We show that the sublinear bound of the bad Green’s functions implies explicit logarithmic bounds of moments for long range operators in arbitrary dimension.

In this paper, we are interested in the quantum dynamics of long range operators on the lattice Zd. For a self-adjoint operator H on 2(Zd),ϕ2(Zd) and p > 0, let |XH|ϕp(t) be the pth moment of the position operator,

|XH|ϕp(t)=nZd|n|p|(eitHϕ,δn)|2,
(1)

and |X̃H|ϕp(T) be the time-averaged pth moment of the position operator,

|X̃H|ϕp(T)=2T0e2t/TnZd|n|p|(eitHϕ,δn)|2dt.
(2)

The moments |X̃H|ϕp(T) and |XH|ϕp(t) characterize how fast does eitHϕ spread out, which are closely related to the spectral measure μϕ. For example, dynamical localization, namely, for any ϕ, |XH|ϕp(t) is uniformly bounded, implies that only the pure point spectrum of H1 and continuity (with respect to the Hausdorff measure) of the spectral measure μϕ leads to a power law lower bound of |X̃|ϕp(T) (see Ref. 2 and references therein).

For the Anderson model with large disorder or the spectral edges (for the one dimensional case, it holds for the full spectrum), Anderson/dynamical localization holds.3–26 For other random models, localization/delocalization has been extensively studied as well.27–34 It has been conjectured that the Anderson model in any dimension d ≥ 3 has the localization (pure point spectrum)–extended state (absolutely continuous spectrum) transition, which is still quite open. The transitions of moments are usually easier to study.35–40 For the random operators, the work of Klein has been crucial in establishing the current state of the art.

Unlike random operators, spectral types (singular continuous spectrum or pure point spectrum) of one-dimensional quasi-periodic Schrödinger operators in the positive Lyapunov exponent regime depend on the arithmetics of frequencies and phases.41–47 The estimates on moments are often more stable, namely, less sensitive to arithmetics of frequencies and phases. Study in this line expects to obtain asymptotics of moments for all phases under suitable conditions on frequencies48–53 (restrictions on frequencies are necessary due to the work of Jitomirskaya and Zhang54). In this paper, we will study upper bounds of moments of long range operators, particularly when H is a quasi-periodic operator (defined on the lattice Zd and driven by base dynamics on the torus Tb).

For one-dimensional (d = 1) quasi-periodic Schrödinger operators with the shift dynamics on the torus T, the celebrated work of Damanik–Tcheremchantsev implies that when the Lyapunov exponents are positive (in their setting, the potential is a trigonometric function), the moments have sub-polynomial growth in time.48,49 Jitomirskaya and Mavi51 improved their results to rough potentials. Han and Jitomirskaya50 generalized earlier works to the torus in arbitrary dimension with more general base dynamics. In Ref. 52, Jitomirskaya and Powell combined techniques of Damanik–Tcheremchantsev48 with estimates developed from the proof of Anderson localization for quasi-periodic Schrödinger operators and obtained a logarithmic bound of moments (related to an earlier work of Landrigan–Powell about logarithmic dimensions of spectral measures). In all their work, the transfer matrices and an idea from Jitomirskaya–Last developed to study spectral dimensions55 play crucial roles.

Recently, Jitomirskaya and Liu56 introduced a new approach-Green’s function estimates and show the sub-polynomial growth of moments (by modification, their arguments could lead to logarithmic bounds), which works for long range operators as well.

All the aforementioned work focuses on the lattice Z. Very recently, Shamis and Sodin53 developed an approach to establish power law logarithmic bounds of moments based on large deviation estimates of Green’s functions in arbitrary dimension. Using the author’s earlier results on large deviation theorems (LDTs) of Green’s functions,57 they obtained power law logarithmic bounds for various long range operators on the lattice Zd.

