{'fraction_non_alphanumeric': 0.08680317429215244, 'fraction_numerical': 0.017634956402468895, 'mean_word_length': 3.3419821352615906, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Systems of Dyadic Cubes of Complete, Doubling, Uniformly Perfect Metric Spaces without Detours

Systems of dyadic cubes are the basic tools of harmonic analysis and geometry, and this notion had been extended to general metric spaces. In this paper, we construct systems of dyadic cubes of complete, doubling, uniformly perfect metric spaces, such that for any two points in the metric space, there exists a chain of three cubes whose diameters are comparable to the distance of the points. We also give an application of our construction to previous research of potential analysis and geometry of metric spaces.


Introduction
The standard system of dyadic cubes of R d , is a basic tool of analysis on the Euclidean spaces. Constructing the counterparts of dyadic cubes of general metric spaces were started by David [7,8] and Christ [5] for metric measure spaces. Hytönen and Kairema [11] and Kaënmäki, Rajala and Suomala [12] extended these results for metric spaces without measures. Systems of generalized dyadic cubes were used for various studies of harmonic analysis (e.g. [1,2]) and analysis on metric spaces (e.g. [6,13]). From the viewpoint of discrete approximation of a metric space, it is important whether the structure of a system of dyadic cubes is comparable to that of the original metric space or not. For example, (1.1) satisfies for any x, y ∈ R d and k ∈ Z with |x − y| R d ≤ 2 −k , there exist Q x , Q y ∈ S k such that x ∈ Q x , y ∈ Q y and Q x ∩ Q y = ∅.
x y d(x, y) Figure 1: Two close points without short chains However, a system of dyadic cubes of a metric space may not satisfy such a condition, for instance, two close points may not have any short chain of dyadic cubes (see Figure 1). Hence it is a natural question when a system of dyadic cubes S k (k ∈ Z) of a metric space (X, d), satisfying the following conditions for some C, C ′ , r > 0 and M ∈ N exist: • C −1 r k ≤ diam(Q, d) ≤ Cr k for any Q ∈ S k • for any x, y ∈ X and k ∈ Z with d(x, y) The aim of this paper is to give an answer to the question in a constructive way, under certain conditions. Let (X, d) be a metric space, B(x, r) be an open ball of radius r, and A (resp. intA) be the closure (resp. interior) of a set A ⊂ X. Throughout this paper, we assume (X, d) is complete, doubling and uniformly perfect. Our main result is the following.
for some r ∈ (0, r 0 ], then there exist This theorem is based on the results and proofs of [5] and [11], and our main contribution is to develop the proof in order to satisfy the additional condition (D5). In [11], Hytönen and Kairema proved that there exist , which satisfy (D1) to (D4) for each i and for any x, y ∈ X and k ∈ Z with d(x, y) ≤ Cr k , only using the doubling condition, but i in (1.4) may be different for each point.
In order to state an example of previous results which use a system of dyadic cubes with (D5), we prepare some conditions. Definition 1.4 (Ahlfors regular). Let (Y, ρ) be a metric space and µ be a Borel measure on it. We say µ is α-Ahlfors regular with respect to (Y, ρ) if there exists C > 0 such that where r x = r x,ρ = inf y∈Y \{x} ρ(x, y). (Y, ρ) is called α-Ahlfors regular if there exists a Borel measure µ such that µ is α-Ahlfors regular with respect to (Y, ρ).
Ahlfors regularity is sometimes used in studies of harmonic analysis via dyadic cubes (e.g. [9]). We also note that if (Y, ρ) is an α-Ahlfors regular space without isolated points, then the Hausdorff dimension of (Y, ρ) coincides to α, and (Y, ρ) is uniformly perfect. Definition 1.5 (Quasisymmetry, in the sense of [13]). Let Y be a set and ρ, δ be distances on Y. We say that ρ is quasisymmetric to δ if there exists a homeomorphism θ : [0, ∞) → [0, ∞) such that for any x, y, z ∈ Y with x = z, (δ(x, y)/δ(x, z)) ≤ θ(ρ(x, y)/ρ(x, z)). Definition 1.6 (Ahlfors regular conformal dimension). The Ahlfors regular conformal dimension (ARC dimension in short) of (Y, ρ) is defined by dim AR (Y, ρ) = inf{α | there exists a metric δ on Y such that (Y, δ) is α-Ahlfors regular and δ is quasisymmetric to ρ}, Quasisymmetry was introduced by Tukia and Väisäla in [15] as a condition of embedding maps between metric spaces, and it is a generalization of quasiconformal mappings on the complex plane. ARC dimension was implicitly introduced by Bourdon and Pajot [4], and named by Bonk and Kleiner [3]. In [3], the ARC dimension is related to Cannon's conjecture, which is famous in the study of hyperbolic groups: it claims that for any hyperbolic group G whose boundary is homeomorphic to S 2 , there exists a geometric action of G on the hyperbolic space H 3 . For compact metric spaces, in [13] Kigami showed that if the counterpart of a system of dyadic cubes satisfying (D1) to (D5), called "partition satisfying basic framework" exists, then dim AR (Y, ρ) is characterized by the p-energies (p > 0) on the graphs defined by the system of dyadic cubes (see Theorem 3.7). In Section 3, we introduce the detailed conditions and results of [13] and its extension to σ-compact metric spaces and graphs, [14]. We also state the application of our main result to them in that section.

