Intermediate dimensions under self-affine codings

Intermediate dimensions were recently introduced by Falconer, Fraser, and Kempton [Math. Z., 296, (2020)] to interpolate between the Hausdorff and box-counting dimensions. In this paper, we show that for every subset $ E $ of the symbolic space, the intermediate dimensions of the projections of $ E $ under typical self-affine coding maps are constant and given by formulas in terms of capacities. Moreover, we extend the results to the generalized intermediate dimensions in several settings, including the orthogonal projections in Euclidean spaces and the images of fractional Brownian motions.


Introduction
The study on the dimensions of projections of sets has a long history.For a survey of this topic, please refer to [11].In this paper, we focus on the intermediate dimensions of projections of sets under the coding maps associated with typical affine iterated function systems.
In what follows, we fix a family of d×d invertible real matrices T 1 , . . ., T m with T j < 1 for 1 ≤ j ≤ m.Let a = (a 1 , . . ., a m ) ∈ R dm .By an affine iterated function system (affine IFS) we mean a finite family F a = {f a j } m j=1 of affine maps taking the form f a j (x) = T j x + a j for 1 ≤ j ≤ m.Here we write f a j instead of f j to emphasize its dependence on a.It is well known [15] that there exists a unique non-empty compact set K a such that We call K a the self-affine set generated by F a .Write Σ := {1, . . ., m} N .The (self-affine) coding map π a : Σ → R d associated with It is well known [15] that K a = π a (Σ).
There is an amount of work studying various dimensional properties of projected sets and measures under typical coding maps [6,13,16,17,18,19,22].Let L d denote the Lebesgue measure on R d .In his seminal paper [6], Falconer showed that the Hausdorff and box-counting dimensions of self-affine sets K a = π a (Σ) remain as a common constant for L dm -a.e. a provided that T j < 1/3 for all j.The upper bound in this norm condition was later relaxed to 1/2 by Solomyak [22].Assuming T j < 1/2 for all j, very recently Feng, Lo, and Ma [13] showed that for every Borel set E ⊂ Σ, each of the Hausdorff, packing, upper, and lower box-counting dimensions of π a (E) is constant for L dm -a.e.a.In this paper, letting E ⊂ Σ, we obtain an analogous constancy result about the intermediate dimensions of π a (E) for L dm -a.e.a.
Intermediate dimensions were introduced by Falconer, Fraser, and Kempton [12] to interpolate between the Hausdorff and box-counting dimensions, see [10] for a survey.Despite their extremely recent introduction, the intermediate dimensions have already seen interesting applications, for example [5,Section 6] and [2].To avoid problems of definition, throughout the paper we assume that all the sets, whose dimensions are considered, are non-empty and bounded.Denote the diameter of a set U ⊂ R d by |U|.Definition 1.1.Let F ⊂ R d .For 0 ≤ θ ≤ 1, the upper θ-intermediate dimension of F is defined by dim θ F = inf{s ≥ 0 : for all ε > 0, there exists r 0 ∈ (0, 1] such that for all r ∈ (0, r 0 ), there exists a cover {U i } of F such that r 1/θ ≤ |U i | ≤ r for all i and i |U i | s ≤ ε} and the lower θ-intermediate dimension of F is defined by dim θ F = inf{s ≥ 0 : for all ε > 0 and r 0 ∈ (0, 1], there exists r ∈ (0, r 0 ) and a cover {U i } of F such that r 1/θ ≤ |U i | ≤ r for all i and It is immediate that the Hausdorff dimension dim H F , the upper box-counting dimension dim B F , and the lower box-counting dimension dim B F are the extreme cases of the θ-intermediate dimensions.Specifically, dim H F = dim 0 F = dim 0 F, dim B = dim 1 F, and dim B F = dim 1 F.
Below we state our first main result on the θ-intermediate dimensions of π a (E) for E ⊂ Σ in terms of the capacity dimensions dim C,θ E, dim C,θ E whose rigorous definitions are given in Definition 2.4.Theorem 1.2.Let 0 < θ ≤ 1 and E ⊂ Σ.Then the followings hold.
(ii) Assume T j < 1/2 for 1 ≤ j ≤ m.Then for L dm -a.e. a ∈ R dm , dim θ π a (E) = dim C,θ E and dim θ π a (E) = dim C,θ E. Theorem 1.2 is proved through a capacity approach by adapting and extending some ideas in [5,9,13].Our definitions of kernels are inspired by, but different from that of Burrell, Falconer, and Fraser [5] where the projection theorems are established for the θ-intermediate dimensions under the orthogonal projections in Euclidean spaces.It is these new kernels that reveal a unified computational scheme and pave the way for the extensions to the generalized intermediate dimensions.
It is natural to ask whether there are some results analogous to Theorem 1.2 for the Φ-intermediate dimensions.Our answer is affirmative.Moreover, our strategy can be exploited to study the Φ-intermediate dimensions in several settings, including the orthogonal projections in Euclidean spaces and the images of fractional Brownian motions.For the clarity of illustration, we separately state the settings where we study the Φ-intermediate dimensions.
As is noted by Banaji, a question not explored in [1] is that whether the potential-theoretic methods in [4,5] can be adapted to study the Φ-intermediate dimensions.This question is answered affirmatively by Theorem 1.6 based on the kernels in Definition 5.3 and the condition (1.3).Note that (1.3) holds if lim inf r→0 log r/ log Φ(r) > 0, which is satisfied by Φ(r) = r 1/θ (0 < θ < 1) and Φ(r) = −r/ log r.There are more general functions satisfying (1.3), for example, Φ(r) = r − log r .
The paper is organizied as follows.In Section 2, we provide the definitions of the intermediate and capacity dimensions.Then the proofs of (i) and (ii) of Theorem 1.2 are respectively given in Section 3 and Section 4. After introducing the generalized capacity dimensions and the generalized dimension profiles, we prove Theorem 1.6 in Section 5. Finally, a few remarks are given in the last section.