In the present work, we introduce a different approach to study the power law logarithmic bounds of moments. Our idea is inspired by the localization proof of quasi-periodic Schrödinger operators developed by Bourgain and his collaborators58–62 (see a recent survey63 for more details) and a generalization by the author.57 Research in this direction starts with the work of Bourgain and Goldstein,60 where the authors studied the one-dimensional quasi-periodic Schrödinger operators. With the development in the subsequent work by Bourgain, Goldstein, and Schlag,58–61,64,65 it becomes a robust approach to study many spectral problems of quasi-periodic operators. Recently, Jitomirskaya, Liu, and Shi66 extended (also streamlined Bourgain’s proof) Bourgain’s results in Ref. 59 to arbitrary dimension of frequencies, and the author proved a quantitative and non-self-adjoint version of the work in Ref. 61 in arbitrary lattice dimension.57 Thanks to all previous works, such as Cartan’s estimates and techniques from semi-algebraic sets, the localization proof boils down to establish a sublinear bound (a discrepancy problem) of the bad Green’s functions. Our main result shows that the sublinear bound immediately implies explicit power law logarithmic bounds of moments.

Finally, we want to compare logarithmic bounds obtained in this paper and by Jitomirskaya–Powell52 and Shamis–Sodin.53 Assume that the sublinear bound is N1−δ [see (13) for the precise definition]. Roughly speaking (see Corollary 2.3), our main theorem says that the rate of power law logarithmic bounds is 1δ. The rate of power law logarithmic bounds in Ref. 53 by Shamis–Sodin comes from large deviation estimates of Green’s functions. That says the bounds in the present paper and in Ref. 53 are building on two different assumptions. We remark that the currently available large deviation estimates for multi-dimensional operators rely on the sublinear bounds, multi-scale analysis, and Cartan’s estimates. Since multi-scale analysis and Cartan’s estimates bring extra dimension loss (see Remark 10 in Ref. 57 for the details of dimension loss), our bounds for multi-dimensional operators are usually better than those obtained in Ref. 53. See item 2 in Remark 4 and item 2 in Remark 6. The results in Ref. 52 only work for Schrödinger operators (not long range operators) on the lattice Z since their arguments are based on transfer matrices. As mentioned in Ref. 52 (Remark 6), they obtain the (implicit) rate C(b)δ, where C(b) is a constant depending on the dimension of the torus. Our rate shows that C(b) can be 1.

Let H be a long range operator acting on u={un}nZd in the following form:

(Hu)n=nZdH(n,n)un.
(3)

Assume that H satisfies the following:

  • for any n,nZd,
    |H(n,n)|C1ec1|nn|,C1>0,c1>0,
    (4)
    where |n|max1id|ni| for n=(n1,n2,,nd)Zd;
  • for any n,nZd,
    H(n,n)=H(n,n)̄.
    (5)

For d = 1, the elementary region of size N centered at 0 is given by

QN=[N,N].

For d ≥ 2, denote by QN an elementary region of size N centered at 0, which is one of the following regions:

QN=[N,N]d

or

QN=[N,N]d\{nZd:niςi0,1id},

where for i = 1, 2, …, d, ςi ∈ { < , >,∅} and at least two ςi are not ∅.

Denote by EN0 the set of all elementary regions of size N centered at 0.

Let

EN{n+QN:nZd,QNEN0}.

We call elements in EN elementary regions.

Let RΛ be the operator of restriction (projection) to ΛZd. Define the Green’s function by

GΛ(z)=(RΛ(HzI)RΛ)1.
(6)

Set G(z) = (HzI)−1. Clearly, both GΛ(z) and G(z) are always well defined for zC+{zC:Iz>0}. Sometimes, we drop the dependence on z for simplicity.

We say an elementary region ΛEN is in class G (Good) if

|GΛ(n,n)|ec2|nn| for |nn|N10,
(7)

where 0 < c2c1.

Since the self-adjoint operator H given by (3) is bounded, there exists a large K > 0 such that σ(H) ⊂ [−K + 1, K − 1].

For ΛZd, denote by Λ its boundary.

