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ONSET OF THE WAVE TURBULENCE DESCRIPTION OF THE LONGTIME

BEHAVIOR OF THE NONLINEAR SCHRODINGER EQUATION

T. BUCKMASTER, P. GERMAIN, Z. HANI, J. SHATAH

Abstract. Consider the nonlinear Schrodinger equation set on a d-dimensional torus, with datawhose Fourier coefficients have phases which are uniformly distributed and independent. We showthat, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-calledwave kinetic equation, predicted in wave turbulence theory, on a nontrivial time scale.

1. Introduction 21.1. The Kinetic Equation 21.2. Background 31.3. Why is this problem difficult? 41.4. Main result 51.5. Main Components of the proof 61.6. Organization of the paper 61.7. Notations 71.8. Acknowledgements 72. A more general result 82.1. The Equidistribution parameter ν 82.2. The Strichartz parameter θr 82.3. The approximation theorem 93. Formal derivation of the kinetic equation 104. Feynman diagrams: bounding the k first terms on average 134.1. Expansion of the solution in the data 134.2. Bound on the correlation 164.3. Cancellation of degenerate interactions 174.4. Estimate on non-degenerate interactions 185. Deterministic local well-posedness 225.1. The local well-posedness theorem 225.2. Rescaled Strichartz estimate 235.3. The a priori energy estimate 245.4. The a priori Zs estimate 245.5. The fixed point argument: proof of Theorem 5.1, Proposition 5.2 and Proposition 5.3 256. Improved integrability through randomization 277. Proof of the main theorem 288. Number theoretic results 308.1. Upper bounds 308.2. Equidistribution 32References 47

Contents1

2 T. BUCKMASTER, P. GERMAIN, Z. HANI, J. SHATAH

1. Introduction

1.1. The Kinetic Equation. The central theme in the theory of non-equilibrium statisticalphysics of interacting particles is the derivation of a kinetic equation that describes the distributionof particles in phase space. The main example here is Boltzmann’s kinetic theory: rather thanlooking at the individual trajectories of N -point particles following N−body Newtonian dynam-ics, Boltzmann derived a kinetic equation that described the effective dynamics of the distributionfunction in a certain large-particle limit (so-called the Boltzmann-Grad limit).

A parallel kinetic theory for waves, being as fundamental as particles, was proposed by physicists inthe past century. Much like the Boltzmann theory, the aim is to understand the effective behaviorand energy-dynamics of systems where many waves interact nonlinearly according to time-reversibledispersive or wave equations. The theory predicts that the macroscopic behavior of such nonlinearwave systems is described by a wave kinetic equation that gives the average distribution of energyamong the available wave numbers (frequencies). Of course, the shape of this kinetic equationdepends directly on the particular dispersive system/PDE that describes the reversible microscopicdynamics.

The aim of this work is to start the rigorous investigation of such passage from a reversible nonlineardispersive PDE to an irreversible kinetic equation that describes its effective dynamics. For this,we consider the cubic nonlinear Schrodinger equations on a generic torus of size L (with periodicboundary conditions) and with a parameter λ > 0 quantifying the importance of nonlinear effects(or equivalently via scaling, the size of the initial datum):i∂tu−

12π∆βu = −λ2|u|2u, x ∈ TdL = [0, L]d,

u(0, x) = u0(x).(NLS)

The spatial dimension is d ≥ 3. Here, and throughout the paper, we denote

∆β :=

d∑i=1

βi∂2i ,

where β := (β1, . . . , βd) ∈ [1, 2]d, and we denote ZdL := 1LZ

d, the dual space to TdL.

Typically in this theory, the initial data are randomly distributed in an appropriate fashion. Forus, we consider random initial data of the form

u0(x) =1

Ld

∑k∈ZdL

√φ(k)e2πi[k·x+ϑk(ω)], (1.1)

for some nice (smooth and localized) deterministic function φ : Rd → [0,∞). The phases ϑk(ω)are independent random variables, uniformly distributed on [0, 1]. Notice that the normalizationof the Fourier transform is chosen so that

‖u0‖L2 ∼ 1.

Filtering by the linear group and expanding in Fourier series, we write

u(t, x) =1

Ld

∑k∈ZdL

ak(t)e2πi[k·x+tQ(k)], where Q(k) :=

d∑i=1

βi(ki)2. (1.2)

ONSET OF WAVE TURBULENCE FOR NLS 3

The main conjecture of wave turbulence theory is that as L → ∞ (big box limit) and λ2

Ld→ 0

(weakly nonlinear limit), the quantity

ρLk (t) = E|ak(t)|2

converges to the solution of a kinetic equation. More precisely, it is conjectured that, as L → ∞and λ2

Ld→ 0, ρLk (t) ∼ ρ(t, k) as t→∞, where ρ : R× Rd → R+ satisfies the wave kinetic equation∂tρ = 1

τ T (ρ), where τ ∼(Ld

λ2

)2,

ρ(0, k) = φ(k).(WKE)

and furthermore

T (ρ)(k) =

˚

(Rd)r

δ(Σ)δ(Ω)ρ(k)3∏i=1

ρ(ki)

[1

ρ(k)+

3∑i=1

(−1)i

ρ(ki)

]dk1 . . . dk3

with Σ = Σ(k, k1, . . . , k3) = k +

3∑i=1

(−1)iki

Ω = Ω(k, k1, . . . , k3) = Q(k) +3∑i=1

(−1)iQ(ki).

1.2. Background. In the physics literature, the wave kinetic equation (WKE) was first derived byPeierls [33] in his investigations of solid state physics; it was discovered again by Hasselmann [24, 25]in his work on the energy spectrum of water waves. The subject was revived and systematicallyinvestigated by Zakharov and his collaborators [38], particularly after the discovery of special power-type stationary solutions for the kinetic equation that serve as analogs of the Kolmogorov spectraof hydrodynamic turbulence. These so-called Kolmogorov-Zakharov spectra predict steady statesof the corresponding microscopic system (possibly with forcing and dissipation at well-separatedextreme scales), where the energy cascades at a constant flux through the (intermediate) frequencyscales. Nowadays, wave turbulence is a vibrant area of research in nonlinear wave theory withimportant practical applications in several areas including oceanography and plasma physics tomention a few. We refer to [31, 32] for recent reviews.

The analysis of (WKE) is full of very interesting questions, see [17, 23, 34] for recent develop-ments, but we will focus here on the problem of its rigorous derivation. Several partial or heuristicderivations have been put forward for (WKE), or the closely related quantum Boltzmann equa-tions [1, 2, 3, 14, 10, 18, 28, 30, 36]. However, to the best of our knowledge, there is no rigorousmathematical statement on the derivation of (WKE) from random data. The closest attempt inthis direction is due to Lukkarinen and Spohn [29], who studied the large box limit for the discretenonlinear Schrodinger equation at statistical equilibrium (corresponding to a stationary solution to(WKE)).

In preparation for such a study, one can first try to understand the large box and weakly nonlinearlimit of (NLS) without assuming any randomness in the data. In the case where (NLS) is set on arational torus, it is possible to extract a governing equation by retaining only exact resonances [19,22, 21, 7]. Note that the limiting equation is then Hamiltonian and dictates the behavior of themicroscopic system (NLS on TdL) on the time-scales L2/λ2 (up to a log loss for d = 2) and forsufficiently small λ. It is worth mentioning that such a result is not possible if the equation is seton generic tori, since most of the exact resonances are destroyed there.


Finally, we point out that there are very few instances where the derivation of kinetic equationshas been done rigorously. The fundamental result of Lanford [27], later clarified in [20], dealswith the N -body Newtonian dynamics, from which emerges, in the Grad limit, the Boltzmannequation. This can be understood as a classical analog of the rigorous derivation on (WKE).Another instance of such success was the case of random linear Schrodinger operators (Anderson’smodel) [13, 15, 16, 35]. This can be understood as a linear analog of the problem of rigorouslyderiving (WKE).

1.3. Why is this problem difficult? There are several difficulties in proving the validity of(WKE) which we now enumerate:

(a) In physics textbooks, the wave kinetic equation is derived under the assumption that theindependence of the phases propagates for all time. This assumption cannot be verified forany nonlinear model. A way around this difficulty is to derive the equation for the profile ak,and expand the integral representation of ak in terms of the initial data. Such expansions canbe arranged in Feynman diagrams, and give the solution as a power series expansion in termsof the initial data. As a result, the terms in this expansion retain the statistical informationin the initial data, and thus when computing the expected value of |ak|2, many interactionsvanish. Assuming that such an expansion holds, one can control the errors in the derivationof the kinetic equation (WKE). These calculations are presented in Section 4.

