On the general dyadic grids on R d

. Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson, and Wei), we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here, we extend this work to all dimensions, which turns out to have many surprising difficultiesduetothefactthat d + 1,not2 d ,gridsistheoptimalnumberinanadjacentdyadicsystem in R d . As a byproduct, we show that a collection of d + 1 dyadic systems in R d is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on R . The underlying geometric structures that arise in this higher-dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and n -adic, for any n ) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific exa.


Introduction
The purpose of this paper is to give an optimal description of adjacent dyadic systems (or more generally, adjacent n-adic systems) in R d . Dyadic systems are ubiquitous in harmonic analysis, as well as many other fields. Oftentimes, one wants to understand a continuous operator or object via its dyadic counterparts; our goal is to say, in an optimal and precise fashion, exactly what these dyadic counterparts are.
The study of continuous objects via dyadic ones is a central theme in analysis and its application to many different areas of mathematics. For instance, dyadic decompositions and partitions underlie the study of singular integral operators and maximal functions (among others), weight and function classes, partial differential equations, and number theory; our bibliography lists a few out of many references here. In our recent paper [2] joint with Jiang, Olson, and Wei, we gave a necessary and sufficient condition on characterizing the adjacent n-adic systems on R. Here, we generalize these results to higher dimensions. Although we use ideas from [2], the construction of the analogous objects in R d is not trivial; indeed, we have to adapt our techniques from [2] to a way that is compatible with the underlying lattice structure inherent in the construction of adjacent n-adic systems in R d .
Let us begin with the definition of n-adic systems in R d , which is our main object of study.  where > 0 is the sidelength of such a cube) is called a general dyadic grid with base n (or n-adic grid) if the following conditions are satisfied: (i) For any Q ∈ G, its sidelength (Q) is of the form n k , k ∈ Z.
(iii) For each fixed k ∈ Z, the cubes of a fixed sidelength n k form a partition of R d .
In particular, if n = 2, we also refer to such a collection a dyadic grid, which is usually denoted by D.
The defining property of such a structure is a certain dyadic covering theorem. The one that we use is due to Conde Alonso [5], and is optimal in terms of the number of grids required.  (B). The number of dyadic systems is optimal.
We make a remark that the optimal number d + 1 in Theorem 1.2 plays an important role throughout this paper. Motivated by Theorem 1.2, we introduce the following definition of adjacent n-dic systems in R d , our main object of study. Definition 1. 3 Given d + 1 many G 1 , . . . , G d+1 n-adic grids, we say that they are adjacent if, for any cube Q ⊆ R d (or any ball), there exists i ∈ {1, . . . , d + 1}, and R ∈ G i , such that: (1) Q ⊆ R; (2) (R) ≤ C d ,n (Q), where C d ,n is a dimension constant that only depends on d and n.
This characterizing property of adjacent dyadic systems is sometimes referred to as Mei's lemma due to the work [13] on the torus (hence, the definition we use is sometimes called the optimal Mei's lemma). This property has been widely explored in a wide array of contexts and settings (see [9,12,14]), and has a long history; see the introduction of [2] and also the monographs [6,11] for details. The applications of Mei's lemma are vast; adjacent dyadic systems are crucially used in the area of sparse domination (initiated by Lerner to prove the A 2 theorem in [10]; see also [4,7] among others), functional analysis [8,12,14], and measure theory [3].
Note that Conde Alonso's theorem only guarantees the existence of a collection of adjacent dyadic systems in R d ; it does not say how to construct such systems in general nor how to tell if a system is adjacent. Inspired by [2], we ask "what are the necessary and sufficient conditions so that a given collection of d + 1 n-adic grids in R d is adjacent?" In [2], we give a complete answer to this question on the real line, which we will briefly review in Section 2. In order to extend these results in [2] to higher dimensions, we must deal with how d + 1 n-adic grids, instead of only 2, interact with each other. The main idea to overcome such a difficulty is to work on a certain quantified version of the n-adic systems. We introduced a one-dimensional analog of this in [2]; however, extending this concept to higher dimensions requires many new ideas. The geometric structures that we define to quantify adjacency collapse into much simpler concepts on the real line, and we provide perspective on this throughout the paper.

Statement of the main result
Suppose we are given d + 1 n-adic grids G 1 , . . . , G d+1 in R d . Here is how to verify whether they are adjacent or not.