Definition 2.1.
Fix ς ∈ (0, 1). Let Λ0EN be an elementary region. Given ξ with 0 < ξ < 1, we say Λ0 satisfies the sublinear bound property with the parameter ξ if for any family F of pairwise disjoint elementary regions in Λ0 with size M = ⌊Nξ⌋,
#{ΛF:Λ is not in class G}NςNξ.
(8)

Theorem 2.2.
Suppose there existϵ0 > 0 andN0 > 0 such that the following is true. Letz = E + with |E| ≤ Kand 0 < ϵϵ0. For anynZdwith |n| ≥ N0, there existsΛ0ENsuch that|n|100N|n|10,n ∈ Λ0,dist (n,Λ0)N5, and Λ0satisfies the sublinear bound property with the parameterξ. Then, for anyϕwith compact support and anyɛ > 0, there existsT0 > 0 (depending ond, p, ϕ, K,ς, ξ, ϵ0,c1,c2,C1,N0, andɛ) such that for anyTT0,
|X̃H|ϕp(T)(lnT)pξ+ε,
(9)
and for anytT0
|XH|ϕp(t)(lnt)pξ+ε.
(10)

Fixed 0 < σ < 1, we say an elementary region ΛEN is in class SGN (strongly good with size N) if

GΛeNσ
(11)

and

|GΛ(n,n)|ec2|nn| for |nn|N10,
(12)

where 0 < c2c1.

Corollary 2.3.
DefineBN,N1as
BN,N1={n[N,N]d: there exists QN1EN10 such that n+QN1SGN1}
Assume that there existsϵ0 > 0 such that for anyz = E + with |E| ≤ Kand 0 < ϵϵ0and arbitrarily smallɛ > 0,
#BN,NεN1δ when NN0.
(13)
(N0may depend onɛ). Then, for anyϕwith compact support and anyɛ > 0, there existsT0 > 0 (depending ond, p, ϕ, K,σ, δ, ϵ0,c1,c2,C1,N0, andɛ) such that for anyTT0,
|X̃H|ϕp(T)(lnT)pδ+ε
(14)
and for anytT0,
|XH|ϕp(t)(lnt)pδ+ε.
(15)

Remark 1.

In applications, ς = 1 − ɛ with arbitrarily small ɛ > 0. Then, the upper bound in (8) equals N1−ξɛ. Both N1−ξɛ and N1−δ in (13) are referred to as the sublinear bound property of (bad) Green’s functions.

Let us first recall some notations from Ref. 57.

The width of a subset ΛZd is defined by maximum MN such that for any n ∈ Λ, there exists M̂EM such that

nM̂Λ

and

 dist (n,Λ\M̂)M/2.

A generalized elementary region is defined to be a subset ΛZd of the form

ΛR\(R+y),

where yZd is arbitrary and R is a rectangle,

R={n=(n1,n2,,nd)Zd:|n1n1|M1,,|ndnd|Md}.

For ΛZd, denote by diam(Λ) = supn,n′∈Λ |nn′| its diameter.

Denote by RN all generalized elementary regions with diameter less than or equal to N. Denote by RNM all generalized elementary regions in RN with width larger than or equal to M.

Let us collect and define some notations, which will be used throughout the proof:

  1. σ(H) ⊂ [−K + 1, K − 1].

  2. z = E + ϵi, |E| ≤ K, and 0 < ϵϵ0.

  3. ϵ=1T.

  4. GΛ=GΛ(z)=(RΛ(Hz)RΛ)1=(RΛ(HEϵi)RΛ)1.

  5. G = (HzI)−1.

  6. supp ϕ[K1,K1]d.

  7. C(c) is a large (small) constant.

We remark that K1 is independent of K.