(b) The wave kinetic equation induces an O(1) change on its initial configuration at a time scaleof τ . Thus, before we can even talk about the validity of this equation at this timescale, weneed to establish that the Feynman diagram expansions above of the solution to the (NLS)converge up to time τ . This is an existence time which is several orders of magnitude longerthan what is known. In fact, in dimensions d > 3, even the existence of solutions up to thistime is not known given the available techniques. This shortcoming is the main reason whyour argument cannot reach the kinetic time-scale τ , and we have to contend with a derivationover time-scales where the kinetic equation only affects a relatively small change on the initialdistribution, and as such coincides (up to negligible errors) with its first time-iterate.

Therefore, the most pressing issue is to improve the radius of convergence of such expansions,which can be also thought as the time T over which one has an expression of the solution asa power series expansion in terms of the initial data. For deterministic data, the best knowresults that give effective bounds in terms of L come from our previous work [7] which gives adescription of the solution up to times ∼ L2/λ2 (up to a logL loss for d = 2) and for λ 1.Such time scale would be way too short for our purposes, and as we shall see λ needs to bechosen ≥ 1 in order to observe the kinetic description.

To increase this radius of convergence, we have to rely on the randomness of the initial data.Roughly speaking, for a random field that is normalized to 1 in L2(TdL), its L∞ norm can be

heuristically bounded on average by L−d/2. Therefore, regarding the nonlinearity λ2|u|2u as anonlinear potential V u with V = λ2|u|2 and ‖V ‖L∞ . λ2Ld, one would hope that this shouldget a convergent expansion on an interval [0, T ] provided that Tλ2Ld 1, which amounts toT ≤

√τ . This is our target in this work.

This heuristic can be implemented by relying on Khinchine-type improvements to the Strichartznorms of a linear solution eit∆u0,ω with random initial data u0,ω. Similar improvements havebeen used to lower the regularity threshold for well-posedness of nonlinear dispersive PDE.Here, the aim is to elongate the interval of existence (and power-series convergence). Thiswould require propagating such estimates to the nonlinear solution, which in turn requires anappropriate bound on the Duhamel term that does not lose too much when applied on very


long time intervals. By duality, this is intimately related to having an improved deterministicStrichartz estimate for ‖eit∆ψ‖Lp([0,T ]×Td) with ψ ∈ L2(Td) over long time intervals [0, T ] that

does not lose too much in T . Here, the genericity of the (βi) is crucial, and allows (as was first

observed in [11]) to beat inescapable (and brutal for our purposes!) T 1/p growth that happenson the rational torus. Rather unfortunately, the available estimates here (including those in[11]) are not optimal for some ranges of the parameters λ and L, which is why our result inTheorem 1.1 below falls a bit short of the time-scale

√τ ∼ L3/λ (in d = 3).

(c) Having such an expansion in hand, one can write down the an expansion for ρLk (t) = E|ak(t)|2.To derive the kinetic equation in the large box limit, one has to prove equidistribution theoremsfor the quasi-resonances over a very fine scale (basically T−1 scales). Since T could be L2,such scales are much finer than the any equidistribution scale that can hold on the rationaltorus. Again, here the genericity of the (βi) is crucial. For this we use and extend a recentresult of Bourgain on pair correlation for irrational quadratic forms [5].

1.4. Main result. We shall state in Section 2 precise statements of our results in arbitrary dimen-sions d ≥ 3. Those statements depend on several parameters coming from the different parts of theproof, as we shall see below. For the purposes of this introductory section, we will be content withthe following slightly less general theorem:

Theorem 1.1. Consider the cubic (NLS) on the three-dimensional torus T3L. Assume that the

initial data are chosen randomly as in (1.1). There exists δ > 0 such that the following holds for L

sufficiently large and 1 ≤ λ ≤ L953 :

E|ak(t)|2 =t

τT (φ)(k) +O`∞

(L−δ

t

τ

), Lδ ≤ t ≤ T, (1.3)

where τ = 12

(Ld

λ2

)2and T ∼ L2.65

λ2.

A more general theorem dealing with other ranges of λ ≥ 1 will be given in Section 2. We notethat the right-hand side of (1.3) is nothing but the first time-iterate of the wave kinetic equation(WKE) with initial data φ (cf. (1.1)) which coincides (up to the error term in (1.3)) with the exactsolution of the (WKE) over times scales that are shorter than the kinetic time scale τ .

A few remarks are in order:

• The time T : Of course, the main shortcoming of this result is that the time-scale does notreach the kinetic timescale τ . As mentioned in the previous subsection, the main problemcomes from the convergence of the power series expansion of the solution for which thistime scale seems beyond the available tools. In dimensions d ≥ 5, it is not even knownthat solutions even exist up to this time-scale. The natural limitations of our argument isT ∼

√τ ∼ Ld/λ2, which corresponds to expressing the (NLS) solution as a convergent power

series that propagates the same space-time Strichartz bounds as those given by Khinchine’slemma for a linear solution with random initial data. While the latter linear estimates areoptimal themselves, it is possible that a different strategy (less reliant on Strichartz norms)could yield a longer time interval over which the power series converges. We are still short

of this limitation of√τ ∼ L3

λ2(if d = 3) mainly due to the sub-optimality of the available

long-time Strichartz estimates. However, the control goes well-beyond the long-time controlavailable in the rational case given by L2/λ2 in [7] (which only dealt with the case λ 1).

• Why λ ≥ 1? The short answer to this question is exact resonances: While working on anirrational torus eliminates the biggest bulk of the possible exact resonances, it leaves somethat would dominate in the regime λ 1. Such resonances are given by rectangles whose


sides are parallel to the coordinate axes. The main contribution to (1.3) comes from theso-called quasi-resonances, which are the interactions who resonance parameter is small butmight not necessarily vanish completely.

1.5. Main Components of the proof. The proof of the theorem can be split into three almostindependent components:

(1) Feynman Diagram Expansions. (Section 4) Here we derive the expansion of the non-linear solution as a power series (more precisely Taylor expansion) in terms of the initialdata. Roughly speaking, we write the Fourier modes of the nonlinear solution ak(t) (see(1.2)) as follows:

ak(t) =N∑n=0

Jn

(t, k; (a(0)

p )p∈ZdL

)+RN

(t, k, (ap(t))p∈ZdL

),

where Jn are (sums of) Feynman diagrams that yield terms of degree 2n+1 in the initial data

(a(0)p )p∈ZdL

, and RN is the remainder which depends on the nonlinear solution (ap(s))p∈ZdLfor 0 ≤ s ≤ t. While we do not study the convergence of the power series in this section, wedo obtain bounds on the terms of the form EJnJn′ which appear in the expansion of E|ak|2.The estimates we obtain for such interaction are good up to times

√τ r which is sufficient

given the restrictions on the time interval of convergence imposed by the second componentbelow. The estimates in this section rely on essentially sharp bounds on quasi-resonantsums of the form ∑

~k∈ZrdL

1(|~k| . 1)1(|Q(k)| ∼ 2−A) . 2−ALrd (1.4)

where 1(S) denotes the characteristic function of a set S and Q is an irrational quadraticform. Since A will be taken as large as T L2, such estimates belong to the realm onnumber theory and will be a consequence the third number theoretic component of thiswork below.

(2) Convergence of Expansion. (Section 5) This is where we construct the solution on thetime interval [0, Tr] via a contraction mapping, which would essentially give as a consequencethe convergence of the power series in the first component above. Here, we rely on a) theKhinchine-improvment to the space-time Strichartz bounds which are propagated to thenonlinear solution using b) the long-time Strichartz estimates on generic irrational toriproved in [11]. It is here that our estimates are most non-optimal, since we do not haveaccess to the full conjectured range of long-time Strichartz estimates.

(3) Number theoretic Component: Equidistribution of irrational quadratic forms.(Section 8) The purpose of this section is two-fold: the first is proving estimates on quasi-resonant sums like those in (1.4) for A as large as possible, and the second (and the moredelicate one) is to extract the exact asymptotic (with effective error bounds) of the mainpart of the expansion that converges to the kinetic equation collision kernel as L→∞.

Here we remark, that if Q is a rational form, then the largest A for which one could hopefor an estimate like (1.4) is A ∼ L2 which reflects the fact that a rational quadratic formcannot be equidistributed at scales smaller than L−2 (at the level of NLS, it would yield atime interval restriction of T . L2 for the rational torus). However, for generic irrationalquadratic forms, Q is actually equidistributed at much finer scales than L−2. Here, we


adapt a recent work of Bourgain [5] which will allow us to reach equidistribution scalesessentially up to L−d.