Algorithm 1.4
Step I: an infinite matrix with d rows, infinitely many columns, and entries belonging to Z n .
Here, the term G(δ i , L ⃗ a i ) is referred as the representation of the n-adic grid G (see Section 3 for more detailed information about this concept).
Note that Theorem 1.5 is sharp, in the sense that the number of the dyadic systems is optimal. The proof of the above theorem uses the idea of representation of n-adic grids, which was introduced in [2]. Moreover, combining with the one-dimensional result (see [2,Theorem 3.8] or Theorem 2.2), Theorem 1.5 is equivalent to the following result. Theorem 1. 6 The collection of n-adic systems G 1 , . . . , G d+1 is adjacent if and only if for any j ∈ {1, . . . , d} and k 1 , k 2 ∈ {1, . . . , d + 1}, k 1 ≠ k 2 , P j (G k1 ) and P j (G k2 ) are adjacent on R.
Here, P j is the orthogonal projection onto the jth axis of R d , and for any n-adic grid G, P j (G) is defined to be the collection of all P j (Q), Q ∈ G.

Remark 1.7
Recall that in the classical approach of constructing an adjacent system in R d , what we usually do is first take any two adjacent dyadic systems (on R) on each coordinate axis, and then take the Cartesian products of these dyadic systems. Note that this will give us a collection of 2 d adjacent dyadic systems in R d . We would like to point out that Corollary 1.6 does not follow from this classical approach. First of all, our result is optimal, in the sense that the number of the n-adic systems is d + 1, rather than 2 d ; moreover, our result provides a necessary and sufficient condition to tell whether a collection of d + 1 n-adic grids are adjacent or not, rather than a single construction.
Another interesting question to ask is whether there is a more inherent geometric approach to study the adjacency of the systems of the n-adic grids. More precisely, can we generalize the one-dimensional result (see Theorem 2.2) in a more parallel way, that respects the underlying geometric structure present in d + 1 adjacent dyadic systems?
In the second part of this paper, we give an affirmative and precise answer to the above question. The key idea is to introduce the so-called fundamental structures of a collection of d + 1 n-adic grids in R d . These basic structures allow us to generalize the one-dimensional result (see Theorem 2.2) in a more heuristic way (see Theorem 8.1), whereas Theorem 1.5 is much less obviously connected with the geometry of adjacent dyadic systems. The intuition for introducing these structures comes from a first, natural attempt to generalize the results in [2] to R d (see Remark 7.10 and Section 8.1). Furthermore, all of these constructions are illustrated by a concrete example, which is elaborated on in detail before the proof of the main result (see Theorem 8.1). This allows the reader to connect the underlying geometry with the results and examples in [2] in a concrete way.
The novelty in this paper is that we generalize the results in [2] via two different ways that retain the key lattice structure implicit in the proof of [5] for d + 1 grids. These generalizations are nontrivial, and motivate us to look at the underlying lattice structures inherent in the construction of d + 1 grids and to expand them in a manner adaptable to the constructions underlying the main result (Theorem 3.8) in [2]. These constructions allow us to better connect the geometry of the lattice with the arithmetic properties outlined in Theorem 8.1, and likely will have applications to a variety of other problems in dyadic harmonic analysis.
The outline of this paper is as follows. Part I begins with a brief reminder of our one-dimensional results, followed by relevant definitions to state and prove our main theorem on necessary and sufficient conditions for adjacency-this statement mirrors the one-dimensional results only in notation, and does not shed light at the interesting geometric interactions taking place. Therefore, Part II is devoted to studying these. Part II fully describes the rich geometry underlying the main result, including the fundamental structures which we define. These descriptions not only motivate a restating of our main result that is geometrically driven, but provide a clear (and unifying) relationship between our one-dimensional result and higher dimensions. They also allow us to comment on the uniformity of such representations. Finally, we illustrate everything with a concrete example, first introduced in Part I and revisited in Part II.

Part 1. Background and the proof of the main result
In the first part of this paper, we first make a short review of the one-dimensional results, which were considered in [2]. Then, using the idea of representation of n-adic grids, we prove Theorem 1.5. Finally, we give an example on how to apply our main result.