Preliminaries
Throughout this paper, we shall mean by a ε b that a ≤ Cb for some positive constant C depending on ε.We write a ≈ ε b if a ε b and b ε a.If it is clear from the context what C should depend on, we may briefly write by a b or a ≈ b.We denote the natural logarithm by log and the natural exponential by exp.By # we denote the cardinality of a finite set.In a metric space, the closed ball centered at x with radius r is denoted by B(x, r), and the closure of a set E is denoted by E.
2.1.Intermediate dimensions.As noted in [5], it is convenient to work with some equivalent definitions of the θ-intermediate dimensions.These definitions are expressed as limits of logarithms of sums over covers.For s ≥ 0, 0 < θ ≤ 1, and E ⊂ R d , define Lemma 2.1 is a direct consequence of the following result. In By convention we let x|0 = ∅ and T ∅ be the identity map on R Following [17], we define for r > 0, For 0 ≤ s ≤ d, 0 < θ ≤ 1, and r > 0, we introduce the kernel Let P(E) denote the set of Borel probability measures supported on a compact set E.
For compact set E ⊂ Σ, the capacity of E is defined by .
By convention we set C s θ,r (E) := C s θ,r (E) for non-compact subset E ⊂ Σ.Thus we can assume, without loss of generality, that the set whose capacities are considered is compact.
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 for the positive symmetric continuous kernels (cf.[14,Theorem 2.4] or [9, Lemma 2.1]).
Lemma 2.3.Let E be a non-empty compact set in a metric space, and let K : E × E → (0, +∞) be a continuous function such that K(x, y) = K(y, x).Then there is some measure µ 0 ∈ P(E) such that , with equality for µ 0 -a.e.x ∈ E.
A measure µ 0 in Lemma 2.3 is called an equilibrium measure for the kernel K. Now we are ready to introduce the capacity dimensions.
In particular, there is a unique s Without loss of generality assume that E is compact.By Lemma 2.3, there exists an equilibrium measure µ 0 on E for the kernel J t θ,r (x ∧ y).Then . By (2.7) and (2.8), taking logarithms and making a rearrangement give (2.5).
Taking limits of the quotients in (2.5) shows that the functions (2.9) On the other hand, since Hence C d θ,r (E) ≤ 1, and so Based on (2.10) and (2.11), the proof is completed by the continuity and strict monotonicity of the functions in (2.9).

Proof of Theorem 1.2(i)
We begin with a simple geometric observation.
for I ∈ Σ * and r > 0, where N r (F ) denotes the minimal number of sets with diameter r needed to cover any bounded set F ⊂ R d .
Next we deduce an upper bound on S s θ,r (π a (E)) from a lower bound on the potentials of a measure with respect to the kernel J s θ,r (x ∧ y).
Proposition 3.2.Let 0 ≤ s ≤ d, 0 < θ ≤ 1, and a ∈ R dm .Let E ⊂ Σ be a non-empty compact set.If for 0 < r ≤ 1 there exist µ ∈ P(E) and γ > 0 such that then for all sufficiently small r > 0, We adapt some ideas from the proof of [5,Lemma 4.4].However, instead of only chopping the balls with too large diameters, we cover each of the balls having relatively large measure with sets of an appropriate diameter provided by the kernel J s θ,r (x ∧ y).
Since {[x|n(x)]} x∈E is a cover of E and n(x) ≤ ℓ, we can find a disjoint subcover Γ by the net structure of Σ.By (3.3), for each I ∈ Γ there exists some δ I ∈ [r 1/θ , r] such that Clearly {π a ([I])} I∈Γ is a cover of π a (E) since Γ covers E. By Lemma 3.1, we can find for each π a ([I]) a cover D I consisting of sets with diameter δ I ∈ [r 1/θ , r] such that .
This finishes the proof since ℓ + 1 θ log(1/r) when r is small.Now we are ready to prove (i) of Theorem 1.2.
Proof of Theorem 1.2(i).Since the θ-intermediate dimensions and capacity dimensions of a set remain the same after taking closure, without loss of generality we can assume that E is compact.
Let 0 ≤ s ≤ d.For 0 < r ≤ 1, by Lemma 2.3 there exists an equilibrium measure µ r ∈ P(E) for the kernel J s θ,r (x ∧ y) such that Applying Proposition 3.2 with µ = µ r gives that for all sufficiently small r > 0, . By taking logarithms and limits, Hence Lemma 2.1 and Definition 2.4 show that This completes the proof of Theorem 1.2(i).