Theorem 3.1.
Letς, σ, ξ ∈ (0, 1). LetΛ0ENbe an elementary region with the property that for all Λ ⊂ Λ0,ΛRLNξ, withNξL ≤ 2 N, the Green’s functionGΛsatisfies
GΛeLσ.
(16)
Assume thatc245c1and for any familyFof pairwise disjoint elementary regions in Λ0with sizeM = ⌊Nξ,
#{ΛF:Λ is not in class G}NςNξ.
(17)
Then, for largeN(depending onC1, c1, ς, σ, ξand the lower bound ofc2),
|GΛ0(n,n)|e(c2Nϑ)|nn| for |nn|N10,
(18)
whereϑ = ϑ (σ, ξ, ς) > 0.

Remark 2.

Theorem 3.1 in the settings of Schrödinger operators (Δ + V) on Z2 was proved in Ref. 61. The author generalized their proof to the settings in Theorem 3.1 in Ref. 57.

Assume that Λ1 and Λ2 are two disjoint subsets of Zd.

Let Λ = Λ1 ∪ Λ2. Suppose that RΛARΛ and RΛiARΛi, i = 1, 2 are invertible. Then,

GΛ=GΛ1+GΛ2(GΛ1+GΛ2)(HΛHΛ1HΛ2)GΛ.

If n ∈ Λ2 and m ∈ Λ, we have

|GΛ(m,n)||GΛ2(m,n)|χΛ2(n)+nΛ1,nΛ2ec1|nn||GΛ(m,n)GΛ2(n,n)|.
(19)

Lemma 3.2.
Fixed anyσ ∈ (0, 1), letN(log1ϵ)1ξσ. Assume thatΛ0EN,n ∈ Λ0,dist (n,Λ0)N5, and Λ0satisfies the sublinear bound property with the parameterξ. Then, for anyjwith |j| ≤ K1,
|((HEϵi)1δj,δn)|Cϵ2ec|n|.
(20)

Proof.
Since H is self-adjoint, one has that for any ΛZd, dist(σ(HΛ), z) ≥ϵ, and hence,
GΛϵ1.
Then, for any Λ ⊂ Λ0, ΛRLNξ, with NξL ≤ 2 N, one has that the Green’s function GΛ satisfies
GΛϵ1eLσ,
(21)
where the second inequality in (21) holds by the assumption. By Theorem 3.1, one has that
|GΛ0(n,n)|ec|nn| for |nn|N10.
(22)
By (19) (applying Λ2 = Λ0 and Λ=Zd) and using that j ∉ Λ0, one has that
|G(j,n)|CnΛ\Λ0,nΛ0ec1|nn||G(j,n)GΛ0(n,n)|.
(23)
If dist (n,Λ0)N1000, then |nn|N1000, and hence, ec1|nn|ecN.

If dist (n,Λ0)N1000, then |nn|N6. By (22), one has that |GΛ0(n,n)|ecN.

Therefore, one concludes that
|((HEϵi)1δj,δn)|CnΛ\Λ0,nΛ0dist (n,Λ0)N1000ec1|nn||G(j,n)GΛ0(n,n)|+CnΛ\Λ0,nΛ0dist (n,Λ0)<N1000ec1|nn||G(j,n)GΛ0(n,n)|Cϵ2nΛ\Λ0,nΛ0|nn|N1000ec|nn|+Cϵ1ecNnΛ\Λ0,nΛ0ec|nn|Cϵ2ecNCϵ2ec|n|.
(24)

The following lemma follows from Lemma 2 in Ref. 49:

Lemma 3.3.
Assume thatϕhas compact support. Then, for any large |n|,
|(eitHϕ,δn)|2ec|n|+1tKKHEit1ϕ,δn2dE.
(25)