Remarks and proof of main theorem
Before the proof of our main theorem, we give some remarks on Theorem 1.3. We first note that the existence of {x ω } ω∈Ω k with (1.2) and (1.3) is easily shown as noted in [11,Subsection 2.12].
Proof. It is shown by the maximal argument with Zorn's lemma.
Remark. We can avoid Zorn's lemma if we know (X, d) is separable. A doubling metric space is separable, but this fact is shown by the existence of points satisfying (1.2) and (1.3) in general.
We also note that the results in [5] and [11] were stated for quasimetrics, that is, for (Y, ρ) satisfying the axioms of a metric spaces but replacing the triangle inequality with the following condition: It is known that if ρ is a quasimetric, then there exists c, β ∈ (0, 1) and a metric ρ ′ such that cρ ′ (x, y) ≤ ρ β (x, y) ≤ c −1 ρ ′ (x, y) for any x, y ∈ Y (see [10,Proposition 14.5]), so we can apply Theorem 1.3 to quasimetric spaces with this fact.

Proof of Theorem 1.3
For simplicity of notation, we write max{a, b} = a ∨ b, min{a, b} = a ∧ b and Ω = k∈Z Ω k . We also write #A for the number of vertices of a set A. We first prove Theorem 1.3 under strong conditions. Since (X, d) is doubling, we may assume Ω k in the assumption of Theorem 1.3 is countable, so we can write We assume that for any r ∈ (0, there exists a map π = π A : Ω → Ω such that for any k ∈ Z, • π(Ω k+1 ) = Ω k and sup Then, Theorem 1.3 holds.
. We first show that x l,m ∈ K k,n for some m ∈ N and l ≥ k, then π l−k (l, m) = (k, n).
Proof. By (T4), there exist n l , o l for l ≥ k such that π l−k (l, n l ) = (k, n), π l−k (l, o l ) = (k, o) and d(x l,n l , x l,o l ) < 2α 3 r l . Since both K k,n and K k,o are complete, and both {x l,n l } l≥k and {x l,o l } l≥k are Cauchy sequences, these sequences converge to the same limit on K k,o ∩ K k,n .
Proof of Proposition 2.2. We inductively define Q k,n .
• Let y ∈ O k,n for some (k, n) ∈ l<0 Ω l . By definition of Q k,n , there exists o ∈ N such that y ∈ Q 0,o ⊂ K 0,o . This with y ∈ O k,n assures π −k (0, o) = (k, n) and O k,n ⊂ Q k,n ⊂ K k,n .
We next prove that the assumptions of Proposition 2.2 actually hold under the assumption of Theorem 1.3. Let γ > 1 be the constant in Definition 1.2, in other words, for any y ∈ X and r > 0 with B(y, r) = X satisfies B(x, γr) \ B(x, r) = ∅. Proposition 2.6. For any α 1 , α 3 with
Therefore (n i,p k+1 , n) ∈ P i k+1 or (n, n i,p k+1 ) ∈ P i k+1 holds if n = n i,p k+1 , and so we obtain desired m 2 and m 3 by definition of π on Ω B,i k+1 , in the same way as the proof of (T4). Theorem 1.3 immediately follows from Propositions 2.2 and 2.6.

Application for evaluations of the Ahlfors regular conformal dimension
Here we introduce the framework and results of [13,14] to give an application of our main theorem.
Definition 3.1. Let T be a countable set and π : T → T be a map such that the following hold: • let F π = {w | π n (w) = w for some n ≥ 1}, then #F π ≤ 1. (H1) • For any w, v ∈ T, there exist n, m ≥ 0 such that π n (w) = π m (v). (H2) Let φ ∈ F π if F π = ∅, otherwise we fix any φ ∈ T. We call the triplet (T, π, φ) a tree with a reference point.
We justify this definition as follows.
(2) (H2) assures that (T, A) is connected. Let (w i ) n i=0 be a simple path from w to v, that is, w 0 = w, w n = v, (w i−1 , w i ) ∈ A for any 1 ≤ i ≤ n and w i = w j for any i = j. Since (w i ) n i=0 is simple, there exist 0 ≤ i * ≤ n such that π(w i ) = w i+1 for any i < i * and w i = π(w i+1 ) for any i ≥ i * . This shows π i * (w) = π n−i * (v), and it also follows that w b(w,v) = w n−b(v,w) because of (1). Therefore i * = b(w, v) and n = b(w, v) + b(v, w) by simplicity, so the simple path from w to v is unique.
Throughout this section, T = (T, π, φ) is a tree with a reference point. We Definition 3.3 (Partition). Let (Y, ρ) be a (σ-compact) metric space without isolated points, and C(Y, ρ) be a set of all nonempty compact subsets of (Y, ρ) except single points. We say K : T → C(Y, ρ) is a partition of Y (parametrized by T ,) if the following conditions hold.
• w∈(T )0 K(w) = Y and for any w ∈ T, (P1) • For any sequence (w k ) k∈Z with π(w k+1 ) = w k for any k ∈ Z, ∩ k∈Z K(w k ) is a single point.
Hereafter, we write K w instead of K(w) for simplicity. In [13, Definition 2.2.1], the following condition is also included in the definition of partition.
• For any w ∈ T, K w has no isolated points.
Proof. Assume x ∈ K w be an isolated point of K w for some w ∈ T. By (P1), we can find (w k ) k∈Z with π(w k+1 ) = w k and x ∈ K w k for any k ∈ Z, and w 0 = w. Since x is an isolated point of K w k ⊂ K w0 for any k ≥ 0 and K w k is not a single point, (K w k l \{x}) k≥0 is a decreasing sequence of nonempty compact sets.
Definition 3.6 (Basic framework). Assume sup w∈T #π −1 (w) < ∞ and let K be a partition of (Y, ρ) such that for any w ∈ T, K w \ v∈(T ) [w] :v =w K v = ∅, and there exists an open set U w ⊃ K w with #{v ∈ (T ) [w] | U w ∩ K v = ∅} < ∞. We say ρ satisfies basic framework with respect to K if the following conditions hold.
The main result of [13] is the following.