Assouad dimension influences the box and packing dimensions of orthogonal projections

We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of $F \subseteq \rn$ is no greater than $m$, then the box and packing dimensions of $F$ are preserved under orthogonal projections onto almost all $m$-dimensional subspaces. We also show that the threshold $m$ for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.


Introduction and results
The relationship between the dimension of a Borel set F ⊆ R n and its projections onto mdimensional subspaces goes back to Marstrand [12] and Mattila [13] who showed that dim H π V F = min{m, dim H F } for almost all V ∈ G(n, m) with respect to the natural invariant measure on the Grassmannian G(n, m), where π V : R n → V is orthogonal projection onto V and dim H is Hausdorff dimension.
Finding the box and packing dimensions of projections of sets is more awkward. For a nonempty bounded F ⊆ R n , let N r (F ) be the minimum number of sets of diameter r that can cover (Note that this gives the same values for the dimensions as taking N r (F ) to be the least number of sets of diameter at most r that can cover F or other equivalent definitions, see [2]). The packing dimension of a (not-necessarily bounded) set may be defined in terms of upper box dimension as see [2]. Although the values of dim B π V F, dim B π V F and dim P π V F are constant for almost all V ∈ G(n, m), this constant can take any value in the range with analogous inequalities for lower box and packing dimensions. These inequalities were established in [5,6,10] using dimension profiles, with examples showing them to be best possible in [5,11], and recently a simpler approach using capacities was introduced [3]. For background on the dimensions of projections, see [4,14], and for general dimension theory, see [2]. In light of the fact that, in general, box and packing dimensions may drop below the upper bound in (1.2) under almost all projections, it may be of interest to find geometric conditions that ensure that such a drop does not occur. In this paper we obtain a result of this type: we show that if the Assouad, or even the quasi-Assouad, dimension of F ⊆ R n is no greater than m, then the box and packing dimensions of F are preserved under orthogonal projection onto almost all m-dimensional subspaces, i.e. there is equality on the right-hand side of (1.2). We also obtain estimates when dim A F > m as well as bounds on the dimension of the exceptional set of subspaces V .
The Assouad dimension of a non-empty set F ⊆ R n is defined by The related upper Assouad spectrum is defined by where θ ∈ (0, 1). If one replaces r R 1/θ with r = R 1/θ , then one obtains the Assouad spectrum dim θ A F , see [8], but it was proved in [7] that and so we are able to rely on the theory of dim θ A F , which is somewhat more developed. The upper Assouad spectrum is clearly non-decreasing in θ but the Assouad spectrum need not be. However, in most commonly studied situations it is non-decreasing and therefore the two spectra coincide. Finally, the quasi-Assouad dimension is defined by Generally, for θ ∈ (0, 1), With these definitions we may state our two basic theorems which will be proved in the next section using dimension profiles. Theorem 1.1. Let 1 m < n and θ ∈ (0, 1). If F ⊆ R n is bounded then, for almost all V ∈ G(n, m),

3)
and the same conclusion holds with dim B replaced by dim B . If F is Borel the conclusion holds with dim B replaced by dim P .
The following statement bounds the Hausdorff dimension of the exceptional set of projections in Theorem 1.1 strictly away from m(n − m), the dimension of G(n, m). Theorem 1.2. Let 1 m < n, s ∈ (0, m) and θ ∈ (0, 1). If F ⊆ R n is bounded then 4) and the same conclusion holds with dim B replaced by dim B . If F is Borel the conclusion holds with dim B replaced by dim P .
The following corollaries follow easily from the theorems by choosing appropriate parameters.
The same conclusion holds with dim B replaced by dim B . If F is Borel the conclusion holds with dim B replaced by dim P .
and the same conclusion holds with dim B replaced by dim B . If F is Borel the conclusion holds with dim B replaced by dim P .
In the absence of a precise result, a natural question is when (1.6) improves on the general lower bounds from (1.2). A careful analysis of the lower bound yields many such situations. We provide one instance, based on a knowledge of the Assouad dimension. We exclude the range dim A F < max{m, dim B F } since this is covered by Corollary 1.3.
then, for almost all V ∈ G(n, m), Therefore by (1.3), for almost all V ∈ G(n, m), as required. The final strict inequality uses the assumption on the Assouad dimension.  The proof of Corollary 1.5 involved choosing a particular θ in the 'worst case scenario'. If dim θ A F is known, then there may be a much better choice leading to better estimates in the particular setting. For example, better choices of θ always exist if F is a Bedford-McMullen carpet, see [9].
We remark that the examples in [6, Lemma 19] of sets F ⊆ R n for which there is almost sure equality on the left-hand inequality of (1.2), all have Assouad dimension dim A F = n, consistent with our estimates.