Proof of Theorem 2.2.
Fix any σ in (0,1). For any j with |j| ≤ K1, let
a(j,n,T)=2T0e2t/T|(eitHδj,δn)|2dt,
(26)
and then,
|X̃H|ϕp(T)C|j|K1nZd|n|pa(j,n,T).
(27)
By the Parseval formula,
a(j,n,T)=1TπHEiT1δj,δn2dE.
(28)
Recall that σ(H) ⊂ [−K + 1, K − 1]. For any E ∈ (−, −K) ∪ (K, ), dist(E+iT,spec(H))1. The well-known Combes–Thomas estimate (e.g., A.11 in Ref. 67) yields
HEiT1δj,δnCec|n|.
(29)
By (28) and (29), one has that
a(j,n,T)Cec|n|+1TπKKHEiT1δj,δn2dE.
(30)
Rewrite (27),
|X̃H|ϕp(T)C|j|K1nZd|n|pa(j,n,T)C|j|K1nZd|n|(logT)1ξσ|n|pa(j,n,T)+C|j|K1nZd|n|(logT)1ξσ|n|pa(j,n,T).
(31)
By Lemma 3.2 and (30), one has that
nZd|n|(logT)1ξσ|n|p|a(j,n,T)nZd|n|(logT)1ξσ|n|pT3ec|n|T3ec(logT)1ξσC.
(32)
Direct computations imply that
CnZd|n|(logT)1ξσ|n|pa(j,n,T)C(logT)pξσnZda(j,n,T)=C(logT)pξσ,
(33)
where (33) holds by the fact that nZda(j,n,T)=1.
By (31)–(33), we conclude that
|X̃H|ϕp(T)C(logT)pξσ.
(34)
By letting σ → 1, we complete the proof of (9).

Replacing (28) with (25) and repeating the proof of (34), we have (10).□

Proof Corollary 2.3.

Let ξ = δ − 2 C(d)ɛ. Let F be any pairwise disjoint elementary regions in [−N, N]d with size ⌊Nξ⌋. By (13), one has that (N1 = ⌊Nɛ⌋) there are at most N1C(d)N1δ=N1δ+C(d)ε elementary regions in F will intersect elementary regions not in SGN1. By resolvent identity arguments (e.g., Ref. 57, Theorem 6.1), any elementary region in [−N, N]d with size ⌊Nξ⌋, without intersecting any non-SGN1 elementary regions, satisfies (7). It implies that (17) is true for ς = 1 − ɛ. Applying Theorem 2.2 and letting ɛ → 0, we obtain Corollary 2.3.□

In this section, the long range operator S on 2(Zd) satisfies the following:

  • for any n,nZd,
    |S(n,n)|C1ec1|nn|,C1>0,c1>0;
    (35)
  • for any n,nZd,
    S(n,n)=S(n,n)̄;
    (36)
  • for any nZd,nZd and kZd,
    S(n+k,n+k)=S(n,n).
    (37)

Let f be a function from Zd×Tb to Tb. Assume that for any m1,m2,,mdZd and n1,n2,,ndZd,

f(m1+n1,m2+n2,,md+nd,x)=f(m1,m2,,md,f(n1,n2,,nd,x)).

Sometimes, we write down fn(x) for f(n, x) for convenience, where nZd and xTb.

Define a family of operators Hx on 2(Zd),

Hx=S+v(f(n,x))δnn,
(38)

where v is a real analytic function on Tb.

In Ref. 57, the author obtained the large deviation estimates for Green’s functions of long range operators in arbitrary dimension, which generalized results in Ref. 61. As a result, he proved the large deviation theorem of Green’s functions and sublinear bound (13) for various models. Therefore, we can apply Corollary 2.3 to obtain logarithmic bounds for many operators studied in Ref. 57. For simplicity, we only discuss applications of Corollary 2.3 to several cases. In this section, estimates on δ in (13) can be found in Ref. 57.

For readers’ convenience, we will provide the sketch of the proof.

Let x1,x2.,xN0,1b and S0,1b. Let A(S;{xj}j=1N) be the number of xj (1 ≤ jN) such that xjS. Let DN(f) be the discrepancy of the sequence {f(n,x)}n=1N,

DN(f)=supxTbsupSCA(S;{f(n,x)}n=1N)NLeb(S),
(39)

where C is the family of all intervals in 0,1b, namely, S has the form of

S=[ϱ1,β1]×[ϱ2,β2]××[ϱb,βb],

with 0 ≤ ϱk < βk < 1, k = 1, 2, …, b.

