Sharp Mei's lemma with different bases
aa r X i v : . [ m a t h . C A ] S e p SHARP MEI’S LEMMA WITH DIFFERENT BASES
THERESA C. ANDERSON AND BINGYANG HU
Abstract.
In this paper, we prove a sharp Mei’s Lemma with assuming thebases of the underlying general dyadic grids are different. As a byproduct, wespecify all the possible cases of adjacent general dyadic systems with differentbases. The proofs have connections with certain number-theoretic properties. Introduction
The purpose of this paper is to give an optimal description of the adjacency ofgeneral dyadic systems in R d with different bases. The study of describing dyadicsystems in a refined manner dates back to the work [6] of Conde Alonso, in which, heproved d + 1 is the optimal number of dyadic systems in R d to guarantee adjacency.However, Conde Alonso’s result only implies the existence of such a collection of d +1 dyadic systems. Our goal is to understand for a given collection of d +1 dyadicgrids (or more general, n -dyadic grids), what the necessary and sufficient conditionsare so that such a collection is adjacent. In our recent paper [4] joint with Jiang,Olson and Wei, we answered this question when d = 1, and later in [2] we extendedthis result to higher dimensions by studying the fundamental structures of d + 1 n -adic systems in R d . Note that in both [4] and [2], the bases of the given d + 1grids are the same.In this paper, we further generalize these results to the case when the bases ofthese d + 1 grids are different. Moreover, we are also able to specify all the possiblecases for adjacent systems with different bases. Let us begin with the definition of n -adic systems in R d , which is our main object of interest in this paper. Definition 1.1.
Given n ∈ N , n ≥
2, a collection G of left-closed and right-opencubes on R d (that is, a collection of cubes in R d of the form[ a , a + ℓ ) × · · · × [ a d , a d + ℓ ) , a i ∈ R , i = 1 , . . . , d, where ℓ > sidelength of such a cube) is called a general dyadic grid withbase n (or n -adic grid) if the following conditions are satisfied:(i). For any Q ∈ G , its sidelength ℓ ( Q ) is of the form n k , k ∈ Z ;(ii). Q ∩ R ∈ { Q, R, ∅} for any Q, R ∈ G ;(iii). For each fixed k ∈ Z , the cubes of a fixed sidelength n k form a partition of R d .Note that when n = 2, the above definition refers to the classical dyadic system in R d , which we denote by D .An important property for such a structure is the following optimal dyadic cov-ering theorem due to Conde [6], which is also known as the optimal Mei’s lemma . Date : September 28, 2020.The first author is funded by NSF DMS 1954407 in Analysis and Number Theory.
Theorem 1.2. [6, Theorem 1.1]
There exists d + 1 dyadic grids D , . . . , D d +1 (withbase ) of R d such that every Euclidean ball B (or every cube) is contained in somecube Q ∈ d +1 S i =1 D i satisfying that diam ( Q ) ≤ C d diam ( B ) . The number of dyadicsystems is optimal. The origin of the adjacency of dyadic systems is obscure but we believe thatcredit should be given to Okikiolu [18] and, for a somewhat weaker version, toChang, Wilson and Wolff [5]. Later in 2013, Hyt¨onen and Per´ez [13] proved thatMei’s lemma holds 2 d dyadic grids with the constant C d = 6, and in the sameyear, Conde Alonso [6] showed the optimal number of the dyadic systems neededis d + 1 but with a larger constant C d ≃ d . Moreover in 2014, Cruz-Uribe [9]gave a short proof of Mei’s lemma for 3 d dyadic grids with a better constant C d =3. We would also refer the reader for [15, 16, 19, 12] and the references therein for more detailed information about the development of this property. Theadjacency of dyadic systems are crucially used in harmonic analysis (for instance,by Lerner to prove the A theorem in [14], among many other recent papers on sparse domination ), functional analysis [7], [10], [16], [19] and measure theory [8].Theorem 1.2 motivates the following definition of the adjacency of a collectionof d + 1 general dyadic grids with different bases. Note that the adjacency we areconsidering in this note is more general than the one in [2] and [4]. Definition 1.3.