Proof of Theorem 1.2(ii)
We begin with a lemma modified from [5,Lemma 5.4], which allows us to control S s θ,r (π a (E)) from below using the upper bounds on the potentials with respect to the kernel ψ s θ,r (|x − y|).For 0 ≤ s ≤ d, 0 < θ ≤ 1 and 0 < r ≤ 1, define (4.1) We can view Lemma 4.1 as a potential-theoretic version of the mass distribution principle (see [7]).Let {U i } be a cover of F with r 1/θ ≤ |U i | ≤ r.Without loss of generality we can pick some Taking infima over all such covers gives This finishes the proof.
The following lemma is contained in the proof of [17,Lemma 5.1] which verifies the so-called self-affine transversality in (4.4).
where B ρ denotes the closed ball in R dm centered at 0 with radius ρ.
Next we exploit Lemma 4.2 to relate the integral of a → ψ s θ,r (|π a (x) − π a (y)|) to the kernel J s θ,r (x ∧ y). Bρ Proof.Write L := L dm for short.By Lemma 4.2, where the last inequality is by s By Lemma 2.3, for each k ∈ N there is an equilibrium measure µ k on E for the kernel J s θ,r k (x ∧ y).Write Let ρ > 0. Proposition 4.3 implies that (4.8) Let ε > 0. Note that there is some A > 0 such that r ε/2 log(1/r) ≤ A for all r > 0. Then summing (4.8) over k ∈ N and using Fubini's theorem lead to Hence for L dm -a.e. a ∈ B ρ , there exists M a > 0 such that Then for each k there exists some Borel The proof for the lower θ-intermediate dimensions is similar.

Generalized intermediate dimensions
In this section, we will prove Theorem 1.6 through a similar strategy of Theorem 1.2.In what follows, let Φ : (0, Y ) → (0, ∞) be an admissible function for some Y > 0.

Generalized intermediate dimensions.
Following [1], we introduce the generalized intermediate dimensions called the Φ-intermediate dimensions.

Definition 5.1 (Φ-intermediate dimensions)
. For E ⊂ R d , its upper Φ-intermediate dimension is defined by dim Φ E = inf{s ≥ 0 : for all ε > 0 there exists r 0 ∈ (0, 1] such that for all r ∈ (0, r 0 ), there exists a cover {U i } of E such that Φ(r) ≤ |U i | ≤ r for all i and i |U i | s ≤ ε} and its lower Φ-intermediate dimension is defined by dim Φ E = inf{s ≥ 0 : for all ε > 0 and r 0 ∈ (0, 1] there exists r ∈ (0, r 0 ) and a cover {U i } of E such that Φ(r) ≤ |U i | ≤ r for all i and We can describe the Φ-intermediate dimensions by employing a similar approach to that used in defining the Hausdorff dimension with the aid of the Hausdorff measures.For s ≥ 0, r > 0, and Proof.Since E is non-empty, we have S 0 Φ,r (E) ≥ 1. Pick any x ∈ E, then E ⊂ B(x, |E|).Since E is bounded, we have for r ≤ |E|, , where the last inequality follows from Lemma 5.4.Note that for 0 ≤ t ≤ s, if lim inf r→0 log r/ log Φ(r) > 0.

Generalized capacity dimensions and dimension profiles.
We begin with the introduction of some appropriate kernels in the corresponding settings.We proceed by defining the capacities with respect to the above kernels.
Definition 5.5 (capacities).Let X be a compact metric space and K : X ×X → (0, +∞) be a continuous function.For each compact set E ⊂ X, the capacity of E with respect to the kernel K is defined by .
By convention we set C K (E) = C K (E) for every non-compact set E ⊂ X.Thus when it comes to capacities, without loss of generality we can assume that the underlying set is compact.In particular, we focus on the following capacities.
• In Setting 1.3, let K(x, y) = J s Φ,r (x ∧ y) (see (5.5)).Define Now we are ready to define the generalized capacity dimensions called the Φ-capacity dimensions and the generalized dimension profiles called the Φ-dimension profiles.
Definition 5.6 and Definition 5.7 are justified as follows.Let K s Φ,r (x, y) = J s Φ,r (x ∧ y) or J s,τ Φ,r (|x − y|).According to Definition 5.3, for 0 A combination of (5.9), (5.10), and (5.11) justifies Definition 5.6 and Definition 5.7 in the same way as the proof of Lemma 5.2.
It is readily checked that Φ α is admissible.