Let ζ, σ ∈ (0, 1). We say the Green’s function of an operator Hx satisfies property LDT (large deviation theorem) in complexified energies (sometimes just say LDT for short) if there exists ϵ0 > 0 and N0 > 0 such that for any NN0, there exists a subset XNTb such that

Leb(XN)eNζ
(40)

and for any xXNmodZb and QNEN0,

GQN(z)eNσ,|GQN(z)(n,n)|ec2|nn| for |nn|N10,

where z = E + , with E ∈ [−K, K] [recall that σ(Hx) ⊂ (−K + 1, K − 1)] and 0 < ϵϵ0.

Theorem 4.1.
Letd = 1. Assume that for anyNN0, the discrepancy
DN(f)Nδ1.
(41)
Assume thatHxsatisfies LDT. Then, for anyϕwith compact support and anyɛ > 0, there existsT0 > 0 (depending onp, N0, ϕ, S, v, fandɛ) such that for anyTT0,
|X̃Hx|ϕp(T)(lnT)pδ1+ε
(42)
and for anytT0,
|XHx|ϕp(t)(lnt)pδ1+ε.
(43)

Proof.

By approximating the analytic function with trigonometric polynomials and using Taylor expansions and standard perturbation arguments, we can assume that XN [given by (40)] is a semi-algebraic set with degree less than NC. By Ref. 58, Corollary 9.7 (also see Ref. 57, Theorem 8.7), one has that (13) holds for any δ<δ1b. Now, Theorem 4.1 follows from Corollary 2.3.□

Remark 3.

The largeness of T0 in this section does not depend on xTb.

Let α=(α1,α2,,αb)0,1b and

f(x)=x+αmodZb,xTb.
(44)

We say that α = (α1, α2, …, αb) satisfies the Diophantine condition DC(κ, τ) if

kατ|k|κ,kZb\{(0,0,,0)}.
(45)

Lemma 4.2
(Ref. 68).Assumeα ∈ DC(κ, τ). Letfbe given by(44). Then,
DN(f)C(b,κ,τ)N1κ(logN)2.

Denote by Δ the discrete Laplacian on 2(Z), that is, for {u(n)}2(Z),

(Δu)n=un+1+un1

Let Hx on 2(Z) be given by

Hx=Δ+v(fn(x))=Δ+v(x1+nα1,x2+nα2,,xb+nαb)δnn,
(46)

where n,nZ and v is real analytic on Tb.

Theorem 4.3.
Letα ∈ DC(κ, τ). LetHxbe given by(46). Assume the Lyapunov exponentL(E) is positive for allER. Then, for anyϕwith compact support and anyɛ > 0, there existsT0 > 0 (depending onp, ϕ, v, α, andɛ) such that for anyTT0,
|X̃Hx|ϕp(T)(lnT)bκp+ε
(47)
and for anytT0,
|XHx|ϕp(t)(lnt)bκp+ε.
(48)

Proof.

Since L(E) > 0 for any ER, by the continuity of Lyapunov exponents (e.g., Refs. 58, 69, or 70), one has that there exists ϵ0 > 0 such that L(E + ) > 0 for any E ∈ [−K, K] and 0 < ϵϵ0. This implies that Hx satisfies LDT (e.g., Ref. 58). Now, Theorem 4.3 follows from Theorem 4.1 and Lemma 4.2.□

Remark 4.

  1. Under a stronger Diophantine condition of frequencies α, the modulus of continuity and large deviation theorem of Lyapunov exponents were first established in Ref. 65 (also see Ref. 71 for a recent generalization).

  2. A larger power law logarithmic bound b3κ2p was established by Shamis–Sodin.53 

  3. An implicit bound C(b)κp [C(b) is a constant depending on b] was obtained in Ref. 52.

When the dimension of the torus is 1, we can improve Theorem 4.3.