Given d + 1 general dyadic grids G , . . . , G d +1 , where the base of G i is n i , i = 1 , . . . , d + 1, we say they are adjacent if for any open cube Q ⊆ R d (orany ball), there exists i ∈ { , . . . , d + 1 } , and D ∈ G i , such that Q is comparable to D , in the sense that(1). Q ⊆ D ;(2). ℓ ( D ) ≤ Cℓ ( Q ), where the constant C only allows to depend on n , . . . , n d +1 and d , in particular, it is independent of the sidelength of the cubes Q and D .The new feature of the adjacency of general dyadic systems with different basescomes from the fact that the cubes from different grids living in different genera-tions start interacting with each other, in both small scale case (where the cubehas sidelength less than or equal to 1) and large scale case (where the cube hassidelength great than 1). This leads to the fact that some generations of the generaldyadic grids make a significant contribution to the adjacency, while some make nocontributions. This is quite different from the case considered in [4] and [2] wherethere is only one base; the adjacency there is decided by cubes from all generationsin different grids. Example . Here is an easy way to produce adjacent general dyadic systems withdifferent bases.To start with, we can take a known example of adjacent grids with the samebases (see, e.g., [6, Page 786–787]), and then change bases by deleting specificgenerations. For example, let D and D be two dyadic grids which are adjacenton R , then we define G := D and(1.1) G := [ i ∈ Z { I ∈ ( D ) i } . Here are some remarks for the above example. (1). It is easy to check that G and G are also adjacent on R , while G is ofbase 16;(2). This construction easily suggests to the following fact: the optimal numberthat is needed to guarantee the adjacency for grids with different bases isalso d + 1;(3). It turns out that this “changing bases trick” is the only possible case foradjacent systems with different bases (see, Theorem 1.14).Finally, note that not all generations in G make a contribution to the adjacency.Indeed, let us define G ′ := [ i ∈ Z { I ∈ ( D ) i } Note that by [4, Theorem 3.8], G ′ and G are adjacent on R (note that both G ′ and G are of base 16). This suggests that in the adjacent pair {G , G } , only the cubesin ( G ) i , i ∈ Z (that is, cubes only every four levels) contribute to the adjacency,while the cubes from other generations are redundant.Let us make the above phenomenon in a quantitative way. To do this, we firstintroduce the following auxiliary function: for any n, n ′ ∈ N with n, n ′ ≥ φ n ; n ′ : N → N is given by φ n ; n ′ ( j ) := (cid:22) j log n log n ′ (cid:23) . Note that φ n,n ( j ) = j for all j ∈ N and n ≥ far ” considered in[2] and [4], which is the first ingredient that we need for our main result. Such ageneralization is two-fold: first of all, we are able to define “ far number with respectto a finite collection of integers ” (see, Definition 1.5); second, we also describe the“ far pair of integer-valued functions with respect to a finite collection of integers ”(see, Definition 1.7). Now let us turn to some details. Definition 1.5.
Let N := { n , . . . , n L } be a collection of positive integers whereeach n ℓ ≥ , ℓ = 1 , . . . , L . Given any δ ∈ R and n, n ′ ∈ N with n, n ′ ≥
2, we say δ is a ( n, n ′ ) -far number with respect to N if there exists C > n ℓ ∈ N , there holds that(1.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ − k n φ nℓ ; n ( m ) − k ( n ′ ) φ nℓ ; n ′ ( m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ Cn mℓ , ∀ m ≥ , k , k ∈ Z , where C only depends on n , n ′ , δ , N , L and any dimension constants, but isindependent of m , k and k . Remark . (1). The concept of far numbers with respect to a set will be usedto deal with the small scale case in most of our applications later, and wewill only consider the case when n, n ′ ∈ N and L = d + 1, which is theoptimal number of general dyadic systems that needed to guarantee theadjacency in R d . Therefore, the constant C in (1.2) later will only dependon δ, N and any dimensional constants;(2). When N = { n } and n = n ′ = n , Definition 1.5 coincides with theclassical definition of n -far number, which was considered in [4] and [2].The concept of far numbers , where N = { } , was introduced by Mei [17] toprove that the one-parameter space BMO( T ), which is the space of bounded THERESA C. ANDERSON AND BINGYANG HU mean oscillation on the torus T , can be written as the intersection of twodyadic product BMO( T ) spaces, with equivalent norms. In 2013, Li, Pipherand Ward [16] generalized Mei’s result to multi-parameter case and a vastclass of function spaces via a more careful study of far numbers. For asystematic study of far numbers, we refer the interested reader to [4] formore details.Next, we define the “far pairs of integer-valued functions with respect to a finiteset”. Definition 1.7.
Let N and L be defined as above. Given any integer valuedfunctions L , L ′ : N → N and n, n ′ ∈ N with n, n ′ ≥
2, we say ( L , L ′ ) is a ( n, n ′ ) -farpair of integer-valued functions with respect to N if there exists a C ′ > J ∈ N sufficiently large, such that for any n ℓ ∈ N , any j ≥ J and any k , k ∈ Z , thereholds that(1.3) (cid:12)(cid:12)(cid:12) L ( φ n ℓ ,n ( j )) + k n φ nℓ ; n ( j ) − L ′ ( φ n ℓ ,n ′ ( j )) − k ( n ′ ) φ nℓ ; n ′ ( j ) (cid:12)(cid:12)(cid:12) ≥ C ′ n jℓ , where C ′ only depends on n, n ′ , L , L ′ , N , L, J and any dimensional constants, butindependent of j , k and k . Remark . The concept of far pairs of integer-valued functions with respect toa finite set will be used to deal with the large scale case in our application later,where again, we will only consider the case when n, n ′ ∈ N and L = d + 1.The second concept that we need is the representation of general dyadic systems .The setting is as follows.(1). δ ∈ R d , which can be interpreted as the “initial point” to build the grid;(2). n ∈ N with n ≥
2, which is the base of the grid;(3). An infinite matrix(1.4) ~ a := { ~a , . . . , ~a j , . . . } , where ~a j ∈ { , , . . . , n − } d , j ≥ location function associated to ~ a : L ~ a : N Z d , which is defined by L ~ a ( j ) := j − P i =0 n i ~a i , j ≥ ~ , j = 0 . Definition 1.9.