Dimension profiles and proofs of theorems
We first review the relationship between the dimension profiles of a set and the box dimensions of its projections. Then estimating the dimension profiles in terms of (quasi-)Assouad dimensions will lead to the theorems in Section 1.
Dimension profiles may be defined in terms of capacities with respect to certain kernels [3]. For s ∈ [0, n] and r > 0 we define the kernel φ s r (x) = min 1, For a non-empty compact F ⊆ R n , the capacity, C s r (F ), of F with respect to this kernel is given by where M(F ) denotes the collection of Borel probability measures supported by F . The double integral inside the infimum is called the energy of µ with respect to the kernel. The capacity of a general bounded set is taken to be that of its closure. For bounded F ⊆ R n and s > 0 we define the lower and upper box dimension profiles of F by and, analogously to the packing dimension (1.1), the packing dimension profile (for F not necessarily bounded) by In particular, by [3, Corollary 2.5] if s n then but for s < n the dimension profiles give the almost sure dimensions of projections of sets as well as information on the size of the set of exceptional projections, as follows.
The following theorem relates the dimension profiles to (quasi-)Assouad dimension. Combined with Theorem 2.1 this will give the bounds stated in Section 1 for the box and packing dimensions of projections.
Theorem 2.2. Let s ∈ (0, n] and θ ∈ (0, 1). If F ⊆ R n is bounded then Proof. We first consider upper box dimensions. We may assume for convenience that |F | < 1/2, where |F | denotes the diameter of F . Throughout this proof we write N r (E) to denote the maximal size of an r-separated subset of E. Fix α > dim θ A F , β > dim A F and let C > 0 be a constant such that for all 0 < r < R < 1 and x ∈ F and for all 0 < r R 1/θ < R < 1 and x ∈ F Let 0 < r < 1 and {x i } be a maximal r-separated set of points in F . Place a point mass of weight 1/N r (F ) at each x i and let the measure µ be the aggregate of these point masses so that µ(F ) = 1.
Write D = log 2 (2|F |r −1 ) and B = (1 − θ) log 2 (r −1 ) noting that for sufficiently small r, 1 B < D. For each i the potential of µ at x i is for a constant c which is independent of r. Summing over the x i , the energy of µ is and so the capacity C s r (F ) satisfies The conclusion for upper dimensions follows on taking α and β arbitrarily close to dim θ A F and dim A F respectively. For the lower box dimension case we take lower limits in the final inequalities.
Finally, we extend the conclusions for upper dimensions to packing dimensions. Given a Borel set F with dim P F > γ there exists a compact E ⊆ F such that dim P (E ∩ U ) > γ for every open set U that intersects E, see for example [ for all i, using the monotonicity of dim θ A and dim A . Let {E j } be any countable cover of E by compact sets. By Baire's category theorem, for some j, E ∩ E j contains a set that is open relative to E, so E ∩ U i ⊆ E ∩ E j for some i. It follows from the definition of the packing dimension profile dim s P (2.2) that Taking γ arbitrarily close to dim P F gives (2.4).

Sharpness of the threshold for the (quasi-)Assouad dimension
To conclude the paper, we show that Corollary 1.3 is sharp in the following sense: Lemma 3.1. For all s ∈ (m, n] and t ∈ (0, s) there exists a compact set F ⊂ R n such that dim P F = dim B F = t, dim A F = s (in particular, dim qA F s), and for every V ∈ G(n, m).
This lemma says that, in order to guarantee that packing or upper box dimensions are preserved under almost all orthogonal projections (or indeed under even one orthogonal projection), it is not enough to assume that dim qA F s, or even that dim A F s, if s > m (while s = m is enough by Corollary 1.3). However, how much the packing or box dimension of a typical projection π V F , V ∈ G(n, m), can drop from dim B F in terms of dim B F and dim A F remains an open problem, since the upper and lower bounds provided by Corollary 1.3 and Lemma 3.1 respectively, are in general quite far apart from each other. We note that when s = n, the upper bound given by Lemma 3.1 agrees with the lower bound in (1.2), and is therefore sharp in this case.
The construction of the set F in the above lemma is based on sets defined by restricting the digits in dyadic expansions. Given a set S ⊆ N, let We write #F to denote the cardinality of a set F . Recall the definition of (upper) density and Banach density of a subset of N: Lemma 3.2. Given S ⊂ N and n ∈ N, where X n S ⊆ [0, 1] n is the n-fold product of X S . Proof. The claim for upper box dimension is almost immediate from the definition, see [1,Example 1.4.2] for details in the case n = 1. The claim for packing dimension follows from the one on upper box dimension and [1, Lemma 2.8.1]. Finally, for the Assouad dimension formula, we note that if 2 − −k r < 2 1− −k 2 − −1 < R 2 − then, for any x ∈ X S , the set X S ∩ B(x, R) can be covered by C n · 2 −n#(S∩{ ,..., +k−1}) balls of radius r and cannot be covered by fewer than a constant (depending on n) multiple of this number, so that dim A (X S ) = d B (S)n.
Proof of Lemma 3.1. Let A ⊆ N be a set with d(A) = d B (A) = s/n; this is easily arranged. Let (k j ) j∈N be a sequence of natural numbers satisfying (k 1 · · · k j−1 )/k j → 0 as j → ∞. and F = X n S . Here A + k j = {a + k j : a ∈ A}. Using (3.1), we see that d(S) is realized along the sequence sk j s−t , and a calculation shows that d(S) = t/n. Also, d B (S) = d B (A) = s/n and hence, thanks to Lemma 3.2, dim P F = dim B F = t, dim A F = s. Now fix V ∈ G(n, m), ε > 0 and k ∈ N. Pick j such that k j k < k j+1 . Provided k (and therefore j) is large enough in terms of ε, n, s and t, the set F can be covered by tk i s−t ) 2 εk j cubes of side-length 2 −k j , where we used (3.1). Hence, π V F can be covered by C n,m 2 m(k−k j ) 2 εk j cubes of side-length 2 −k . On the other hand, if k > sk j /(s − t), then (again assuming k is large enough), F can be covered by 2 (t+ε)sk j /(s−t) cubes of side-length 2 −k , and hence π V F can be covered by at most a constant C n,m multiple of that number. Up to the terms involving ε, the first bound is more efficient when otherwise the second bound is more efficient. Note that 1 + st m(s−t) > s s−t , since s > m. A short calculation shows that, in any case, π V F can be covered by C n,m 2 (d+εCm,s,t)k , d = mst m(s − t) + st , cubes of side-length 2 −k . Since ε > 0 was arbitrary, this concludes the proof.