One-dimensional results and some application
Let us make a brief review of the case d = 1, which was considered in our early work [2]. The main question that was under the consideration in [2] is the following "Given two n-adic grids G 1 and G 2 on R, what is the necessary and sufficient condition so that they are adjacent?" We start with recalling the following definition.
where C may depend on δ but independent of m and k.
The key idea in [2] to study this problem is to quantify each n-adic grids. More precisely, for any n-adic system G on R, we can find a number δ ∈ R, and an infinite sequence and L a (0) = 0.
Given two n-adic systems G 1 and G 2 , let us write them as G 1 = G(δ 1 , L a1 ) and G 2 = G(δ 2 , L a2 ). Here is the main result in [2].

Theorem 2.2 [2, Theorem 3.8]
The n-adic grids G(δ 1 , L a1 ) and G(δ 2 , L a2 ) are adjacent if and only if: be some other representations of G 1 and G 2 , respectively. Then either With the help of Theorem 1.6, we can easily generalize Theorem 2.4 to higher dimensions.

Corollary 2.5
Under the same assumption of Theorem 1.5, let . Proof Corollary 2.5 is an easy consequence of Theorems 1.5 and 1.6, and we would like to leave the detail to the interested reader. ∎

Representation of n-adic grids
Let us extend the concept of the representation of n-adic grids to higher dimension. The setting is as follows.
(1) δ ∈ R d , in particular, δ should be thought as a vertex of some cube belonging to the zeroth generation, and we may think it as the "initial point" of our n-adic system.
On the general dyadic grids on R d

1153
(3) The location function associated to ⃗ a: which is defined by For a vector δ ∈ R d , we use the notation (δ) i , 1 ≤ i ≤ d, refers to the ith component of δ. Note that we will frequently be working with sets of d vectors in R d , which we label δ 1 , . . . , δ d . Therefore, the parentheses distinguish the selection from the components: (δ i ) s is the sth component of the vector δ i .

Definition 3.1
Let δ ∈ R d , ⃗ a, and L ⃗ a be defined as above. Let G(δ, L ⃗ a ) be the collection of the following cubes: We make a remark and sometimes we drop the dependence of the location function here, since location function only contributes to the negative generations.
(2) For m < 0, the mth generation is defined as Or equivalently: (1) For m ≥ 0, the vertices of all mth generation are defined as (2) For m < 0, the vertices of all mth generation are defined as

T. C. Anderson and B. Hu
Note that in the above definition, the term δ + L ⃗ a (−m) can be interpreted as the location of the "initial point" (that is, δ ∈ A(δ, L ⃗ a ) 0 ) after choosing n-adic parents (with respect to the zeroth generation) (−m) times.
Proof If we restrict the grid to each axis, we obtain an n-adic grid with respect to that axis [2]. Since cubes are a one-parameter family, one can easily see (by contradiction) that cubes of a given level tile the space, two cubes are either contained one in the other or disjoint, and each cube has n d children (each with 1/n d of its parent's size) and exactly one parent. ∎ Proposition 3.3 Given any n-adic grid G, we can find a δ ∈ R d and an infinite matrix However, this representation may not be unique.
Proof The proof of this result is an easy modification of [2, Proposition 4.10], and hence we leave the detail to the interested reader. While the fact that such a representation is not unique is also straightforward, one example in R d would be

Proof of the main result
In this section, we prove the main result Theorem 1.5.