Let δ ∈ R d , n ≥ ~ a and L ~ a be defined as above.Let G ( n, δ, L ~ a ) be the collection of the following cubes:(1). For m ≥
0, the m -th generation of G ( δ, L ~ a ) is defined as G ( n, δ ) m := ( (cid:20) ( δ ) + k n m , ( δ ) + k + 1 n m (cid:19) × . . . × (cid:20) ( δ ) d + k d n m , ( δ ) d + k d + 1 n m (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ( k , . . . , k d ) ∈ Z d ) , where here and in the sequel, we use ( δ ) s to denote the s -th component ofa vector δ ∈ R d .Note that all the positive generations (that is, the collection of cubeswith sidelength less or equal to 1) are uniquely determined by the initialpoint δ , and hence the location function L ~ a does not make any contributionfor positive generations;(2). For m <
0, the m -th generation is defined as G ( n, δ, L ~ a ) m := ( (cid:20) ( δ ) + [ L ~ a ( − m )] + k n m , ( δ ) + [ L ~ a ( − m )] + k + 1 n m (cid:19) × . . . × (cid:20) ( δ ) d + [ L ~ a ( − m )] d + k d n m , ( δ ) d + [ L ~ a ( − m )] d + k d + 1 n m (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ( k , . . . , k d ) ∈ Z d ) . To this end, for each m ∈ N , we denote b [ G ( n, δ, L ~ a ) m ] as the collection of allthe boundaries of the cubes in G ( n, δ, L ~ a ) m . Remark .
1. The term δ + L ~ a ( − m ) in Definition 1.9 can be interpretedas the location of δ after choosing n -adic parents (with respect to the 0-thgeneration) m times;2. G ( n, δ, L ~ a ) is a n -adic grid; on the other hand, for any n -adic grid G , itcan be represented as G ( n, δ, L ~ a ), for some δ ∈ R n and infinite matrix ~ a defined in (1.4) (see, [2, Proposition 3.2 and Proposition 3.3]). Moreover,although the representation of a n -adic grid in general is not unique, theyare essentially the “same” from the view of adjacency (see, [4, Theorem 3.14]for both the real line case and [2, Corollary 2.5] for the higher dimensionalcase).We are ready to state our main result, which generalizes [2, Theorem 1.5]. Theorem 1.11.
Let G i := G ( n i , δ i , L ~ a i ) , i = 1 , . . . , d + 1 be a collection of gen-eral dyadic grids, where d, n i , δ i and ~ a i are defined as above. Let further, N := { n , . . . , n d +1 } . Then G , . . . , G d +1 are adjacent if and only if the following condi-tions hold: (1). For any ℓ , ℓ ∈ { , . . . , d + 1 } where ℓ = ℓ , and s ∈ { , . . . , d } , ( δ ℓ ) s − ( δ ℓ ) s is a ( n ℓ , n ℓ ) -far number with respect to N ; (2). For any ℓ , ℓ ∈ { , . . . , d + 1 } , ℓ = ℓ , and s ∈ { , . . . , d } , the pairs ofinteger valued function (cid:16)h L ~ a ℓ ( · ) i s , h L ~ a ℓ ( · ) i s (cid:17) is a ( n ℓ , n ℓ ) -far pair of integer-valued functions with respect to N .Remark .
1. Theorem 1.11 is sharp, in the sense that the number of thegeneral dyadic systems needed to guarantee the adjacency cannot be re-duced;
THERESA C. ANDERSON AND BINGYANG HU
2. Let us include some motivation for the auxiliary function φ . As we havepointed out earlier, the new feature for the general dyadic systems withdifferent bases is that adjacency is given by cubes with different sidelengthsfrom different grids. For example, in [2], the term we have for the secondcondition in our main result is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h L ~ a k ( j ) i s − h L ~ a k ( j ) i s n j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . However, if the general dyadic grids are allowed to have different bases, itis no longer correct to compare the location functions at the same “level” j (that is, ( − j )-th generation), for instance, consider a term like(1.5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h L ~ a k ( j ) i s − h L ~ a k ( j ) i s n jk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Heuristically, let us assume n k ≥ n k , and ~ a k is an infinite matrix suchthat h L ~ a k ( j ) i s ∼ n jk when j is large. However, 0 ≤ h L ~ a k ( j ) i s < n jk , which is negligible com-pared to the term h L ~ a k ( j ) i s . In other words, (1.5) suggests that the adja-cency for the large scale only depends on grids with a larger base, which isnot correct since we can always do the “changing base trick” as in (1.1) tomake the base as large as we want. A similar observation suggests that forthe small scale case , one has to extend the definition of “far with respectto a number” to “far with respect to a set” (see, Definition 1.5).These phenomena suggest that when the bases are different, it is morereasonable to explore how the cubes from different grids with comparablesize interact with each other, rather than the from the same generations.This motivates us to introduce φ to quantify such a phenomenon.To this end, we make a comment that another possible approach tostudy the geometry underlying Theorem 1.11 is to consider the fundamentalstructures of d + 1 general dyadic dyadic systems with different bases , whichwere introduced in [2] to study the adjacency of general dyadic grids withthe same base. It is not hard to see that when the bases are different, thesestructures make sense if and only if the cubes used to build these structuresfrom different grids are of comparable sizes.The following corollary is straightforward from the main result Theorem 1.11(see [2] for a more detailed explanation). Corollary 1.13.