1.6. Organization of the paper. In Section 2, we give a precise statement of the theorem aboveafter introducing all the needed parameters. In the following Section 3, we give the formal derivationof the kinetic equation which will elucidate the emergence of the kinetic timescale τ , as well as theneeded relations between λ, L and the time T . In Section 4, we give the Feynman diagram expansionof the solution and obtain some estimates on the correlations of the terms of the series. In Section 5section, we run a contraction argument that will gaurantee the convergence of the Duhamel iteratesof the solution on the time interval [0, T ] by relying on the Khinchine-type improvements to theStrichartz estimates of the linear part of the solution. Such improvements are reviewed in Section6. Afterwards, we give a proof of the main theorems in Section 7. Finally, in Section 8, we gatherand proof all the number theoretic results that are used crucially in Section 4 and 7.

1.7. Notations. In addition to the notation introduced earlier for TdL = [0, L]d and ZdL = 1LZ

d, we

use standard notations. A function f on TdL and its Fourier transform f on ZdL are related by

f(x) =1

Ld

∑ZdL

fke2πik·x and fk =

ˆ

TdL

e−2πik·xf(x) dx.

Parseval’s theorem becomesˆTdL

|f(x)|2 dx

1/2

= ‖f‖L2(TdL) = ‖f‖`2L(ZdL) =

1

Ld

∑k∈ZdL

|fk|21/2

.

We adopt the following definition for weighted `p spaces: if p ≥ 1, s ∈ R, and b ∈ `p,

‖b‖`p,sL (ZdL) =

1

Ld

∑k∈ZdL

(〈k〉s|bk|)p1/p

.

Sobolev spaces Hs(Td) are then defined naturally by

‖f‖Hs(Td) = ‖〈k〉sf‖`2,s(ZdL).

For functions defined on Rd, we adopt the normalization

f(x) =

ˆ

Rd

e2πiξ·xf(ξ) dξ and f(ξ) =

ˆ

Rd

e−2πik·xf(x) dx.

Finally, we write C for a constant whose value may change from line to line; the notation A . Bmeans that there exists a constant C such that A ≤ CB. We also write

A . Lr+B

if for any ε > 0 there exists Cε such that A ≤ CεLr+εB.

1.8. Acknowledgements. We would like to thank Peter Sarnak for pointing us to the reference[5] and several illuminating discussions.


2. A more general result

Let us start by formulating the equations in Fourier space. Expanding the solutions as in (1.2), weobtain the following equation satisfied by (ak(t))k∈ZdL

:iak = −

(λLd

)2 ∑(k1,...,k3)∈(ZdL)3

k−k1+k2−k3=0

ak1ak2ak3e−2πitΩ(k,k1,k2,k3)

ak(0) = a(0)k =

√φ(k)eiϑk(ω),

(2.1)

where

Ω3(k, k1, k2, . . . , kr) = Q(k)−Q(k1) +Q(k2)−Q(k3) = Q(k) +3∑j=1

(−1)jQ(kj),

and ϑk(ω) are i.i.d. random variables that are uniformly distributed in [0, 2π].

2.1. The Equidistribution parameter ν. The resonance modulus Ω(k, k1, k2, k3) above is anirrational quadratic form. Such quadratic forms can be equidistributed at scales that are muchsmaller than the finest scale ∼ L−2 of rational forms. Roughly speaking, we will denote by ν theimprovement in this scale, that is Ω : Z3d

L → R will be equidistributed down to a scale L−2−ν (upto some Lε losses).

More precisely, ν is defined as the smallest non-negative real number such that: uniformly ink ∈ ZdL, |k| ≤ 1, for all ε > 0, there exists δ > 0 such that, for |a|, |b| < 1 with b− a ≥ L−2−ν−ε,∑a≤Ω(k,k1,k2,k3)≤b|k1|,|k2|,|k3|≤1k−k1+k2−k3=0

1 = (1 +O(L−δ))L2d

ˆ|k1|,|k2|,|k3|≤1

1a≤Ω(k,k1,k2,k3)≤bδ(k − k1 + k2 − k3) dk1 dk2 dk3.

Proposition 2.1. With the above definition for ν, we have

(i) If βi = 1 for all i ∈ 1, . . . , d, ν = 0.

(ii) If the βi are generic, ν = d− 2, and this value is optimal.

Proof. The first assertion, (i), is classical, see for instance [7]. The second assertion is proved inSection 8.

Remark 2.2. If one replaces the dispersion relation Q(k) by a more general function B(k), thecorresponding value of ν will always be ≤ d − 2. This is a due to the presence of the degenerateresonances, for which k1 = k2 or k1 = k. If these are removed, ν would take the above values forQ(k), but could improve for generic B(k).

2.2. The Strichartz parameter θr. As we mentioned in the introduction, we rely on long-timeStrichartz estimates in order to propagate the bounds resulting from the randomization-improvedlinear estimates. Here again, we will rely on the genericness of the (βi)1≤i≤d to obtain crucialimprovements on the rational case. We will introduce a parameter θr to record how the constantin the Strichartz estimates depends on the size of the time interval. This constant also appears inthe work [11] which derived the long-time Strichartz estimates we use here.

One way to define θr is as follows: it is such that the Lr+1t,x Strichartz estimates on [0, 1]×Td, for a

function localized in frequency on a ball of radius N , enjoy the same constant as on [0, Nθr ]× Td.


Therefore, N θr can be thought of as the time it takes for a wave localized on frequencies ≤ N , andinitially focused in space, to focus again.

More precisely, θ3 is the largest real numbers such that, for T,N ≥ 1, and for any ε > 0,

‖eit∆βPNψ‖L4t,x([0,T ]×Td) . N

d4− 1

2+ε

(1 +

T

N θ3

)1/4

‖ψ‖L2(Td)(2.2)

Proposition 2.3. With the above definition for θ3,

(i) For any choice of the βi, 0 ≤ θ3 ≤ d− 2.

(ii) If βi = 1 for all i ∈ 1, . . . , d, θ3 = 0.

(iii) For a generic choice of the (βi), θ3 :=

413 , d = 3(d−2)2

2(d−1) , d ≥ 4

Proof. In the first assertion, (i), the fact that θ3 ≥ 0 follows from the `2 decoupling result ofBourgain and Demeter [6], while the upper bound θ3 ≤ d − 2 follows from the L2 conservation.The second assertion follows immediately by periodicity of the group eit∆. The proof of the thirdassertion can be found in [11], where it is also conjectured that θ3 = d− 2.

Remark 2.4. For generic dispersion relations with quadratic growth, it seems natural to conjecturethat θ3 = d− 2 see [12] for the case of two-dimensional non-rectangular tori.

2.3. The approximation theorem. We are now in a position to state precisely the approximationtheorem for the cubic NLS in arbitrary dimension d ≥ 3.

Theorem 2.5. Set d ≥ 3 and let φ0 : Rd → R+ be a smooth compactly supported function. Supposethat ak(0) =

√φ(k)eiϑk(ω) where ϑk(ω) is a family of i.i.d. random variables uniformly distributed

in [0, 2π]. Let ε0 be a sufficiently small constant, and L sufficiently large depending on ε0.

• There exists a set Eε0,L of measure P(Eε0,L) ≥ 1− e−Lε0 such that: if ω ∈ Eε0,L and λ ≥ 1,

then for any L, the solution ak(t) of the cubic (NLS) equation exists in CtHs([0, T ]× TdL)

for

T ∼

λ−2L

d+2+θ32−4ε0 if 1 ≤ λ ≤ L

d−θ3−24−2ε0 ,

min(L2+θ3 , λ−4Ld−8ε0) if λ ≥ Ld−θ3−2

4−2ε0 .

(Recall that θ3 is defined in (2.2) and given in Proposition 2.3).

• Moreover, there holds

E[|ak(t)|21Eε0,L

]=t

τT3(φ)(k) +O`∞

(L−ε0

t

τ

), Lε0 ≤ t ≤ T, and τ =

L2d

2λ4.

-For d = 3, 4, the solutions exist globally in time [4, 26], and one has the same estimatewithout multiplying with 1Eε0

inside the expectation.

A couple of remarks are in order for this theorem and the following:

(1) The condition that φ ∈ C∞0 is only imposed to simplify certain technical arguments ofthe proof, particularly when bounding the Feynman diagrams. The theorem also holds forφ ∈ S, and φ ∈ HS , with S sufficiently large. Similarly, the error could be controlled in amuch stronger norm than `∞.

(2) Other randomizations of the data are possible (complex Gaussians for instance) withoutany significant change for the proof.


