilibrium measure µ ∈ M(E) and a set E 0 with µ(E 0 ) = 1 such that , δ)).
Let {U i } i be a finite cover of E by sets of diameters r ≤ |U i | ≤ r θ and define I = {i : which yields the desired result upon taking the infimum over all such covers.
Note that by comparing kernels, C s,m r,θ (E) ≤ C s,n r,θ (E) for m ≤ n so (4.2) implies the weaker conclusion that r s C s,m r,θ (E) ≤ S s r,θ (E). In the following proof, we use potential estimates to find a Besicovitch cover of E by balls of relatively large measure. The Besicovitch covering lemma gives a bounded number of families of disjoint such balls with their union covering E. The balls with diameters between r and r θ , together with covers of any larger balls by balls of diameters at most r θ , provide efficient covers for estimating the sums S s r,θ (E). Additionally, in the next section, Lemma 4.4 will be important when considering intermediate dimensions of projections. for all x ∈ E, then there is a number r 0 > 0 such that for all 0 < r ≤ r 0 , S s r,θ (E) ≤ a n ⌈log 2 (|E|/r) + 1⌉ r s γ where the constant a n depends only on n. In particular, S s r,θ (E) ≤ a n ⌈log 2 (|E|/r) + 1⌉C s,n r,θ (E)r s .
Proof. To avoid ambiguity we will assume that θ ∈ (0, 1), though the proof is virtually the same when θ = 1, essentially by taking M = 0; this 'box-counting dimension' case is also covered in [3].
We choose r 0 sufficiently small to ensure that 2 ≤ M ≤ D − 2 for all 0 < r ≤ r 0 . For x ∈ E, using (4.4) and estimating the kernel φ s,n r,θ (x − y) given by (3.1) over consecutive annuli Hence, for each x ∈ E, there exists some integer 0 ≤ k(x) ≤ D such that one of the above summands is at least the arithmetic mean of the sum. There are three cases. We will use that there are numbers d n depending only on n such that every ball of radius ρ in R n may be covered by at most λ −n d n balls of diameter λρ for all 0 < λ ≤ 1 (d n = 3 n n n/2 will certainly do).
The cover of E by the balls B = {B(x, 2 k(x) r) : x ∈ E} is a Besicovitch cover, that is each point of E is at the centre of some ball in the collection. The Besicovitch covering theorem, see for example [13, Theorem 2.7], allows us to extract subcollections C 1 , . . . , C cn of disjoint balls from B where c n depends only on n and such that E ⊂ i B∈Ci B. Let From (4.5) each B ∈ C i \ (E i ∪ F i ) has diameter at most r θ . Also, for each B = B(x, 2 k(x) r) ∈ E i let D B denote a collection of at most (2 M r/r θ ) n d n ≤ 2 n d n balls of diameter r θ that cover B, and for each B = B(x, 2 k(x) r) ∈ F i let D B denote a collection of at most 2 k(x) r/r θ n d n balls of diameter r θ that cover B.

Intermediate dimensions of Projections
Our main theorem in this section is that the intermediate dimension profiles dim m θ E and dim Theorem 5.1. Let E ⊂ R n be bounded. Then, for all V ∈ G(n, m) Moreover, for γ n,m -almost all V ∈ G(n, m), To prove Theorem 5.1 we begin with some technical lemmas relating the kernel φ s,m r,θ to the integral over V of certain kernels defined on V ∈ G(n, m). We derive this from a standard estimate on integrals of the characteristic functions of slabs, which has been used in several results on projections, see for example [14,Lemma 3.11] and [3]. The next lemma states this standard fact; we indicate the proof for the lower bound which does not seem readily accessible. For this we use the kernels for r > 0 and m > 0 which were used in [3] in connection with box-counting dimensions of projections. Proof. The right-hand inequality is given in [14,Lemma 3.11]. The left-hand inequality is obvious when |x| ≤ r, otherwise we may adapt the proof of [14,Lemma 3.11] by using the estimate σ n−1 y ∈ S n−1 : where σ n−1 denotes the normalised surface measure on S n−1 , α(n) is the volume of the unit ball in R n and L n is n-dimensional Lebesgue measure.
It is convenient to introduce further kernels φ s r,θ on m-dimensional subspaces, where 0 < r < 1, θ ∈ (0, 1] and 0 < s ≤ m where V ∈ G(n, m) is some m-dimensional subspace. The motivation for this is that whilst φ s r,θ is of the same form as φ s,m r,θ (x) in the key region |x| ≤ r θ integrating φ s r,θ (π V x) over V ∈ G(n, m) gives a kernel comparable to φ s,m r,θ (x). For brevity, we write ≃ to mean that the ratio of the two sides is bounded away from 0 and infinity by constants that are uniform in x, r and θ. Lemma 5.3. For all m ∈ {1, . . . , n − 1} and 0 ≤ s < m there exist constants a, b > 0, depending only on n, m and s, such that for all x ∈ R n , θ ∈ (0, 1] and 0 < r < 1 2 , a n,m φ s r,θ (π V x)dγ n,m (V ) ≤ φ s,m r,θ (x) ≤ b n,m φ s r,θ (π V x)dγ n,m (V ).