Let G , . . . , G d +1 be defined as in Theorem 1.11. G , . . . , G d +1 isadjacent in R d if and only if the projection of any two of them onto any coordinateaxis is adjacent in R . A second application of our main result Theorem 1.11 is that the typical examplesprovided by the “changing bases trick” in Example 1.4 are actually the only possiblecases for adjacent general dyadic systems with different bases. More precisely, wehave the following result.
Theorem 1.14.
Let G , . . . , G d +1 be adjacent on R d . Then there exists an integer n ≥ , and s i ∈ N , s i ≥ , i = 1 , . . . , d + 1 , such that n i = n s i , i = 1 , . . . , d + 1 . Remark . Note that if such an n does not exist, then this means that log n i log n j is irrational for some i = j , whereas if n does exist, then log n i log n j is always rational.This is related to numbers normal to different bases, that is, log n log n is rational if andonly if every number that is normal to base n is also normal to base n [20]. Also,via work of Wu [21], log n log n being rational means that the null sets for the n -adicdoubling measures and n -adic doubling measures are equal. This in turn looselyrelates to our recent work [3].The structure of the paper is as follows: Section 2 is devoted to prove the mainresult Theorem 1.11. Moreover, we also show Theorem 1.11 is independent ofthe representation. While in Section 3, we prove Theorem 1.14 and connect thediscoveries therein with other recent work.2. Proof of Theorem 1.11
In this section, we prove our main result Theorem 1.11.2.1.
Necessity.
Suppose G = G ( n , δ , L ~ a ) , . . . , G d +1 = G ( n d +1 , δ d +1 , L ~ a d +1 ) areadjacent. We prove the result by contradiction.Assume condition (1) fails. This means we can take some ℓ , ℓ ∈ { , . . . , d + 1 } with ℓ = ℓ and s ∈ { , . . . , d } , such that for each N ≥
1, there exists some m ≥ K , K ∈ Z and n ℓ ∈ N , such that(2.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( δ ℓ ) s − ( δ ℓ ) s − K n φ nℓ ; nℓ ( m ) ℓ − K n φ nℓ ; nℓ ( m ) ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < N n mℓ . This implies the distance between the hyperplane ( ( x ) s = ( δ ℓ ) s − K n φnℓ ; nℓ m ) ℓ ) and the hyperplane ( ( x ) s = ( δ ℓ ) s + K n φnℓ ; nℓ m ) ℓ ) is less than N n mℓ .Observe that ( x ) s = ( δ ℓ ) s − K n φ nℓ ; nℓ ( m ) ℓ ⊂ b h ( G ℓ ) φ nℓ ; nℓ ( m ) i and ( x ) s = ( δ ℓ ) s + K n φ nℓ ; nℓ ( m ) ℓ ⊂ b h ( G ℓ ) φ nℓ ; nℓ ( m ) i . Therefore, the estimate (2.1) suggests that we can take two sufficiently close points,with the first one located on b h ( G ℓ ) φ nℓ ; nℓ ( m ) i , and the second one on b h ( G ℓ ) φ nℓ ; nℓ ( m ) i , THERESA C. ANDERSON AND BINGYANG HU respectively. More precisely, we may assume s = ℓ = ℓ = 1 and ℓ = 2 for sim-plicity. We consider the points p and p , which are given by intersection of d non-parallel hyperplanes as follows: p := (cid:26) ( x ) = ( δ ) − K n m (cid:27) ∩ { ( x ) = ( δ ) } ∩ · · · ∩ { ( x ) d = ( δ d +1 ) d } and p := ( ( x ) = ( δ ) + K n φ n n ( m )2 ) ∩ { ( x ) = ( δ ) } ∩ · · · ∩ { ( x ) d = ( δ d +1 ) d }} . Note that p and p enjoy the following properties:(a). p ∈ b [( G ) m ] ∩ (cid:18) d +1 T t =3 b [( G t ) ] (cid:19) and p ∈ b h ( G ) φ n n ( m ) i ∩ d +1 T t =3 b [( G t ) ];(b). dist( p , p ) < N n m .Note that the second property above allows us to choose an open cube Q of side-length N n m containing both p and p as interior points; while the first propertyasserts that if there is a dyadic cube D ∈ G ℓ , ℓ ∈ { , . . . , d + 1 } covering Q , thenthe sidelength of D is at least n m . Indeed, if D ∈ G , then since D ∩ b [( G ) m ] = ∅ , D has to belong to ( G ) m ′ for some m ′ > m , this suggests ℓ ( D ) > n m ; if D ∈ G ,then similarly we have ℓ ( D ) > n φ n n ( m )2 = 1 e log n · j m log n n k ≥ e log n · m log n n = 1 n m . (2.2)Finally, if D ∈ G i , i ∈ { , . . . d + 1 } , a similar argument suggests that ℓ ( D ) ≥ ℓ ( D ) > N · ℓ ( Q ) . This will contradict adjacency if we choose N sufficiently large.Next, we assume condition (2) fails. This means that there is some ℓ , ℓ ∈{ , . . . , d + 1 } with ℓ = ℓ and s ∈ { , . . . , d } , such that for each N ≥