3. Formal derivation of the kinetic equation

In this section, we present the formal derivation of the kinetic equation, whose basic steps we shallfollow in the proof. We will perform this formal argument for the cubic equation only. The startingpoint is the equation (2.1) for the modes ak(t) which when integrated gives:

ak(t) = a0k +

tˆ

0

λ2

L2d

∑(k1,k2,k3)∈(ZdL)3

k−k1+k2−k3=0

ak1ak2ak3e−2πisΩ(k,k1,k2,k3) ds (3.1)

The derivation of the kinetic equation proceeds as follows:

Step 1: expanding in the data. Noting the symmetry in the variables k1 and k3, we have uponexpanding (3.1) that

ak(t) =a0k (3.2a)

+λ2

L2d

∑k−k1+k2−k3

a0k1a

0k2a0k3

1− e−2πitΩ(k,k1,k2,k3)

2πΩ(k, k1, k2, k3)(3.2b)

+ 2λ4

L4d

∑k−k1+k2−k3=0k1−k4+k5−k6=0

a0k4a

0k5a0k6a

0k2a0k3

1

2πΩ(k1, k4, k5, k6)

[e−2πitΩ(k,k4,k5,k6,k2,k3) − 1

2πΩ(k, k4, k5, k6, k2, k3)− e−2πitΩ(k,k1,k2,k3) − 1

2πΩ(k, k1, k2, k3)

](3.2c)

+λ4

L4d

∑k−k1+k2−k3=0k2−k4+k5−k6=0

a0k1a

0k4a0k5a

0k6a0k3

1

2πΩ(k2, k4, k5, k6)

[e−2πitΩ(k,k1,k4,k5,k6,k3) − 1

2πΩ(k, k1, k4, k5, k6, n)− e−2πitΩ(k,k1,k2,k3) − 1

2πΩ(k, k1, k2, k3)

](3.2d)

+ higher order terms. (3.2e)

where we denoted Ω(k, k1, k2, k3, k4, k5) = Q(k)−Q(k1)+Q(k2)−Q(k3)+Q(k4)−Q(k5); we also used

the convention that, if a = 0, e2πita−1

2πa = it, while, if a = b = 0, 12πa

(e2πit(a+b)−1

2π(a+b) − e2πita−12πa

)= −1

2 t2.

Step 2: parity pairing. We now compute E|ak|2, where the expectation E is understood with respectto the random phases (random parameter ω). The key observation is,

E(a0k1 . . . a

0ksa

0`1. . . a0

`s) =

φk1 . . . φks if there exists a permutation ν such that kν(i) = `i0 otherwise.

(for k ∈ ZdL, we write φk = φ(k)). Computing E(|ak|2

)with the help of the above formula, we see

that, there are no terms of order λ2. Second, terms of order λ4 can be obtained in different ways:either by pairing the term of order λ2, namely (3.2b), with its conjugate, or by pairing one of the


terms of order λ4, (3.2c) or (3.2d), with the term of order 1, namely a0k. Overall, this leads to1

E|ak|2(t) = φk +2λ4

L4d

∑k−k1+k2−k3=0

φkφk1φk2φk3

[1

φk− 1

φk1+

1

φk2− 1

φk3

] ∣∣∣∣sin(tπΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2+ higher order terms+ degenerate cases.

The details of the computation are as follows:

(1) Consider first E|(3.2b)|2 = E(3.2b)(3.2b), and denote k1, k2, k3 the indices in (3.2b) and

k′1, k′2, k′3 the indices in (3.2b). There are several possibilities:

• k1, k3 = k′1, k′3, in which case k2 = k′2, and Ω(k, k1, k2, k3) = Ω(k, k′1, k′2, k′3).

• (k2 = k1 or k3) and (k′2 = k′1 or k′3), in which caseΩ(k, k1, k2, k3) = Ω(k, k′1, k′2, k′3) = 0.

Overall, we find, neglecting degenerate cases (which occur for instance if k, k1, k2, k3 arenot distinct),

E|(3.2b)|2 =2λ4

L4d

∑k−k1+k2−k3

φk1φk2φk3

∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2 +4λ4

L4dt2∑k1,k3

φkφk1φk2 .

(2) Consider next the pairing of a0k with (3.2c), which contributes 2ERe

[(3.2c)a0

k

]. The possible

pairings are

• k, k2 = k4, k6, implying k3 = k5, and leading toΩ(k1, k4, k5, k6) = −Ω(k, k1, k2, k3),and Ω(k, k4, k5, k6, k2, k1) = 0.

• (k3 = k2 or k) and (k5 = k4 or k6) in which case Ω(k, k1, k2, k3) = Ω(k1, k4, k5, k6) = 0.

This gives, neglecting degenerate cases,

2ERe[a0k(3.2c)

]=

8λ4

L4d

∑k−k1+k2−k3

φkφk2φk3Re

[e−2πitΩ(k,k1,k2,k3) − 1

4π2Ω(k, k1, k2, k3)2

]− 8λ4

L4dt2∑k1,k3

φkφk2φk3

= −2λ4

L4d

∑k−k1+k2−k3

φkφk1φk2φn

[1

φk1+

1

φk3

] ∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2 − 8λ4

L4dt2∑k1,k3

φkφk2φk3 ,

where we used in the last line the symmetry between the variables k1 and k3, as well as theidentity Re(eiy − 1) = −2| sin(y/2)|2, for y ∈ R.

(3) Finally, the pairing of a0k with (3.2d) can be discussed similarly, to yield

2ERe[a0k(3.2d)

]=

2λ4

L4d

∑k−k1+k2−k3

φkφk1φk3

∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2 +4λ4

L4dt2∑k1,k3

φkφk2φk3 ,

Summing the above expressions for E|(3.2b)|2, 2ERe[a0k(3.2c)

]and 2ERe

[a0k(3.2d)

]gives the

desired result.

Step 3: the big box limit L→∞. Assuming that Ω(k, k1, k2, k3) is equidistributed on a scale

L−2−ν3 1

t, (3.3)

1Degenerate cases, like higher order terms, have smaller order of magnitude, on the time scales we consider as willbe illustrated in Section 4.


we see that, as L→∞,∑k−k1+k2−k3

φkφk1φk2φn

[1

φk− 1

φk1+

1

φk2− 1

φk3

] ∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2∼ L2d

ˆδ(Σ)φ(k)φ(k1)φ(k2)φ(k3)

[1

φ(k)− 1

φ(k1)+

1

φ(k2)− 1

φ(k3)

] ∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2 dk1 dk2 dk3.

Step 4: the large time limit t→∞ Observe first that´ (sinx)2

x2dx = π2,2 so that, in the sense of

distributions, ∣∣∣∣sin(πtΩ)

πΩ

∣∣∣∣2 ∼ tδ(Ω) as t→∞.

Therefore, as t→∞,∑k−k1+k2−k3

φkφk1φk2φn

[1

φk− 1

φk1+

1

φk2− 1

φk3

] ∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2∼ tL2d

ˆδ(Σ)δ(Ω)φ(k)φ(k1)φ(k2)φ(k3)

[1

φ(k)− 1

φ(k1)+

1

φ(k2)− 1

φ(k3)

]dk1 dk2 dk3

= tL2dT (φ, φ, φ).

Conclusion: relevant time scales for the problem. Overall, we find, assuming that the above limitsare justified

E|ak|2(t) = φk + 2λ4

L2dtT (φ, φ, φ) + lower order terms. (3.4)

This suggests that the actual time scale of the problem is

τ =L2d

2λ4,

and that, setting s = tτ , the governing equation should read

∂sφ = T (φ, φ, φ) (3.5)

In which regime is this approximation expected? Let T be the time scale over which we considerthe equation.

• In order for (3.4) to hold, the condition (3.3) has to hold, and the limits L→∞ and T →∞have to be taken: one needs

T L2+ν3 , L 1, and T 1.

• In order for the nonlinear evolution of (3.5) to affect an O(κ) change on the initial data,the two conditions above should be satisfied; in addition T should be of the order of κτ(equivalently s ∼ κ). Thus we find the conditions

1 T ≈ κτ L2+ν3 and κ14Ld/2 λ κ

14Ld/2−(2+ν3)/4.

2This follows from Plancherel’s theorem, and the fact that the Fourier transform of 1π

sin xx

is the characteristic

function of [− 12π, 12π

].


4. Feynman diagrams: bounding the k first terms on average

4.1. Expansion of the solution in the data. We will perform the calculations and expansionsin this section for the cubic case. We follow mostly the notations in Lukkarinen-Spohn [29], Section3.

The iterates of φ, considered in the previous section, can be represented through diagrams (at leastup to lower order error terms). To arrive at these diagrams, let us start with the equation satisfiedby the frequency modes ak which we write in the following form

d

dta(t)(k, σ) =

iσλ2

L2d

∑k′=(k′1,k

′2,k′3)∈(ZdL)3

δ(σ(k − k′2) + (k′1 − k′3)

)a(t)(k′1,−1)a(t)(k′2, σ)a(t)(k′3,+1)e−2πitΩ(k,k′,σ), (4.1)

where we denote

a(k, σ) =

a(k) if σ = 1

a(k) if σ = −1

and

Ω(k,k′, σ) := σ(Q(k)−Q(k′2)) + (Q(k′1)−Q(k′3)).

