Dimensions of fractional Brownian images

This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential theoretic methods are used to produce dimension bounds for images of sets under H\"older maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$\alpha$ fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of H\"older images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.


Introduction
The growing literature on dimension spectra is beginning to provide a unifying framework for the many notions of dimension that arise throughout the field of fractal geometry. Suppose you are given two notions of dimension, dim X and dim Y , with dim X E ≤ dim Y E for all E ∈ R n . Dimension spectra aim to provide a continuum of dimensions, perhaps denoted dim θ and parametrised by θ ∈ [0, 1], such that dim 0 = dim X and dim 1 = dim Y . This is of interest for a number of reasons. For example, dim X and dim Y under stochastic processes, such as index-α fractional Brownian motion. Recall that a map f : E → R m is α-Hölder on E ⊂ R n if there exists c > 0 and 0 < α ≤ 1 such that for all x, y ∈ E. This scheme of work continues a tradition of Xiao [16,18], who used dimension profiles almost immediately after their introduction in 1997 [7] to consider the packing dimensions of sets under fractional Brownian motions. Unexpectedly, obtaining bounds on the dimension of fractional Brownian images allowed us to quickly establish continuity of the profiles for arbitrary Borel sets. Moreover, this led to an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. Both of these applications followed from a method which suggests a more general philosophy; dimensional information in a general setting can be obtained by transporting information back from a well-chosen fractional Brownian image.
Finally, we return to the setting of projections where our main results may be applied to bound the Hausdorff dimension of the exceptional sets, see Theorem 3.9. That is, the dimension of the family of sets whose projection has unusually small dimension. There is a long history of interest in this topic, see [2,14,13]. Throughout, we adopt a capacity theoretic approach to intermediate dimension profiles, as in [1], while adapting this strategy to meld it with ideas from [4].

Setting and Preliminaries
In this section we will define the necessary tools and concepts used throughout. This section is intentionally brief, and the interested reader is directed to [1] for a more elaborate discussion of the material and [3] for a gentle introduction to dimension theory. We begin with the precise formulation of the intermediate dimensions. Throughout, all sets are assumed to be non-empty, bounded and Borel.
For E ⊂ R n and 0 < θ ≤ 1, the lower intermediate dimension of E may be defined as dim θ E = inf s ≥ 0 : for all ǫ > 0 and all r 0 > 0, there exists 0 < r ≤ r 0 and a cover {U i } of E such that where |U | denotes the diameter of a set U ⊂ R n . If θ = 0, then we recover the Hausdorff dimension in both cases, since the covering sets may have arbitrarily small diameter. Moreover, if θ = 1, then we recover the lower and upper box-counting dimensions, respectively, since sets within admissible covers are forced to have equal diameter. While the above makes the interpolation intuitive, for technical reasons it is practical to use an equivalent formulation. First, for bounded and non-empty E ⊂ R n , θ ∈ (0, 1] and s ∈ [0, n], define It is proven in [1, Section 2] that The first step of a capacity theoretic approach is to define an appropriate kernel for the setting. For each collection of parameters θ ∈ (0, 1], t > 0, 0 ≤ s ≤ t and 0 < r < 1, define φ s,t r,θ : R n → R by In addition, for Lemma 3.2 and Theorem 3.3, we will require a set of modified kernels φ s r,θ : R m → R (m ∈ N) given by where 0 < r < 1, θ ∈ (0, 1] and 0 < s ≤ m. Using the first of these kernels, we define the capacity of a compact set E ⊂ R n to be where M(E) denotes the set of probability measures supported on E. For a set that may be bounded, but not closed, the capacity is simply defined to be that of its closure.
A measure that obtains the infimum in the definition of capacity is known as an equilibrium measure.
The existence of such measures 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, [5, Lemma 2.1].
In [1] In [1], only integer t ≤ n was required, as this corresponded to the topological dimension of the subspace being projected onto. However, as we shall see, it is necessary and possible to consider dimension profiles for non-integer and arbitrarily large t in the more general setting of Theorems 3.1, 3.3 and 3.4. In fact, to ensure that the above profiles exist, we require the following lemma, which allows [1, Lemma 3.2] to be easily extended for this greater range of t.
In particular, there exists a unique s ∈ [0, t] such that Proof. It suffices to show that (2.5) C t,t r,θ (E) ≤ cr −t for some fixed c > 0 depending only on E and t. For 0 < r < 1, let µ be the equilibrium measure associated with φ t,t r,θ . Since E is bounded, there exists a constant B > 1 such that for all x, y ∈ E. Directly from the definition, for all x, y ∈ E. Hence, from which (2.5) follows. The final part of the lemma may then be deduced since log C s,t r,θ (E) − log r − s is continuous and strictly monotonically decreasing in s by a trivial extension to [1,Lemma 3.2]. We may similarly argue for the upper limits.
To conclude this section, we briefly recall the definition of index-α fractional Brownian motion (0 < α < 1), which we denote B α : R n → R m for m, n ∈ N. In particular, are normally distributed with mean 0 and variance |x − y| 2α for all x, y ∈ R n . Moreover, B α,i and B α,j are independent for all i, j ∈ {1, . . . , m}. It immediately follows that for Borel A ⊂ R, The reader may enjoy the classical text of Kahane [12] for a more detailed account of index-α fractional Brownian motion and related stochastic processes.