Necessity
Suppose We prove the necessary part by contradiction. Assume condition (a) fails, that is, there exists some 1 , 2 ∈ {1, . . . , d + 1} with 1 ≠ 2 and s ∈ {1, . . . , d}, such that for each N 1 ≥ 1, there exists some m 1 ≥ 0 and K ∈ Z, such that which implies that the distance between the hyperplane {(x) s = (δ 1 ) s } and the hyperplane {(x) s = (δ 2 ) s + K/n m1 } is less than 1/ (N 1 n m1 ). On the other hand, note that and ] is the union of all the boundaries of the cubes in G 1 with sidelength 1/n m1 , and similarly for the term b [G 2 ,m1 ]. Now, the idea is to find two sufficiently closed points on the intersection of the boundaries of these n-adic grids. Without loss of generality, let us assume s = 1 = 1 and 2 = 2. Now, let us consider the points which satisfy the following properties: Note that property (b) above allows us to choose an open cube Q of radius 1/(N 1 n m1 ) that containing both p 1 and p 2 , whereas property (a) asserts that if there is a dyadic cube D ∈ G , ∈ {1, . . . , d + 1} covering Q, then (D) > 1/n m1 , and hence This will contradict to the second condition in Definition 1.3 if we choose N 1 sufficiently large (see Figure 1).
Next, expecting a contradiction again, we assume (b) fails. The proof for this part is very similar to the previous one. Let us consider two different cases.
Case I: There exists some k 1 , k 2 ∈ {1, . . . , d + 1}, k 1 ≠ k 2 , and s ∈ {1, . . . , d}, such that Again, for simplicity, we may assume s = k 1 = 1 and k 2 = 2. By (4.1), for any ε > 0, there exists some j 1 sufficiently large, such that which implies that since we assume j 1 is sufficiently large. Now, we can exactly follow the idea in part (a) now. More precisely, we define which enjoy similar properties as p 1 and p 2 : Then desired contradiction will follow by taking an open cube with sidelength 2ε ⋅ n j1 containing both q 1 and q 2 , where ε is sufficiently small.
Case II: There exists some k 1 , k 2 ∈ {1, . . . , d + 1}, k 1 ≠ k 2 , and s ∈ {1, . . . , d}, such that lim sup The proof for the second case is an easy modification of the first one. Indeed, (4.2) implies that for any ε > 0, there exists some j 2 sufficiently large, such that either holds, where in the above estimates, ⃗ e s refers to the stand unit vector in R d with the sth entry being 1. The rest of the proof is the same as Case I.
On the general dyadic grids on R d 1157
Case II: m 0 ≤ −N. Again, we wish to show that Q is contained in some cubes in G k,m0 for some k ∈ {1, . . . , d + 1} and we prove it by contradiction. Following the argument in Case I above, we see that there exists some k 1 , k 2 ∈ {1, . . . , d + 1} with k 1 ≠ k 2 and j * ∈ {1, . . . , d}, such that This implies that there exists some K 3 , K 4 ∈ Z, such that . Note that since we can always choose N sufficiently large, we can indeed reduce the above estimate to We claim that K 3 − K 4 ∈ {−1, 0, 1}. Indeed, by the choice of C, the right-hand side of the above estimate is bounded by 1/5; on the other hand, by the definition of the location function, we have The desired claim then follows from these observations and the fact that K 3 − K 4 is an integer. Therefore, the estimate (4.6) implies either which contradicts (4.6).
Case III: −N < m 0 ≤ 0. Indeed, we can "pass" the third case to the second case, by taking a cube Q ′ containing Q with the sidelength is n N . Applying the second case to Q ′ , we find that there exists some D ∈ G k for some k ∈ {1, . . . , d + 1}, such that Q ′ ⊂ D and (D) ≤ C 4 (Q ′ ), which clearly implies The proof is complete. ∎

An illustrated example
We now take some time to illustrate the effects of this theorem with a concrete example. To begin with, let We will show that the grids G 1 = G(δ 1 , L ⃗ a1 ), G 2 = G(δ 2 , L ⃗ a2 ), and G 3 = G(δ 3 , L ⃗ a3 ) form adjacent dyadic systems in R 2 . Note that this example is optimal, in the sense that any two dyadic systems are not adjacent in R 2 . We start by verifying Condition (i) in Theorem 1.5, which is straightforward. Clearly, it suffices to show the numbers , and 2 3 − 1 5 = 7 16 are 2-far (in the sense of Definition 2.1). This is an easy exercise due to [2, Proposition 2.4] (see also [1,Lemma 3]).
While for the second condition, let us compute all the location functions. Indeed, the cases j = 1, 2 provide the key for the computations.
, j even, 3 ) , j even, and The second condition can be easily verified, and we would like to leave the detail to the interested reader.

Part 2. A geometric approach
In the second part of this paper, we provide an alternative way, based on the underlying geometry of n-adic systems, to generalize the one-dimensional result Theorem 2.2. This approach is much more intuitive and unifies the proof of both the cases d = 1 and d > 1.

Notations
For A, B ⊂ R d , we write the distance between them by Definition 6.1 Let x ∈ R d and A ⊆ R d , and the natural deviation between x and A is defined to be where dist ⃗ e k (A, x) denotes the distance between x and A along the direction ⃗ e k .