1, thereexists some n ℓ ∈ N , j sufficiently large and K , K ∈ Z , such that(2.3) (cid:12)(cid:12)(cid:12)(cid:12)h L ~ a ℓ (cid:0) φ n ℓ ; n ℓ ( j ) (cid:1)i s + K n φ nℓ ; nℓ ( j ) ℓ − h L ~ a ℓ (cid:0) φ n ℓ ; n ℓ ( j ) (cid:1)i s − K n φ nℓ ; nℓ ( j ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) < n jℓ N . Without loss of generality, we consider the case when s = ℓ = ℓ = 1 and ℓ = 2.Therefore, (2.3) can be simplified as(2.4) (cid:12)(cid:12)(cid:12) [ L ~ a ( j )] + K n j − [ L ~ a ( φ n ; n ( j ))] − K n φ n n ( j )2 (cid:12)(cid:12)(cid:12) < n j N . This gives (cid:12)(cid:12)(cid:12) [ δ + L ~ a ( j )] + K n j − [ δ + L ~ a ( φ n ; n ( j ))] − K n φ n n ( j )2 (cid:12)(cid:12)(cid:12) < n j N , since we can pick j is sufficiently large. To this end, let us rewrite the above estimateas follows(2.5) (cid:12)(cid:12)(cid:12)h δ + L ~ a ( j ) + K n j ~e i − h δ + L ~ a ( φ n ; n ( j )) + K n φ n n ( j )2 ~e i (cid:12)(cid:12)(cid:12) < n j N , where for s ∈ { , . . . , d } , ~e s is the standard unit vector in R d with s -th entry being1. Similarly as (2.1), the estimate (2.5) can also be interpreted geometrically. In-deed, let q := n ( x ) = h δ + L ~ a ( j ) + K n j ~e i o ∩ d \ t =2 (cid:8) ( x ) t = (cid:2) δ t +1 + L ~ a t +1 (cid:0) φ n ; n t +1 ( j ) (cid:1)(cid:3) t (cid:9) and q := n ( x ) = h δ + L ~ a ( φ n ; n ( j )) + K n φ n n ( j )2 ~e i o ∩ d \ t =2 (cid:8) ( x ) t = (cid:2) δ t +1 + L ~ a t +1 (cid:0) φ n ; n t +1 ( j ) (cid:1)(cid:3) t (cid:9) . which satisfies the following properties:(a). q ∈ b h ( G ) − j i ∩ (cid:18) d +1 T k =3 b h ( G k ) − φ n nk ( j ) i(cid:19) and q ∈ d +1 T k =2 b h ( G k ) − φ n nk ( j ) i ;(b). dist( q , q ) < n j N .Note that the reason for us to consider the negative generations in property (a)above is that the cubes underlying the condition (2) is of side length greater than1 (namely, the large scale case ).To derive the desired contradiction, we again start from the second conditionabove by taking a cube Q containing both q and q , with ℓ ( Q ) = n j N . Now we let D ∈ G i for some i ∈ { , . . . , d + 1 } which contains Q . However, since Q ∩ b h ( G i ) − φ n ni ( j ) i = ∅ , (note that φ n ; n ( j ) = j for j large), it follows that ℓ ( D ) > n φ n ni ( j ) i = exp (cid:18) log n i · (cid:22) j log n log n i (cid:23)(cid:19) ≥ exp (cid:18) log n i · (cid:18) j log n log n i − (cid:19)(cid:19) = n j n i . (2.6)Therefore, we have ℓ ( D ) > N ℓ ( Q )2 n i , which is again a contradiction if we choose N sufficiently large.The proof for the necessity is complete. Sufficiency.
Suppose conditions (1) and (2) hold, that is,(1). For any ℓ , ℓ ∈ { , . . . , d + 1 } where ℓ = ℓ , and s ∈ { , . . . , d } , thereexists some constant C ( ℓ , ℓ , s ) >
0, such that for any m ≥
0, any n ℓ ∈ N and k , k ∈ Z , there holds(2.7) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( δ ℓ ) s − ( δ ℓ ) s − k n φ nℓ ; nℓ ( m ) ℓ − k n φ nℓ ; nℓ ( m ) ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C ( ℓ , ℓ , s ) n mℓ . (2). For any ℓ , ℓ ∈ { , . . . , d + 1 } , ℓ = ℓ , and s ∈ { , . . . , d } , there exists a C ′ ( ℓ , ℓ , s ) > J ( ℓ , ℓ , s ) ∈ N sufficiently large, such that for any n ℓ ∈ N , j > J ( ℓ , ℓ , s ) and any k , k ∈ Z , there holds that (cid:12)(cid:12)(cid:12)(cid:12) h L ~ a ℓ (cid:0) φ n ℓ ,n ℓ ( j ) (cid:1)i s + k n φ nℓ,nℓ ( j ) ℓ − h L ~ a ℓ (cid:0) φ n ℓ ,n ℓ ( j ) (cid:1)i s − k n φ nℓ,nℓ ( j ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) ≥ C ′ ( ℓ , ℓ , s ) · n jℓ . (2.8)Denote C := min ≤ ℓ = ℓ ≤ d +11 ≤ s ≤ d C ( ℓ , ℓ , s ) , C ′ := min ≤ ℓ = ℓ ≤ d +11 ≤ s ≤ d C ′ ( ℓ , ℓ , s ) . and(2.9) J := max ≤ ℓ = ℓ ≤ d +11 ≤ s ≤ d J ( ℓ , ℓ , s ) . It is clear that C , C ′ > G , . . . , G d +1 is adjacent on R d . Takesome C > < C < min (cid:26) C , C ′ (cid:27) . Let Q be any cube in R d and let m ∈ Z , such that(2.10) Cn m +11 ≤ ℓ ( Q ) < Cn m . Let us consider several cases.