This gives the following formula for the product

d

dt

n∏i=1

a(t)(ki, σi) =

(iλ2

L2d

) n∑j=1

σj∏i 6=j

a(t)(ki, σi)∑

k′=(k′1,k′2,k′3)∈(ZdL)3

[δ(σ(kj − k′2) + (k′1 − k′3)

)×

a(t)(k′1,−1)a(t)(k′2, σ)a(t)(k′3,+1)e−2πitΩ(kj ,k′,σj)

]. (4.2)

and as a result, we get that

n∏i=1

a(t)(ki, σi) =

n∏i=1

a(0)(ki, σi) +

(iλ2

L2d

) tˆ

0

n∑j=1

σj∏i 6=j

a(s)(ki, σi)

∑k′=(k′1,k

′2,k′3)∈(ZdL)3

δ(σ(kj − k′2) + (k′1 − k′3)

)a(s)(k′1,−1)a(s)(k′2, σ)a(s)(k′3,+1)e−2πisΩ(kj ,k

′,σj)ds.

(4.3)

Using this formula to iterate the expression for ak(t) obtained from (4.3) with n = 1, we obtain thefollowing expression for ak(t)

ak(t) =

N∑n=0

Jn(t, k)(a(0)) +RN+1(t, k)(a(t)), (4.4)

where Jn contains all diagrams of depth n, degree 2n + 1, in the variable a(0) = (a(0)k )k∈ZdL

and

RN (t, k) is the remainder which is of degree 2n+ 3 in a(t) = (ak(t))k∈ZdL, as we shall explain below.


We start by describing those terms more precisely. For n small, explicit expressions were given inSection 3:

J0 = ak0, J1 = (3.2b), J2 = (3.2c) + (3.2d).

For general n we will further decompose

Jn =∑`

Jn,`,

where the index ` describes the interaction history.

+

+ +

+

+

+

++

+

+

-

- -

--

-

Figure 1. Diagram of depth 3.

Referring to the diagram in Figure 1, the interaction index ` functions as follows. Ascending upthe graph with top vertex k = k2n+1,1, from level 2n − 2 to 2n − 1, with interacting frequenciesk2n−2,4, k2n−2,5, and k2n−2,6, then the frequencies at level 2n−1 have values, and labeled, as follows:

k2n−1,1 = k2n−2,1, k2n−1,2 = k2n−2,2, k2n−1,3 = k2n−2,3,

k2n−1,4 = k2n−2,5 ± (k2n−2,6 − k2n−2,4), depending on the parity of ak2n−2,4 ,

k2n−1,5 = k2n−2,7.

For Figure 4.1, we depicted J3,(2,3,1). More generally, we have

Jn,`(t, k) =

(iλ2

L2d

)n n∏i=1

σi,`i∑

k∈(ZdL)2n+1

δ(k − kn,1)2n+1∏j=1

a0(k0,j , σ0,j)

ˆ

(R+)n+1

n∏m=1

e−2πitm(s)Ωm(k,σ)δ

(t−

n∑0

si

)ds

(4.5)

where

• a0(k, σ) =

a0k if σ = 1

a0k if σ = −1

• k = (k0,1, . . . , k0,2n+1) ∈ (ZdL)2n+1

• k = kn,1 ∈ ZdL• s = (s0, . . . , sn) ∈ (R+)n+1


• σ = (σ0,1, . . . , σ0,n) ∈ ±12n+1

• ` ∈ 1, . . . , 2n − 1 × 1, . . . , 2n − 3 × · · · × 1, 2, 3 × 1 describes the history of theinteractions.

• kj,m, with 1 ≤ j ≤ n and 1 ≤ m ≤ 1 + 2(n− j), is defined recursively by the conditions kj,m = kj−1,m for m ≤ `j − 1kj,m = kj−1,m+2 for m ≥ `j + 1kj,`j = kj−1,`j+1 + σj,`j (kj−1,`j+2 − kj−1,`j )

• σj,m, with 0 ≤ j ≤ n and 1 ≤ m ≤ 1 + 2(n− j), is fully determined by the requirement that

σn,1 = 1

and by the conditions, for 1 ≤ j ≤ n,σj,m = σj−1,m for m ≤ `j − 1σj,m = σj−1,m+2 for m ≥ `j + 1σj−1,`j = −1σj−1,`j+1 = σj,`jσj−1,`j+2 = 1.

• tj(s) =j−1∑k=0

sk, 1 ≤ j ≤ n.

• Ωj(k, σ) = σj,`j (Q(kj,`j )−Q(kj−1,`j+1)) +Q(kj−1,`j )−Q(kj−1,`j+2); 1 ≤ j ≤ n.

Finally, there remains to define

Rn(t, k)(a) =∑`

tˆ

0

Rn,`(t, s0; k)(a(s0))ds0,

where

Rn,`(t, s0; k)(b) =

(iλ2

L2d

)n n∏i=1

σi,`i∑

k∈(ZdL)2n+1

δ(kn,1 = k)2n+1∏j=1

b(k0,j , σ0,j)

ˆ

(R+)n

n∏j=1

e−2πitj(s)Ωj(k,σ)

δ

(t− s0 −

n∑1

si

)ds. (4.6)

Remark 4.1. Another way to arrive at the same expression for Jn,` is by using normal formcalculation as follows:

(a) Write the integral equation for a(t, k),

a(t)(k, 1) = a0(k, 1) +

tˆ

0

iλ2

L2d

∑k′=(k′1,k

′2,k′3)∈(ZdL)3

k−k′2+k′1−k′3=0

a(s)(k′1,−1)a(s)(k′2, 1)a(s)(k′3, 1)e−2πisΩ(k,k′,1)ds,

(b) Integrate by parts, writing F(t)s =

´ ts e−2πit′nΩ(k,k′,1)dt′n,

a(t)(k, 1) = a0(k, 1) +iλ2

L2d

∑k′=(k′1,k

′2,k′3)∈(ZdL)3

k−k′2+k′1−k′3=0

a0(k′1,−1)a0(k′2, 1)a0(k′3, 1)F(t)0 +R2(t, k)(a),


where R2 = iλ2

L2d

∑k′=(k′1,k

′2,k′3)∈(ZdL)3

σ(k−k′2)+k′1−k′3=0

´ t0 I + II + IIIds,

I = a(s)(k′1,−1)a(s)(k′2, 1)a(s)(k′3, 1)F (t)s ,

with similar expressions for II and III. Renaming,

k′1 → kn−1,1, k′2 → kn−1,2, k′3 → kn−1,3,

and Ω(k,k′, 1)→ Ωn, we get the expression for J1.

(c) The expression for J2 consists of a sum of three diagrams coming from the terms I, II, andIII respectively. For example, the diagram coming from I is obtained by using the differentialequation for a(t)(k, σ) to substitute

a(s)(k′1,−1) = − iλ2

L2d

∑n′=(n′1,n

′2,n′3)∈(ZdL)3

−(k′1−n′2)=n′3−n′1

a(s)(n′1,−1)a(s)(n′2, 1)a(s)(n′3, 1)e−2πisΩ(k′1,n′,−1)ds,

and relabeling

n′1 → kn−2,1, n′2 → kn−2,2, n′3 → kn−2,3, k′2 → kn−2,4, k′3 → kn−2,5,

we get a sums of terms

a(s)(kn−2,1, 1)a(s)(kn−2,2,−1)a(s)(kn−2,3,−1)e−2πisΩn−1(k,σ)a(s)(kn−1,4, 1)a(s)(kn−1,5, 1)F (t)s .

(d) Integrate by parts, writing F(t)s =

´ ts

´ tt′n−1

e−2πit′n−1Ωn−1(k,σ)e−2πit′nΩn(k,σ)dt′ndt′n−1, we get the

quintic term that I contribute

a0(kn−1,1, 1)a0(kn−1,2,−1)a0(kn−1,3,−1)a0(kn−1,4, 1)a0(kn−1,5, 1)F(t)0 ,

which gives the term J2,(1,1) in the expansion, along with three remainder diagrams that con-tribute to R3. Here we use the fact that

ˆ

(R+)n+1

n∏m=1

e−2πitm(s)Ωm(k,σ)δ

(t−

n∑0

si

)ds =

ˆ t

0

ˆ t

t1

. . .

ˆ t

tn−1

n∏m=1

e−2πitmΩm(k,σ)dtndtn−1 . . . dtt.

(e) Repeating the integration by parts n times gives Jn,`(t, k)(a(0)) where t′j = tj(s), as definedabove.