Statement and Discussion of Results
In this section, we collect and discuss the main results of the paper, the proofs of which may be found in later sections. Our first result establishes an upper bound on the intermediate dimensions of Hölder images using dimension profiles. Recalling that the m-intermediate dimension profiles intuitively tell us about the typical size of a set from an m-dimensional viewpoint for m ∈ {1, . . . , n}, it is interesting to note how the Hölder exponent dictates which profile appears in the bound. This is in contrast to the setting of projections [1], where the profile appearing in the upper-bound is simply the topological dimension of the codomain.
For certain families of mappings, such as fractional Brownian motion, we are able to obtain almost-sure lower bounds for the dimension of the images in terms of profiles too. Let (Ω, F , P ) denote a probability space with each ω ∈ Ω corresponding to a σ({F ×B : where B denotes the Borel subsets of R n . In order for this problem to be tractable, some condition must be placed on the set of functions. Specifically, we need to assume a relationship between and the kernels (2.1). This is analogous to Matilla's result [15,Lemma 3.11], which covers the special case where f ω denote orthogonal projections and Ω = G(n, m), the Grassmannian of m dimensional subspaces of R n . However, such a result does not hold in general and so must be included as a hypothesis. This allows us to prove the following lemma that is a critical component of the following proofs. Essentially, it says that the integral of the modified kernels (2.2) over the probability space is bounded above by the kernels (2.1).
for all x, y ∈ E and r > 0, then there exists C s,m > 0 such that This allows us to obtain the desired almost-sure lower bound.
for all x, y ∈ E and r > 0, then Fractional Brownian motion is known to be (α − ε)-Hölder and is shown to satisfy condition (3.4) in Section 6. Thus, a combination of Theorem 3.1 and Theorem 3.3 yields our main result.
Theorem 3.4. Let θ ∈ (0, 1], m, n ∈ N, B α : R n → R m be index-α fractional Brownian motion (0 < α < 1) and E ⊂ R n be compact. Then Recent literature has sought to identify situations in which the intermediate dimensions are continuous at θ = 0, for example, see [1,6]. Theorem 3.1 implies that this continuity is preserved under index-α fractional Brownian motion. Corollary 3.5. Let E ⊂ R n be bounded and B α : R n → R m denote index-α fractional Brownian motion with mα ≤ n. If dim θ E is continuous at θ = 0, then dim θ B α (E) is almost surely continuous at θ = 0. The analogous result holds for upper dimensions.
Furthermore, Theorem 3.1 together with Corollary 3.5 have a surprising application to the box and Hausdorff dimensions of sets with continuity at θ = 0. In the following, we use the notation dim nα B E = dim nα 1 E, since our profiles extend the box dimension profiles dim m B of Falconer [5] to non-integer values of m when θ = 1 (and similarly for the upper dimensions).
On the other hand, if α ≤ 1 The analogous result holds for upper dimensions.
In particular, since dim H E ≤ dim B E, the first part of Corollary 3.6 shows us that dim nα B E is strictly less than the trivial upper bound of nα implied by Lemma 2.2 for and similarly for dim B E. Furthermore, Corollary 3.6 may immediately be translated into the context of fractional Brownian motion by Theorem 3.4.
Corollary 3.7. Let E ⊂ R n be a bounded set such that dim θ E is continuous at θ = 0 and B α : R n → R n denote index-α Brownian motion. A further implication of Theorem 3.4 is that an inequality derived from a slight modification of the proof allows us to show in Section 7.3 that the dimension profiles are continuous for any Borel set E ⊆ R n .
One final application concerns the Hausdorff dimension of the set of exceptional sets in the projection setting. The proof is based on an application of Theorem 3.3, which allows the proof of [5, Theorem 1.2 (ii), (iii)] to be generalised from box dimension (the case where θ = 1) to all intermediate dimensions.
Theorem 3.9. Let E ⊂ R n be compact, m ∈ {1, . . . , n} and 0 ≤ λ ≤ m, then Recall that dim λ θ E and dim λ θ E decrease as λ decreases. Thus, Theorem 3.9 tells us that the there is a stricter upper bound on the dimension of the exceptional set the larger the drop in dimension from the expected value. We conclude by posing a slightly different question which is a slight strengthening of Theorem 3.9, an analogy of which was considered in [5, Theorem 1.3 (ii), (iii)].
The method in [4] for box dimensions relied on Fourier transforms and approximating the potential kernels by a Gaussian with a strictly positive Fourier transform. However, the natural family of kernels appropriate for working with intermediate dimension have a more complex shape, which complicates matters. A significantly different, but perhaps interesting, approach may be required.

Proof of Theorem 3.1
To prove Theorem 3.1 we use the following result [1,Lemma 4.4], which is stated here for convenience.
for all continuous functions g and by extension. This verifies that f (E) supports a measure satisfying the condition of Lemma 4.1. Hence, for sufficiently small r > 0, S s r,θ (f (E)) ≤ a m ⌈log 2 (|E|/r) + 1⌉r s C sα,mα and thus we may set sα = dim mα θ E. It follows and so by Fubini's theorem which must be evaluated in three cases.
Moreover, for r ≤ u ≤ |x − y| 1/γ we have To conclude, we deduce from Case 1, Case 2 and Case 3 that as required.

Proof of Theorem 3.3.
Let E ⊂ R n be compact, θ ∈ (0, 1], γ ≥ 1, m ∈ N and 0 ≤ s < m. Choose a sequence (r k ) k∈N such that 0 < r k < 2 −k and Moreover, define a sequence of constants β k by where µ k is the equilibrium measure from Lemma 2.1 on E associated with the kernel φ Hence, by (3.4) and Lemma 3.2 we have Then, for each ε > 0, Hence, for P -almost all ω ∈ Ω, there exists M ω > 0 such that This is true for all ǫ > 0, so using (5.3), Hence, for s/γ = dim The argument for dim θ f ω E is similar, although it suffices to set r k = 2 −k .

Acknowledgement
The author thanks the Carnegie Trust and London Mathematical Society for financially supporting this work, as well as Kenneth Falconer and Jonathan Fraser for insightful discussion and comments.