Case I: m > . We have the following claim: there exists some ℓ ∈ { , . . . , d +1 } ,such that Q ∈ ( G ℓ ) φ n nℓ ( m ) . Proof of the claim:
We prove the claim by contradiction. If Q is not containedin any cubes from ( G ℓ ) φ n nℓ ( m ) for any ℓ ∈ { , . . . , d + 1 } , then for each ℓ ∈{ , . . . , d + 1 } , we can find a s ℓ ∈ { , . . . , d } , such that P s ℓ ( Q ) ∩ P s ℓ (cid:16) V ℓ,φ n nℓ ( m ) (cid:17) = ∅ , where P s is the orthogonal projection onto the s -th coordinate axis in R d for s ∈{ , . . . , d } , and for ℓ ∈ { , . . . , d + 1 } and m > V ℓ,m := (cid:26) δ ℓ + ~vn mℓ : ~v ∈ Z d (cid:27) is the collection of all the vertices of the cubes in ( G ℓ ) m . By pigeonholing, thereexists some ℓ , ℓ ∈ { , . . . , d + 1 } with ℓ = ℓ , but s ∗ := s ℓ = s ℓ , such that P s ∗ ( Q ) ∩ P s ∗ (cid:16) V ℓ ,φ n nℓ m (cid:17) , P s ∗ ( Q ) ∩ P s ∗ (cid:16) V ℓ ,φ n nℓ m (cid:17) = ∅ , which implies that there exists some K , K ∈ Z , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( δ ℓ ) s ∗ + K n φ n nℓ ( m ) ℓ − ( δ ℓ ) s ∗ − K n φ n nℓ ( m ) ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ ( Q ) < Cn m < C ( ℓ , ℓ , s ∗ ) n m , which contradicts (2.7). Case II: m ≤ − J , where J is defined in (2.9) . In this case, our goal is to showthat Q ∈ ( G k ) − φ n nℓ ( − m ) for some ℓ ∈ { , . . . , d + 1 } . We prove it by contradiction again. Following theargument in Case I above, we see that there exists some ℓ , ℓ ∈ { , . . . , d + 1 } with ℓ = ℓ and s ∗ ∈ { , . . . , d } , such that P s ∗ ( Q ) ∩ P s ∗ ( V ℓ , − φ n nℓ ( − m ) ) , P s ∗ ( Q ) ∩ P s ∗ ( V ℓ , − φ n nℓ ( − m ) ) = ∅ . This implies there exists some K , K ∈ Z , such that (cid:12)(cid:12)(cid:12)(cid:12) h δ ℓ + L ~ a ℓ (cid:0) − φ n ; n ℓ ( − m ) (cid:1)i s ∗ + K n φ n nℓ ( − m ) ℓ − h δ ℓ + L ~ a ℓ (cid:0) − φ n ; n ℓ ( − m ) (cid:1)i s ∗ − K n φ n nℓ ( − m ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) < Cn m . Note that since we can always choose J sufficiently large, we can indeed reduce theabove estimate to (cid:12)(cid:12)(cid:12)(cid:12) h L ~ a ℓ (cid:0) − φ n ; n ℓ ( − m ) (cid:1)i s ∗ + K n φ n nℓ ( − m ) ℓ − h L ~ a ℓ (cid:0) − φ n ; n ℓ ( − m ) (cid:1)i s ∗ − K n φ n nℓ ( − m ) ℓ (cid:12)(cid:12)(cid:12)(cid:12) < Cn m . This gives the desired contradiction to (2.8), since
C < C ′ ≤ C ′ ( ℓ ,ℓ ,s ∗ )2 . Case III: − J < m ≤ . Indeed, we can “pass” the third case to the second case,by taking a cube Q ′ containing Q with the sidelength is n J . Applying the secondcase to Q ′ , we find that there exists some D ∈ G k for some k ∈ { , . . . , d + 1 } , suchthat Q ′ ⊂ D and ℓ ( D ) ≤ C ℓ ( Q ′ ), which clearly implies(1). Q ⊂ D ;(2). ℓ ( D ) ≤ C n J ℓ ( Q ).The proof is complete. (cid:3) Finally, we make a remark that Theorem 1.11 is independent of the choice of therepresentation. Note that by Corollary 1.13, it suffices to consider the case when d = 1. Let G := G ( n , δ , L a ) and G := G ( n , δ , L b ), where a = ( a , . . . , a i , . . . )is an infinite sequence of integers where a i ∈ { , . . . , n − } and b can be definedsimilarly with n being replaced by n . Then Theorem 1.11 asserts that G and G are adjacent on R if and only if (1). there exists some C >