Formulas (4.4) and (4.6) can be derived by a similar computation (using induction on n). Indeed, asimilar integration by parts as above yields that Rn,` = Jn,` + (2n+ 1)-terms contributing to Rn+1.

4.2. Bound on the correlation. Our aim is to prove the following proposition.

Proposition 4.2. If t < Ld−ε0, then∣∣∣∣∣∣∑

n+n′=S

∑`,`′

E(Jn,`(t, k)Jn′,`′(t, k))

∣∣∣∣∣∣ .S log t

(t√τ

)S 1

t. (4.7)

Remark 4.3. The trivial estimate would be that∣∣∣E(Jn,`(t, k)Jn′,`′(t, k))∣∣∣ . ( t√

τ

)n+n′

.


Indeed, Jn,`Jn′,`′ comes with a prefactor(λ2

L2d

)n+n′

; the size of the domains where the time integra-

tion takes place is . tn+n′; and the summation over k and k′ is over 2d(n+n′+1) dimensions, halfof which are canceled by the pairing (see below), out of which d further dimensions are canceled by

the requirement that kn,1 = k. Overall, this gives a bound(λ2

L2d

)n+n′

×tn+n′×Ld(n+n′) =(

t√τ

)n+n′

.

Therefore, the above proposition essentially allows to gain 1t over the trivial bound.

Before we start the proof of Proposition 4.2, we shall split Jn,` into a totally degenerate part whichcorresponds to having all the interactions depicted in the diagram degenerate, in the sense thatkj,`j ∈ kj−1,`j+1, kj−1,`j+1+σj,`j

.

The contribution of such interactions to Jn,` is given by

Dn,`(t, k) =

(iλ2

L2d

)n( n∏i=1

σi,ì

) ∑k∈(ZdL)2n+1

δ(kn,1 = k)

n∏j=1

δ(kj,`j ∈ kj−1,`j+1, kj−1,`j+1+σj)×

2n+1∏j=1

a0(k0,j , σ0,j)

ˆ(R+)n+1

δ

(t−

n∑0

si

)ds = 2n

tn

n!

(iλ2

L2d

)n( n∏i=1

σi,ì

) ∑k∈(ZdL)n

n∏j=1

|a0kj|2 a0

k.

We shall define Jn,` via the relation

Jn,`(t, k) = Jn,`(t, k) +Dn,`(t, k) (4.8)

4.3. Cancellation of degenerate interactions. As can be seen from a simple computation inthe formula for Dn,`, the contribution of each E(Dn,`(t, k)Dn′,`′(t, k)) to the sum in (4.7) is of size

∼(

t√t

)S, which is too large in absolute value. Luckily, all those terms cancel out as the following

lemma shows:

Lemma 4.4. For any S ≥ 2 ∑n+n′=S

∑`,`′

E(Dn,`(t, k)Dn′,`′(t, k)) = 0.

Proof. First notice that, since∑

j σi,j = σn,1 = 1, one has

∑`

n∏i=1

σi,ì =

2n−1∑j1=1

2n−3∑j2=1

. . .

1∑jn=1

σ1,j1σ2,j2 . . . σn,jn = σnn,1 = 1.

Therefore, ∑`

Dn,`(t, k) = 2ntn

n!

(iλ2

L2d

)n ∑k∈(ZdL)n

n∏j=1

|a0kj|2 a0

k

which gives that∑n+n′=S

∑`,`′

E(Dn,`(t, k)Dn′,`′(t, k)) = 2StS(λ2

L2d

)S ∑k∈(ZdL)S

S∏j=1

|a0kj|2 |a0

k|2( ∑n+n′=S

in−n′

n!n′!

).


The result will follow once we show that ∑n+n′=S

in−n′

n!n′!= 0.

This follows by parametrizing the above sum as (n, n′) = (S − j, j) : j = 0, . . . S, which gives∑n+n′=S

in−n′

n!n′!= iS

S∑j=0

(−1)j

(S − j)!j!=iS

S!

S∑j=0

(−1)jS!

(S − j)!j!=iS

S!(1 + x)S

∣∣x=−1

= 0.

4.4. Estimate on non-degenerate interactions. Proposition 4.2 now follows from the followinglemma:

Lemma 4.5. Suppose Gn′,`′(t, k) ∈ Jn′,`′(t, k)), Dn′,`′(t, k)), then for 0 < t < Ld−ε0,∣∣∣E(Jn,`(t, k)Gn′,`′(t, k))∣∣∣ .n (log t)2

(t√τ

)n+n′ 1

t.

Proof. We will only consider the case of Gn′,`′(t, k) = Jn′,`′(t, k)) as the other one is easier. Wenote that using the identity

δ

(t−

n∑0

sj

)=

1

2π

ê−iα(t−

∑nj=0 sj) dα ,

we have that for any (e0, . . . , en) ∈ Rn+1 and η > 0

ˆ(R+)n+1

n∏j=0

e−isjejδ

(t−

n∑0

si

)ds =

eηt

2π

ˆRe−iαt

n∏j=0

i

α− ej + iηdα (4.9)

We choose in the following η = 1t . This gives

Jn,`(t, k) =ei

2π

(− λ2

L2d

)n n∏i=1

σi,ì∑

k∈(ZdL)2n+1

δ(kn,1 = k)∆(k)2n+1∏j=1

a0(k0,j , σ0,j)

ê−iαtdα

(α−Ω1 −Ω2 − · · · −Ωn + it) . . . (α−Ωn + i

t)(α+ it).

where we simply denoted Ωj = 2πΩj(k,σ) and ∆(k) = 1−∏nj=1 δ(kj,`j ∈ kj−1,`j+1

, kj−1,`j+1+σj).

Introduce now the class P = P(n, n′,σ,σ′) of pairings ψ, namely maps from 1, . . . , 2n+ 2n′ + 2to itself such that (denoting for brevity here σj := σn,j for 1 ≤ j ≤ 2n + 1 and σj = σn′,j−2n−1 ifj ≥ 2n+ 2)

• ψ(j) = ` ⇔ ψ(`) = j

• σψ(j) = −σj if j, ψ(j) both in 1, . . . , 2n+ 1 or 2n+ 2, . . . , 2n+ 2n′ + 2

• σψ(j) = σj otherwise.

Let furthermore

Γψ(k,k′) =

2n+2n′+2∏j=1

δ(k0,j = k0,ψ(j)),


where k0,j = k′0,j−2n−1 if j > 2n+ 1. With this definition, by independence of the (θk0,j (ω))∣∣∣∣∣∣Eω2n+1∏j=1

eiσ0,jθk0,j (ω)

2n′+1∏j′=1

e−iσ0,j′θk0,j (ω)

∣∣∣∣∣∣ .∑ψ∈P

Γψ(k,k′).

We can then estimate∣∣∣E(Jn,`(t, k)Jn′,`′(t, k))∣∣∣

.

(λ2

L2d

)n+n′∑ψ∈P

∑k∈(ZdL)2n+1

k′∈(ZdL)2n′+1

δ(kn,1 = k′n′,1 = k)∆(k)∆(k′)Γψ(k,k′)2n+1∏j=1

√φ(k0,j)

2n′+1∏j′=1

√φ(k′0,j′)

∣∣∣∣ˆ e−iαtdα

(α−Ω1 − · · · −Ωn + it) . . . (α−Ωn + i

t)(α+ it)ˆ

eiα′tdα′

(α′ −Ω′1 − · · · −Ω′n′ +it) . . . (α

′ −Ω′n′ +it)(α

′ + it)

∣∣∣∣.Observe that by Holder’s inequality, for any m ≥ 1 and b1, . . . , bn′+1 ∈ R,ˆ

R

dα′

|α′ − b1 + it | . . . |α′ − bm+1 + i

t |. tm . (4.10)

Applying this bound to the α′ integral yields∣∣∣E(Jn,`(t, k)Jn′,`′(t, k))∣∣∣

. tn′(λ2

L2d

)n+n′∑ψ∈P

∑k∈(ZdL)2n+1

k′∈(ZdL)2n′+1

δ(kn,1 = k′n′,1 = k)∆(k)∆(k′)Γψ(k,k′)2n+1∏j=1

√φ(k0,j)

2n′+1∏j′=1

√φ(k′0,j′)

∣∣∣∣∣ˆ

e−iαtdα

(α−Ω1 − · · · −Ωn + it) . . . (α−Ωn + i

t)(α+ it)

∣∣∣∣∣Let p = p(k) be the smallest integer such that kp+1,`p /∈ kp,`p+1, kp,`p+1+σp+1,`p

. This is the first

non-degenerate interaction in the Feynman diagram Jn,` which is not totally degenerate; noticethat 0 ≤ p ≤ n− 1 if ∆(k) 6= 0. Then∣∣∣E(Jn,`(t, k)Jn′,`′(t, k))

∣∣∣. tn

′(λ2

L2d

)n+n′∑ψ∈P

∑k∈(ZdL)2n+1

k′∈(ZdL)2n′+1

δ(kn,1 = k′n′,1 = k)∆(k)∆(k′)Γψ(k,k′)

2n+1∏j=1

√φ(k0,j)

2n′+1∏j′=1

√φ(k′0,j′)

∣∣∣∣∣ˆ

e−iαtdα

(α−Ωp+1 − · · · −Ωn + it)p+1 . . . (α−Ωn + i

t)(α+ it)

∣∣∣∣∣=∑ψ

∑p

Ip,ψ


We now set

I1 = `p, I2 = `p + 1, and I3 = `p + 2

and

J1 = ψ(I1), J2 = ψ(I2), and J3 = ψ(I3).