Theorem 4.4.
Letd = b = 1. Letα ∈ DC(κ, τ). LetHxbe given by(46)(withb = 1). Assume that the Lyapunov exponentL(E) is positive for anyER. Then, for anyϕwith compact support and anyɛ > 0, there existsT0 > 0 (depending onp, ϕ, v, α, andɛ) such that for anyTT0,
|X̃Hx|ϕp(T)(lnT)p+ε
(49)
and for anytT0,
|XHx|ϕp(t)(lnt)p+ε.
(50)

Proof.

Similar to the Proof of Theorem 4.3, Hx satisfies LDT. Similar to the Proof of Theorem 4.1, we can assume that XN [given by (40)] is a semi-algebraic set with degree less than NC. Then, (13) holds for any δ < 1. Now, Theorem 4.4 follows from Corollary 2.3.□

Remark 5.

In Theorem 4.4, the arithmetic condition, namely, Diophantine condition, on frequencies α is necessary. For some non-Diophantine α, |X̃Hx|ϕp(T) could have the power law lower bound (|X̃Hx|ϕp(T)Tγ for some γ > 0).54 

Let

fn(x)=x+nα=x+n1α1+n2α2++ndαdmodZ,

where n=(n1,n2,,nd)Zd and xT. Let Hx on 2(Zd) be given by

Hx=λ1S+v(fn(x))δnn=λ1S+v(x+n1α1+n2α2++ndαd)δnn,
(51)

where v is a non-constant real analytic function on T.

Theorem 4.5.
Letα ∈ DC(κ, τ) andHxbe given by(51). Then, there existsλ0 = λ0(κ, τ, C1, c1, v) such that for anyλ > λ0, the following holds. For anyɛ > 0 and anyϕwith compact support, there existsT0 > 0 (depending onp, ϕ, S, v, αandɛ) such that for anyTT0,
|X̃Hx|ϕp(T)(lnT)p+ε
(52)
and for anytT0,
|XHx|ϕp(t)(lnt)p+ε.
(53)

Proof.

By Ref. 57, Theorem 3.11, Hx satisfies LDT. We can assume that XN [given by (40)] is a semi-algebraic set with degree less than NC. Then, (13) holds for any δ < 1. Now, Theorem 4.5 follows from Corollary 2.3.□

Assume v is real analytic on T2. Let

fn(x)=(x1+n1α1,x2+n2α2)modZ2,

where n=(n1,n2)Z2, α=(α1,α2)R2, and x=(x1,x2)T2. Let Hx on 2(Z2) be given by

Hx=λ1S+v(fn(x))δnn=λ1S+v(x1+n1α1,x2+n2α2)δnn.
(54)

Theorem 4.6.
LetHxbe given by(54). Supposevis non-constant on any line segment contained in0,12,α1 ∈ DC(κ, τ) andα2 ∈ DC(κ, τ) with1κ<1312. Then, there existsλ0 = λ0(κ, τ, c1, C1, v) such that for anyλ > λ0, the following holds. For anyϕwith compact support and anyɛ > 0, there existsT0 > 0 (depending onp, ϕ, S, v, α, andɛ) such that for anyTT0,
|X̃Hx|ϕp(T)(lnT)4p1312κ+ε
(55)
and for anytT0,
|XHx|ϕp(t)(lnt)4p1312κ+ε.
(56)

Proof.

Recall that under the assumption in Theorem 4.6, Hx satisfies LDT (see Ref. 57, Theorem 3.20). By Ref. 62, Theorem 5.1 (also item 1 of Remark 11 in Ref. 57), (13) holds for any δ with 0<δ<1343κ.□

Remark 6.

  1. The LDT of the operator (54) was established by the author in Ref. 57, which builds on an earlier work of Bourgain–Kachkovskiy.62 

  2. A larger power law logarithmic bound (41312κ)2p was established by Shamis–Sodin.53 

This paper is dedicated to Abel Klein on the occasion of his 75th birthday.

W. Liu was supported by Grant Nos. NSF DMS-2000345 and DMS-2052572.

The authors have no conflicts to disclose.

Wencai Liu: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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