0, such that for any m ≥ k , k ∈ Z , thereholds(2.11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ − δ − k n φ nℓ ; n ( m )1 − k n φ nℓ ; n ( m )2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ Cn mℓ , ℓ = 1 , C ′ > J ∈ N sufficiently large, such that for any j > J and any k , k ∈ Z , there holds (cid:12)(cid:12)(cid:12) L a ( φ n ℓ ; n ( j )) + k n φ nℓ ; n ( j )1 − L b ( φ n ℓ ; n ( j )) − k n φ nℓ ; n ( j )2 (cid:12)(cid:12)(cid:12) ≥ C ′ n jℓ , ℓ = 1 , , in other words, (cid:12)(cid:12) δ + L a ( φ n ℓ ; n ( j )) + k n φ nℓ ; n ( j )1 − δ − L b ( φ n ℓ ; n ( j )) − k n φ nℓ ; n ( j )2 (cid:12)(cid:12) ≥ C ′ n jℓ , ℓ = 1 , . (2.12)The goal now is to show that the constants C and C ′ defined in above, respec-tively, are independent of the choice of the representation. Let G ( n , δ ′ , L a ′ ) and G ( n , δ ′ , L b ′ ) be some other representations of G and G , respectively. Note thefollowing two facts: δ ′ = δ + N , δ ′ = δ + N for some N , N ∈ N , and L a ′ ( j ) = L a ( j ) − N + d ( j ) n j , L b ′ ( j ) = L b ( j ) − N + d ( j ) n j , where d ( · ) , d ( · ) : N → Z ; the result follows from plugging these new quantitiesappropriately into the above quantities (2.12), and we leave the details to theinterested reader. 3. Proof of Theorem 1.14
The goal of this section is to prove Theorem 1.14, which states that if G , . . . , G d +1 are adjacent on R d , then their bases are closely related. Definition 3.1.
Let { n , . . . , n d +1 } be a collection of integers with n i ≥ , i =1 , . . . , d + 1. We call such a collection fine if there exists an adjacent collectionof general dyadic systems {G , . . . , G d +1 } on R d , where G i = G ( n i , δ i , L ~ a i ) , i =1 , . . . , d + 1.Note that Theorem 1.14 then asserts that if { n , . . . , n d +1 } is fine, then thereexists an integer n ≥
2, and s i ∈ N , s i ≥ , i = 1 , . . . , d + 1, such that n i = n s i , i = 1 , . . . , d + 1 . We begin with the case when d = 1. Proposition 3.2. If { n , n } is fine, then there exists some n ∈ N , n ≥ , suchthat n i = n s i , i = 1 , , for some integers s , s ≥ . Proof.
We prove by contradiction by assuming such an n does not exist, whichmeans that log n log n is irrational (see the related comments following the theoremstatement in the Introduction). We will crucially use this fact in the proof.Since { n , n } is fine, by Theorem 1.11, for any k , k ∈ Z , we know the estimate(2.11) holds, in particular, this implies that there exists some C >
0, such that forany m ≥ k , k ∈ Z ,(3.1) (cid:12)(cid:12)(cid:12) δn m n φ n n ( m )1 − k n m − k n φ n n ( m )1 (cid:12)(cid:12)(cid:12) ≥ Cn φ n n ( m )1 , where we denote δ := δ − δ . Let us consider the following set A ( n , n ; m ) := n k n m + k n φ n n ( m )1 : k , k ∈ Z o . Note that A ( n , n ; m ) = n k · gcd (cid:16) n m , n φ n n ( m )1 (cid:17) : k ∈ Z o , which suggests that we can find K , K ∈ Z such that(3.2) (cid:12)(cid:12)(cid:12) δn m n φ n n ( m )1 − K n m − K n φ n n ( m )1 (cid:12)(cid:12)(cid:12) ≤ gcd (cid:16) n m , n φ n n ( m )1 (cid:17) . To use this, let us write n = p a p a . . . p a L L and n = p a ′ p a ′ . . . p a ′ L L , where { p , . . . , p L } is a finite collection of primes and a ℓ , a ′ ℓ ∈ N , ℓ = 1 , . . . , L . Notethat the a i ’s and a ′ i ’s only depend on n and n , and independent of m .We now consider two different cases. Case I:
There exists infinitely many m ∈ N and some ℓ ∈ { , . . . , L } such that(1). a ℓ ≥ p a ℓ φ n n ( m ) ℓ ≥ p a ′ ℓ mℓ .We make a remark that here one may presumably assume that ℓ may depend onthe choice of m . However, since there are only finitely many choices for ℓ , bypigeonholing, simply pick a fixed ℓ ∈ { , . . . , L } and restrict attention to the sub-sequence of m which satisfy Case I .For simplicity, denote C I := { m ∈ N : m satisfies the assumption of Case I } . For
Case I , we note that contribution of the prime p ℓ to the term gcd (cid:16) n m , n φ n n ( m )1 (cid:17) is at most p a ′ ℓ mℓ . On the other hand, we define a function Ψ : N → Z given byΨ ( m ) = a ℓ φ n ; n ( m ) − a ′ ℓ m. Note that if m ∈ C I , then(a). Ψ ( m ) ≥ (cid:16) n m , n φ n n ( m )1 (cid:17) ≤ n φn n m )1 p Φ1( m ) ℓ . We have the following claim : Ψ ( m ) is unbounded. Proof of the claim:
Since,Ψ ( m ) = a ℓ · (cid:22) m log n log n (cid:23) − a ′ ℓ m we have(3.3) (cid:18) a ℓ log n log n − a ′ ℓ (cid:19) · m − a ℓ ≤ Ψ ( m ) ≤ (cid:18) a ℓ log n log n − a ′ ℓ (cid:19) · m however, since log n log n is irrational, using assertion (a), we can indeed conclude that a ℓ log n log n − a ′ ℓ > . This, together with the estimate (3.3), clearly implies the desired claim.By (a) and the claim above, there exists a m ∈ C I sufficiently large, such that(3.4) p − Ψ ( m ) ℓ < C , where we recall that C is defined in (3.1). Therefore, we have (cid:12)(cid:12)(cid:12) δn m n φ n n ( m )1 − K n m − K n φ n n ( m )1 (cid:12)(cid:12)(cid:12) ≤ gcd (cid:16) n m , n φ n n ( m )1 (cid:17) ≤ n φ n n ( m )1 p Ψ ( m ) ℓ < Cn φ n n ( m )1 , where in the first line above, we use (3.2), in the second to last estimate, we useassertion (b) above and in the last estimate, we use (3.4). This clearly contradicts(3.1). Case II:
Suppose
Case I fails. This means there exists infinitely many m ∈ N and some s ∈ { , . . . , L } (independent of the choice of m ), such that(i). a ′ s ≥ p a ′ s ms ≥ p a s φ n n ( m ) s .Similarly, we denote C II := { m ∈ N : m satisfies the assumption of Case II } . The proof for the second case is similar to the first one, and we only sketch it here.We define Ψ : N → Z by Ψ ( m ) := a ′ s m − a s φ n ; n ( m )Note that if m ∈ C II , then(c). Ψ ( m ) ≥ (cid:16) n m , n φ n n ( m )1 (cid:17) ≤ n m p Ψ2( m ) s . Similar as above, we can show that Ψ ( m ) is unbounded. Now to use the assertion(d) above, we rewrite (3.1) a little bit by(3.5) (cid:12)(cid:12)(cid:12) δn m n φ n n ( m )1 − k n m − k n φ n n ( m )1 (cid:12)(cid:12)(cid:12) ≥ e Cn m . This is because n φ n n ( m )1 ≥ n m n and we may let e C = Cn .Finally, let us take m ∈ C II sufficiently large, such that(3.6) p − Ψ ( m ) s ≤ e C (cid:3) Finally we turn to the proof of Theorem 1.14.
Proof of Theorem 1.14.
Theorem 1.14 is an easy consequence of Proposition 3.2 andCorollary 1.13, and we would like to leave the details to the interested reader. (cid:3)
Remark . Notice that (3.1) reduces to (cid:12)(cid:12)(cid:12) δn m n φ n n ( m )1 − k · gcd (cid:16) n φ n n ( m )1 , n m (cid:17)(cid:12)(cid:12)(cid:12) , for some k ∈ Z . If we assume adjacency, this means that δ · lcm (cid:16) n φ n n ( m )1 , n m (cid:17) − k = 0 , that is, δ = k lcm (cid:16) n φn n m )1 ,n m (cid:17) , a natural analogue of our previous work [4]. Sincewe now know that n and n are powers of the same base, this exactly replicatesthe fact that kn m are not n -far, which appeared in ([4], Corollary 2.10). References [1] T.C. Anderson, A framework for Calder´on-Zygmund Operators on Spaces of Homoge-neous Type. Ph.D. thesis, Brown University, 2015.[2] T.C. Anderson, B. Hu, On the general dyadic grids on R d , arXiv, preprint, 2020.[3] T.C. Anderson, B. Hu, Dyadic analysis meets number theory, arXiv, preprint, 2020.[4] T.C. Anderson, B. Hu, L. Jiang, C. Olson and Z. Wei. On the translates of generaldyadic systems on R . Mathematische Annalen , January, 2, 2020.[5] S.–Y. A. Chang, J. M. Wilson, and T. H. Wolff. Some weighted norm inequalitiesconcerning the Schr¨odinger operators.
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Theresa C. Anderson: Department of Mathematics, Purdue University, 150 N. Uni-versity St., W. Lafayette, IN 47907, U.S.A.
E-mail address : [email protected] Bingyang Hu: Department of Mathematics, Purdue University, 150 N. University St.,W. Lafayette, IN 47907, U.S.A.
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