We will distinguish several cases depending on the values of these numbers. Note that, by definitionof p,

I1, I2, I3 ∩ J1, J2, J3 = ∅.

Case 1: two of J1, J2, J3 are ≥ 2n+ 2. Suppose first that J1, J2, J3 ≥ 2n+ 2. After summing overthe degenerate pairings on the lines 0 ≤ i ≤ p− 1, we obtain

Ip,ψ . tn′Ldp

(λ2

L2d

)n+n′ ∗∑ ∑kp,I1 ,kp,I2 ,kp,I3

δ(kn,1 = k)Γψ(k,k′)

2(n−p)+1∏j=1

√φ(kp,j)

2n′+1∏j′=1

√φ(k′0,j′)∣∣∣∣∣

ˆe−iαtdα

(α−Ωp+1 − · · · −Ωn + it)p+1 . . . (α−Ωn + i

t)(α+ it)

∣∣∣∣∣ ,where

∑∗ stands for the sum over kp,j , where 1 ≤ j ≤ 2(n − p) + 1 and j /∈ I1, I2, I3, and k′0,j′for 1 ≤ j′ ≤ 2n′ + 1 with j′ /∈ J1, J2, J3.

Since the function φ is compactly supported, the resonance moduli Ωi are bounded.

The contribution of the above integral is acceptable as long as the denominator is O(〈α〉−2). There-fore, it suffices to prove the desired bound when the domain of integration reduces to α ∈ [−R,R],

for some R > 0. Bounding furthermore the integrand by tn−1

|α−Ωp+1−···−Ωn+ it||α+ i

t| , matters reduce to

estimating

tn′+n−1Ldp

(λ2

L2d

)n+n′ ∗∑ ∑kp,I1 ,kp,I2 ,kp,I3

δ(kn,1 = k)Γψ(k,k′)

2(n−p)+1∏j=1

√φ(kp,j)

2n′+1∏j′=1

√φ(k′0,j′)

ˆ R

−R

e−iαtdα

|α−Ωp+1 − · · · −Ωn + it ||α+ i

t |,

This can also be written

tn′+n−1Ldp

(λ2

L2d

)n+n′ ∗∑Γ ∗ψ(k,k′)

∗∏√φ(kp,j)

√φ(k′0,j′)∑

kp,I1 ,kp,I2 ,kp,I3

δ(kn,1 = k)φ(kp,I1)φ(kp,I3)

ˆ R

−R

e−iαtdα

|α−Ωp+1 − · · · −Ωn + it ||α+ i

t |,

where Π∗ and Γ ∗ include the same variables as∑∗. Due to the identity

∑2(n−p)+1j=1 σp,jkp,j = kn,1,

this can also be written

tn′+n−1Ldp

(λ2

L2d

)n+n′ ∗∑Γ ∗ψ(k,k′)

∗∏√φ(kp,j)

√φ(k′0,j′)

ˆ R

−R

∑kp,I1 ,kp,I2 ,kp,I3

δ(kp,I1 − kp,I3 ± kp,I2 + x)φ(kp,I1)φ(kp,I3)

|α−Ωp+1 − · · · −Ωn + it |

dα

|α+ it |,

where x depends on k and the variables kp,j with j /∈ I1, I2, I3.


After setting kp,I2 = kp,I1 ± kp,I3 + x, notice that

Ωp+1 + . . .+Ωn = Q(k)−2(n−p)+1∑

j=1

σp,jQ(kp,j)

= −σp,I1Q(kp,I1)− σp,I3Q(kp,I3)− σp,I2Q(kp,I1 − kp,I3 + x) + C

where C depends only on k and the variables kp,j with j /∈ I1, I2, I3. We now use that, since

t < L2+ν3 , and uniformly in N ∈ ZdL and C ∈ R,∑P,R∈ZdL

1∣∣−Q(P ) +Q(R)−Q(N + P −R) + C + it

∣∣√φ(P )φ(R) .

∑1t<2n.1

2−n∑

|−Q(P )+Q(R)−Q(N+Q−R)+C|∼2n

1 . L2d∑

1t<2n.1

1 . L2d log t.(4.11)

where the second inequality follows by definition of ν3.

Therefore, we can bound the above by

. (log t)tn′+n−1Ld(p+2)

(λ2

L2d

)n+n′ ∗∑Γ ∗ψ(k,k′)

∗∏√φ(kp,j)

√φ(k′0,j′)

ˆ R

−R

dα

|α+ it |,

On the one hand,´ R−R

dα|α+ i

t| . log t. On the other hand, the sum

∑∗ is over 2(n + n′ − p − 2)

variables; through the pairing Γ ∗ψ, half of them drop, so that the remaining sum is . Ld(n+n′−p−2).Therefore, the above is

. (log t)2tn′+n−1Ld(n+n′)

(λ2

L2d

)n+n′

,

which is the desired estimate.

There remains the case where only two of J1, J2, J3 are ≥ 2n + 2 ; Suppose for instance thatJ2 ≤ 2n + 1. Then, there exists I4 ≤ 2(n − p) + 1 such that ψ(I4) = J4 ≥ 2n + 2 (such an indexexists because there is an odd number of elements in the set of elements in 1, . . . , 2(n − p) +1 \ I1, I2, I3, J2, so they cannot be paired together completely). One can then follow the aboveargument replacing I2 by I4.

Case 2: two of J1, J2, J3 are ≤ 2n+ 1 Assume for instance that J1, J3 ≤ 2n + 1. Proceeding as inCase 1, it suffices to bound

Ldptn′(λ2

L2d

)n+n′ ∗∑ ∑kp,I1 ,kp,I3

δ(kn,1 = k)Γ ∗ψ(k,k′)φ(kp,I1)φ(kp,I3)

∗∏√φ(k0,j)

ˆ R

−R

e−iαtdα

|α−Ωp+1 − · · · −Ωn + it |p+1 . . . |α−Ωn + i

t ||α+ it |,

where Σ∗ is the sum over kp,j , with j ∈ 1, . . . , 2(n − p) + 1 \ I1, I3, J1, J3, and over k0,j′ , withj′ ∈ 1, . . . , 2n′ + 1; and

∏∗ is defined similarly.

A crucial observation is that, since∑2(n−p)+1

j=1 σp,jkp,j = kn,1, the condition kn,1 = k is independentof kp,I1 and kp,I3 as soon as kp,I1 = kp,J1 and kp,I3 = kp,J3 . Furthermore, p ≤ n − 2 since J1, J3 ≤


2n+ 1, which enables us to bound the integrand by tn−1

|α−Ωp+2−···−Ωn+ it||α+ i

t| . Overall, we can bound

the above by

. . . . Ldptn+n′−1

(λ2

L2d

)n+n′ ∗∑δ(kn,1 = k)Γ ∗ψ(k,k′)

∗∏√φ(k0,j)

ˆ R

−R

∑kp,I1 ,kp,I3

φ(kp,I1)φ(kp,I3)

|α−Ωp+2 − · · · −Ωn + it |

dα

|α+ it |.

Recall here that Ωp+1 + . . .+Ωn = Q(k)−∑2(n−p)+1

j=1 σp,jQ(kp,j), which implies that

Ωp+2 + . . .+Ωn = ±(Q(kp,I1)−Q(kp,I3)±Q(kp,I1 − σp,I2kp,I2 − kp,I3)) + C,

where C only depends on the variables in∑∗. Applying (4.11) enables us to bound the inner sum

by . L2d log t. We use furthermore that´ R−R

dα|α+ i

t| . log t. Finally, the number of variables in∑∗ is 2(n + n′ − p − 1); through the pairing, there remains only n + n′ − p − 1 of them; finally

the δ(kn,1 = k) brings their number down to n + n′ − p − 2. This means that∑∗ will contribute

. Ld(n+n′−p−2). Overall, we obtain the bound

. (log t)2tn′+n−1Ld(n+n′)

(λ2

L2d

)n+n′

,

which is the desired estimate.