Remark 6.2
Let us make some remarks for the above definition.
(1) The word "natural" refers to the fact that we take the natural basis in the definition. In general, one can replace {⃗ e 1 , . . . , ⃗ e d } by any other basis in R d ; however, it is enough to take the natural basis in this paper.
(2) The word "deviation" refers to the fact that we take the maximal directional distance along all the directions ⃗ e 1 , . . . , ⃗ e d . (3) In our application later, A will be either a corner set (see Definition 7.1), a modulated corner set (see Definition 7.17), or a union of them.
Let us include an easy example for these definitions (see Figure 2). In this example, we consider the case d = 2, x is the point ( 1 2 , 3 10 ), and A is the red part (which we will refer as a corner set later). Then it is clear that We frequently use the terms offspring and ancestor to refer to the generations of a given dyadic cube. In particular, we often use the letter m to refer to any generation (that is both offspring and ancestor) while the letter j is specifically reserved for ancestors. This use comes from the roles of m and j in the shift and location, parameters described below.

Fundamental structures of d + 1 n-adic systems in R d
In this section, we introduce several basic structures of a collection of d + 1 n-adic systems, where we recall that d + 1 is optimal in the sense that any d n-adic systems in R d are not adjacent, while there exists a collection of d + 1 n-adic grids, which are adjacent. Using these basic structures, we are able to generalize Theorem 2.2 to higher dimensions in a natural way. This section consists of three parts, which are the all generations case, the smallscale case, and the large-scale case. Here, the word "small scale" refers to all the offspring generations of the zeroth generation, whereas the word "large scale" refers to all the ancestor generations of the zeroth generation, together with itself. Moreover, we also introduce the concept of n-far vector, which generalize the early definition of n-far number in one-dimensional case (see Definition 2.1). Finally, there are also many concrete examples given for the purpose of understanding these structures better.
Here is a list on all the structures that we are going to introduce in this section.

Corner sets
In the first part of this section, let us introduce an important structure for G(δ, L ⃗ a ), which can be viewed as a "generator" of G(δ, L ⃗ a ).

Definition 7.1
Let G(δ, L ⃗ a ) be an n-adic grid in R d . For ∈ Z, we define the th corner operation where for any x ∈ R d and A ⊂ R d , set associated to x with parameters (δ, ⃗ a), and we refer x as the corner of the corner set Below is an example of a corner set with its corner x = ( 1 3 , 1 3 ) when d = 2, n = 2, and = 1 (see Figure 3).

Small-scale lattices and n-far vectors
The goal of the second part of section is to generalize the concept of far numbers in one dimension to higher dimension. It turns out that the correct thing to do is looking at the so-called small-scale lattices with respect to d many vectors in R d . We point out that such a construction is implicitly mentioned in Conde Alonso's proof of showing that d grids are never adjacent [5].
Let us first state an easy fact for the adjacent n-adic systems. G 1 , . . . , G d+1 be adjacent, with each of them having a representation

Lemma 7.3 Let
Then, for each k ∈ {1, . . . , d}, Proof To start with, we may assume that all δ i 's are of unit size. Otherwise, we consider δ ′ i ∶= δ i (mod 1), where the modulus 1 is taken over all the coordinate components.
We prove it by contradiction. Without loss of generality, we may assume On the general dyadic grids on R d 1163 This implies that where bG 1,0 means the boundary of all cubes in the zeroth generation of G 1 . This further shows that for any m ≥ 0, the set for all m ≥ 0. While for a proof of (7.1), it suffices to note that the set To complete the proof, it suffices to follow Conde Alonso's argument. Namely, we pick a point a ∈ d+1 ⋂ i=1 bG i ,0 and we take a small cube centered at a. It is clear that the only way to cover this ball by using the cubes from d+1 ⋃ i=1 G i is to use a cube whose sidelength is at least n. Shrinking the sidelength of the cube as small as we want, this gives a contradiction to the second property in Definition 1.3. ∎ Therefore, this easy lemma suggests the following definition.