Upper bound.
We begin with a lemma about the behavior of the capacities under the Hölder continuous maps.
Without loss of generality we assume that E is compact.By Lemma 2.3, there is an equilibrium measure ν ∈ P(f (E)) for the kernel J s,m Φ,r (|u − v|).Then [21, Theorem 1.20] gives some µ ∈ P(E) such that ν = f µ.Hence (5.16) This completes the proof.
We now demonstrate how the capacities behave under a map with the modulus of continuity similar to that of the fractional Brownian motion.Lemma 5.9.Let τ > 0 and 0 ≤ s ≤ τ .Let f : R d → R m be a map.If there exist some 0 < α < 1 and 0 < ∆ < 1 such that then for all sufficiently small r > 0 and x, y ∈ R d with |x − y| ≤ ∆, (5.18) Let E ⊂ R d be a bounded set.Suppose further that there is some 0 < β ≤ 1 such that Then for all sufficiently small r > 0, (5.20) Proof.Let x, h ∈ R d with |h| ≤ ∆.According to (5.6), (5.21) This concludes that Together with (5.21), we have Φα,r 1/α (|h|), where the last equality is by taking v = u 1/α .This proves (5.18).
To apply Lemma 5.9, below we recall Lévy's modulus of continuity for the fractional Brownian motion (see e.g., [20,Chp. 18, Eq. (3)]).Remark 5.11.As is in [4,8], the Hölder continuity of the fractional Brownian motion is sufficient for obtaining results about the θ-intermediate dimensions.However, for the Φ-intermediate dimensions, the more precise modulus of continuity in Lemma 5.10 seems necessary.Now we are ready to prove the key ingredients in the proof of the upper-bound part of Theorem 1.6.They are analogous to Proposition 3.2.Proposition 5.12.Let Φ be an admissible function.Proof.(i) follows from a similar proof of Proposition 3.2.
Next we prove (ii) by adapting some ideas of [5] but considering a different kernel J when r is small.This proves (5.25).
This finishes the proof.
As an analog of Proposition 4.3, the following lemma reveals a unified computational scheme for the integrals over parameters in various contexts.
Proof of Theorem 1.6.Based on Proposition 5.12 and Proposition 5.16, the statements of different settings in Theorem 1.6 result from similar arguments.Hence to avoid repetitions while maintaining clarity, we exemplify the arguments by showing (1.4) and (1.5).Without loss of generality, we assume that E is compact.

Final remarks
In the section we give a few remarks.
In our main theorems, the assumption that T j < 1/2 for 1 ≤ j ≤ m can be weaken to max i =j ( T i + T j ) < 1. Indeed the first assumption is only used to guarantee the selfaffine transversality in Lemma 4.2.As pointed out in [3, Proposition 9.4.1], the second assumption is sufficient for the self-affine transversality.
In [1,Definition 2.7], the admissibility of Φ is assumed in the definitions of the Φ-intermediate dimensions in some general metric spaces.However, in Theorem 1.6 concerning the Φ-intermediate dimensions in R d , we only require that Φ is monotone and satisfies 0 < Φ(r) ≤ r instead of the admissibility.
There is no obstruction in adapting the arguments in [13,Section 9] to estimate the Hausdorff dimensions of the exceptional sets for the Φ-intermediate dimensions.For example, below we give one such result.

Setting 1 . 4 .Setting 1 . 5 .
Let G(d, m) be the Grassmannian of m-dimensional subspaces of R d and γ d,m be the natural invariant probability measure on G(d, m).For V ∈ G(d, m), let P V be the orthogonal projection from R d onto V .For 0 < α < 1, the index-α fractional Brownian motion is the Gaussian random function B α : R d → R m that with probability 1 is continuous with B α (0) = 0 and such that the increments B α (x) − B α (y) are multivariate normal with the mean vector 0 ∈ R m and the covariance matrix diag(|x − y| 2α , . . ., |x − y| 2α ) ∈ R m×m .Denote the underlying probability space as (Ω, P).In particular, B α = (B α,1 , . . ., B α,m ), where B α,i : R d → R are independent index-α fractional Brownian motions with distributions given by Now we are ready to present our results for the Φ-intermediate dimensions using the generalized capacity dimensions dim C,Φ E, dim C,Φ E (see Definition 5.6) and generalized dimension profiles dim τ Φ E, dim τ Φ E (see Definition 5.7).