Note that Lemma 5.3 is not quite valid when s = m since a logarithmic term appears in the final integral. However we can avoid this case in our application.
We require one further lemma which is a variant of Lemma 4.3 for the modified kernels φ s r,θ . Lemma 5.4. Let E ⊂ R n be compact, θ ∈ (0, 1], 0 < r < 1 and 0 ≤ s ≤ n. If there exists µ ∈ M(E) and a Borel set F ⊂ E such that given by (2.3).
Proof. As in Lemma 4.3, for all x ∈ F and r ≤ δ ≤ r θ . Let {U i } i be a cover of F by sets with r ≤ |U i | ≤ r θ . We may assume that for each i there is some so taking infima over all such covers, Proof of Theorem 5.1 To prove Theorem 5.1 it suffices to prove the a priori weaker result where we first fix θ ∈ (0, 1] and then establish the result for almost all V . We can do this because the intermediate dimensions are continuous in θ ∈ (0, 1] and are therefore determined by their values on the rationals. Without loss of generality let E ⊂ R n be compact and m ∈ {1, . . . , n − 1}. When θ = 1 Theorem 5.1 reduces to the projection properties for box-counting dimensions, see [3]. Hence we will assume that θ ∈ (0, 1).
By Lemma 3.1, for each 0 ≤ s ≤ m there exists a measure µ ∈ M(E) such that for all where µ V ∈ M(π V E) denotes the image of µ under π V defined by g(w)dµ V (w) = g(π V x)dµ(x) for continuous g and by extension. Then, for each .
Thus π V E ⊂ V supports a measure µ V satisfying the condition of Lemma 4.4 (with n replaced by m and V identified with R m in the natural way). Hence S s r,θ (π V E) ≤ a n ⌈log 2 (|E|/r) + 1⌉r s C s,m r,θ (E) and so lim sup This is true for all ǫ > 0, so using (5.5), Corollary 6.1. Let E ⊂ R n be a bounded set such that dim θ E is continuous at θ = 0. If V ∈ G(n, m) is such that dim H π V E = min{m, dim H E}, then dim θ π V E is continuous at θ = 0. In particular, dim θ π V E is continuous at θ = 0 for γ n,m -almost all V ∈ G(n, m). A similar result holds for the upper intermediate dimensions.
Proof. If m ≤ dim H E, then the result is immediate and so we may assume that m > dim H E. Then, for θ ∈ (0, 1), using (5.1), Lemma 3.3, Theorem 4.1, and the assumption that dim θ E is continuous at θ = 0, we get , which proves continuity of dim θ π V E at θ = 0. The final part of the result, concerning almost sure continuity at 0, follows from the above result together with the Marstrand-Mattila projection theorems for Hausdorff dimensions. Corollary 6.2. Let E ⊂ R 2 be a Bedford-McMullen carpet associated with a regular a×b grid for integers b > a ≥ 2. Then dim θ π V E and dim θ π V E are continuous at θ = 0 for γ 2,1 -almost all V ∈ G(2, 1). In particular, if log a/ log b / ∈ Q, then dim θ π V E and dim θ π V E are continuous at θ = 0 for all V ∈ G(2, 1).   A similar result holds for upper dimensions replacing dim θ E and dim B E with dim θ E and dim B E, respectively.
Proof. One direction is trivial, and holds without the continuity assumption, since, if dim H E ≥ m, then for γ n,m -almost all V ∈ G(n, m). The other direction is where the interest lies. Indeed, suppose dim B π V E = m for γ n,m -almost all V ∈ G(n, m) but dim H E < m. Then Corollary 6.3 implies that dim θ π V E = m for γ n,m -almost all V ∈ G(n, m) and all θ ∈ (0, 1]. Applying the Marstrand-Mattila projection theorem for Hausdorff dimension, it follows that for γ n,m -almost all V ∈ G(n, m) dim θ π V E is not continuous at θ = 0, which contradicts Corollary 6.1.
To motivate Corollary 6.4 we give a couple of simple applications. If E ⊂ R 2 is a Bedford-McMullen carpet satisfying dim H E < 1 ≤ dim B E, then dim B π V E < 1 = min{dim B E, 1} for γ 2,1 -almost all V ∈ G(2, 1). This surprising application seems difficult to derive directly, noting that there is very little known about the box dimensions of projections of Bedford-McMullen carpets, aside from them being almost surely constant. Another, more accessible, example is provided by the sequence sets F p = {n −p : n ≥ 1} for fixed p > 0. It is well-known that dim B F p = 1/(1 + p) and therefore dim B (F p × F p ) = 2/(1 + p) which is at least 1 for p ≤ 1 and approaches 2 as p approaches 0. Continuity at θ = 0 for dim θ F p was established in [4, Proposition 3.1] and it is straightforward to extend this to dim θ (F p × F p ). Therefore, since dim H (F p × F p ) = 0 < 1, we get dim B π V (F p × F p ) < 1 for γ 2,1 -almost all V ∈ G(2, 1). This is most striking when p is very close to 0. A direct calculation, which we omit, in fact reveals that for all V ∈ G(2, 1) apart from the horizontal and vertical projections an entertaining formula which we would not have come across if Corollary 6.4 had not lead us to it, see also [9,Proposition 5.1].