Definition 7.5
Let m ≥ 0 and δ 1 , . . . , δ d ∈ R d be separated. Then the small-scale lattice associated to δ 1 , . . . , δ d at level m is defined as where G(δ) m is defined as in , we can actually motivate our definition from [2]. In the one-dimensional case, the set {δ 1 , . . . , δ d } reduces to the set {0}. Therefore, n-far in the one-dimensional case reduced to checking that for k ∈ Z and m ≥ 0 (see Remark 1.4(2) in [2]). Due to such a simple structure, Theorem 2.8 in [2] asserted that δ ∈ R is n-far if and only if the maximal length of tie, that is, the maximal length of consecutive 0's or consecutive (n − 1)'s in the base n-representation of δ, is finite.
The key point to generalize the concept of n-far number in higher dimension is to realize that δ and k n m are indeed playing different roles in condition (7.3). More precisely, rewrite (7.3) as for k 1 , k 2 ∈ Z and m ≥ 0. Here are the key observations: let m ≥ 0 be fixed, then: (1) when k 1 varies, the set represents the all the boundary points b [G(δ) m ]; (2) when k 2 varies, the set represents the small-scale lattice L(0; m). This is clear from Lemma 7.8.  Hence, an equivalent way to state (7.3) is the following: for any m ≥ 0, which is precisely (7.2). The key here is that the boundary points and the small-scale lattice align in dimension 1.

Example 7.11
We now illustrate via an example why the idea of generalizing the concept of far numbers to pairwise distances is not sufficient to show that d + 1 grids are adjacent. Take δ 1 = (0, 0), δ 2 = (0, 1 3 , ) and δ 3 = ( 1 3 , 1 3 ). Consider the natural pairwise generalization of far where δ and δ ′ are far if (δ) i is far from (δ ′ ) i for some i. Using this definition, we see that each of these points is far from each other, but δ 1 , δ 2 , δ 3 do no form an adjacent n-adic system. In fact, if one looks at L(δ 1 , δ 2 ; 0), one does not get a lattice, but a set of vertical lines (see Figure 5). Therefore, this pairwise comparison between points is not enough to determine an adjacent n-adic system and the lattice structure is needed.

Large-scale sampling, large-scale lattice, and modulated corner sets
The purpose of the last part of section is to introduce all the basic structures with respect to d separated vectors and d location functions, which is used to study the large-scale case. Informally, these structures are the counterparts of the small-scale lattice (see Definition 7.5) and the usual corner sets (see Definition 7.1); however, one new phenomenon for the large scale is that one needs to use the cube [0, n j ) d to quantify the behavior of the location function (see Theorem 8.1(ii)), and this suggests that we need a localized version of those structures in the large-scale case.
In our later application, the modulated corner sets will only be applied to study the ancestors of the zeroth generation, whereas the corner sets defined early will take care of all the generations. More precisely, the modulated corners are important to take care of points near the boundary of the cube [0, n j ) d that might be close to a corner lying just outside [0, n j ) d but far from the modulated corner. This remark will be made clearer in the next section and indeed underlies our use of modulated corners.

An alternative approach
Let us go back to the main question that we are interested in, namely, what the necessary and sufficient conditions are, so that a given collection of d + 1 n-adic grids are adjacent?
The goal of this section is to provide an alternative way to answer the above question via the fundamental structures, and to comment about the uniformity of such a representation. To begin, let us write these d + 1 n-adic grids by their representations, namely G(δ 1 , L ⃗ a1 ), . . . , G(δ d+1 , L ⃗ a d+1 ). Moreover, using Lemma 7.3, we may assume δ 1 , . . . , δ d+1 are separated.

Motivation and the structure theorem of adjacency
As mentioned in the introduction, the intuition behind the next theorem originates back to the one-dimensional result (see Theorem 2.2). The main idea to generalize the first condition is already contained in Remark 7.10. More precisely, we can rewrite condition (1) as: there exists some absolute constant C > 0, such that for any k 1 , k 2 ∈ Z and m ≥ 0, it holds that For simplicity, we may assume both δ 1 , δ 2 ∈ [0, 1), that is, both of them are of unit size (although we do not require such a restriction in our main result). As in Remark 7.10, there are two different ways to interpret the sets The first way is to interpret each set as the set of all the boundary points b [G(δ i ) m ], whereas the second way is to treat it as the small-scale lattice L(δ i ; m). Therefore, we can restate the first condition as δ 1 is n − far with respect to L(δ 2 ) and δ 2 is n − f ar with respect to L(δ 1 ).