5. Deterministic local well-posedness

5.1. The local well-posedness theorem. Define first the following Fourier multipliers: for u afunction on TdL and j ∈ Zd,

uBj (k) = 1Bj (k)u(k).

Here 1Bj is the characteristic function of the unit cube centered at j.

We can now introduce the ZsT norm as follows:

‖u‖ZsT =

∑j∈Zd〈j〉2s‖uBj‖2L4

t,x([0,T ]×TdL)

1/2

, (5.1)

where s ≥ 0.

In the following theorem, we establish a local well-posedness theorem in ZsT , for data of size

A(TL−d)14 . This seemingly strange normalization will actually be well adapted to the physical

situation we consider. Indeed, consider for simplicity initial data f supported on Fourier frequen-cies . 1, of size A in L2, and whose Fourier coefficients have random, uncorrelated phases. Itshould then be expected that eit∆βf will be, in general, evenly spread over TdL. By conservation of

the L2 mass, this corresponds to ‖eit∆βf‖ZsT = ‖eit∆βf‖L4t,x([0,T ]×TdL) ∼ AT

14L−

d4 .

Theorem 5.1. Assume s > d2 . Then the equation

i∂tu− 12π∆βu = −λ2|u|2u

u(0, x) = f(x)(5.2)


is locally well-posed in ZsT . More precisely, for any ε0 > 0, denoting

‖eit∆βf‖Zs = A(TL−d)14 and ρ = ρ(T, L) = C0

⟨ T

L2+θ3

⟩ 12T

12Lε0−

d2

(for a constant C0 to be determined)

(i) A solution u exists in ZsT provided

R := ρ(Aλ)2 ≤ 1. (5.3)

(ii) This solution is such that ‖u‖Zs ≤ 2A(TL−d)14 .

(iii) Finally,

‖u‖L∞t Hsx([0,T ]×TdL) ≤ ‖f‖Hs

x+ Cλ2A3

⟨ T

L2+θ3

⟩1/4T 3/4L−

3d4

+ε0 .

This theorem is proved by finding a fixed point of the map

Φ(u) = eit∆βf + iλ2

tˆ

0

ei(t−s)∆β |u|2u(s) ds.

Classically, this means that u = limN→∞ ΦN (0), where ΦN stands for the N -th iterate of Φ:

Φ0(0) = eit∆βf, ΦN+1(0) = eit∆βf + iλ2

tˆ

0

ei(t−s)∆β |ΦN (0)|2ΦN (0) ds,

Besides the previous theorem, we need to investigate the speed of convergence of ΦN (u) to u. Thisis the content of the following proposition.

Proposition 5.2. Under the conditions of Theorem 5.1, there holds

‖u− ΦN (0)‖L∞Hs ≤ 2RN⟨ T

L2+θ

⟩− 14(A(TL−d)

14

).

Now define

Un,`(t) =1

Ld

∑k∈ZdL

Jn,`(t, k)e2πi(k·x+tQ(k)).

Proposition 5.3. Under the conditions of Theorem 5.1,

‖Un,`‖L∞T Hs . RN⟨ T

L2+θ

⟩− 14(A(TL−d)

14

).

5.2. Rescaled Strichartz estimate. The estimate (2.2) translates into the following estimatesfor linear solutions on the torus TdL:

‖eit∆βPNψ‖Lpt,x([0,T ]×TdL) .Cp(N,T, L)‖ψ‖L2(TdL) with Cp(N,T, L) = LεNd2− d+2

p+ε⟨ T

L2+θpN θp

⟩1/p.

(5.4)

This follows from a simple scaling argument that shows that Cp(T,N,L) = Ld+2p− d

2Cp(TL−2, LN, 1).

We will mostly use the case N = 1:

‖eit∆βP1ψ‖Lpt,x([0,T ]×TdL) . Lε⟨ T

L2+θp

⟩1/p‖ψ‖L2(TdL).


Converting this estimate to its dual, and applying the Christ-Kiselev inequality, one gets∥∥∥∥∥∥T

0

e−is∆P1F (s) ds

∥∥∥∥∥∥L2(TdL)

. Lε⟨ T

L2+θp

⟩1/p‖F‖

Lp′t,x([0,T ]×TdL)

∥∥∥∥∥∥tˆ

0

ei(t−s)∆P1F (s) ds

∥∥∥∥∥∥Lpt,x([0,T ]×TdL)

. Lε⟨ T

L2+θp

⟩2/p‖F‖

Lp′t,x([0,T ]×TdL)

.

5.3. The a priori energy estimate. Here we prove the following estimate on ‖u‖L∞t Hsx:

‖u‖L∞t Hsx([0,T ]×TdL) ≤ ‖f‖Hs

x+ Cλ2Lε0

⟨ T

L2+θ

⟩1/4‖u‖3Zs (5.5)

provided s > d2 .

‖u‖L∞t Hsx≤‖f‖Hs

x+ λ2

∥∥∥∥ˆ t

0ei(t−s)∆β |u|2u ds

∥∥∥∥L∞t H

sx

≤‖f‖Hsx

+ λ2 sup0≤t≤T

sup‖ψ‖

L2x

=1

ˆ t

0

ˆTdL|u|2u eis∆β 〈∇〉sψ dx ds

=‖f‖Hsx

+ λ2 sup0≤t≤T

sup‖ψ‖

L2x

=1

∑j1−j3+j3−j4=O(1)

ˆ t

0

ˆTdLuBj1uBj2uBj3 e

is∆β 〈∇〉sψBj4 dx ds

Using the rescaled Strichartz estimates,∑j1−j3+j3−j4=O(1)

∣∣∣∣∣ˆ t

0

ˆTdLuBj1uBj2uBj3 e

is∆β∇sψBj4 dx ds

∣∣∣∣∣.

∑j1−j2+j3−j4=O(1)

〈j4〉s3∏

k=1

‖uBjk‖L4t,x‖eis∆βψBj4‖L4

t,x

. C(1, T, L)∑

j1−j2+j3−j4=O(1)

〈max(|j1|, |j2|, |j3|)〉s3∏

k=1

‖uBjk‖L4t,x‖ψBj4‖L2

x

. C(1, T, L)

∑j

‖ψBj‖2L2

1/2∑j

〈j〉2s‖uBj‖2L4

1/2∑j

‖uBj‖L4

2

. Lε0⟨ T

L2+θ

⟩1/4‖u‖3ZsT ‖ψ‖L2

x

5.4. The a priori Zs estimate. Here we prove the estimate

‖u‖ZsT ≤ ‖eit∆βf‖ZsT + Cλ2Lε0

⟨ T

L2+θ3

⟩1/2‖u‖3ZsT

Applying PBj to Duhamel’s formula and taking the L4t,x norm,

‖PBju‖L4t,x([0,T ]×Td) ≤ ‖eit∆βPBjf‖L4

t,x([0,T ]×TdL) + λ2

∥∥∥∥ˆ t

0ei(t−s)∆βPBj |u(s)|2u(s)ds

∥∥∥∥L4t,x([0,T ]×TdL)

.


By duality, and using the notation g = e−it∆βv, v(s, x) = eis∆β´ Ts g(t)dt,∥∥∥∥ˆ t


∥∥∥∥L4t,x([0,T ]×TdL)

= sup‖v‖

L4/3t,x

=1

ˆTdL

ˆ T

0

ˆ t

0

[ei(t−s)∆βPBj |u(s)|2u(s)

]v(t, x) ds dt dx

= sup‖v‖

L4/3t,x

=1

ˆTdL

ˆ T

0

ˆ t

0

[e−is∆β |u(s)|2u(s)

]gBj (t, x) ds dt dx

= sup‖v‖

L4/3t,x

=1

ˆTdL

ˆ T

0|u(s)|2u(s)vBj (s, x) ds dx.

Now we notice thatˆ T

0

ˆTdL|u(s)|2u(s)vBj (s, x) ds dx =

∑j1−j2+j3−j=O(1)

ˆ T

0

ˆTdLuBj1uBj2uBj3 vBj ds dx

.∑

j1−j2+j3−j=O(1)

j1,j2,j3∈Zd

3∏k=1

‖uBjk‖L4t,x‖vBj‖L4

t,x. C(1, T, L)2

∑j1−j2+j3−j=O(1)

j1,j2,j3∈Zd

3∏k=1

‖uBjk‖L4t,x.

This gives that∥∥∥∥ˆ t


∥∥∥∥L4t,x([0,T ]×Td)

. C(1, T, L)2∑

j1−j2+j3−j=O(1)

j1,j2,j3∈Zd

3∏k=1

‖uBjk‖L4t,x

and hence we have:∑j∈Zd〈j〉2s

∥∥∥∥ˆ t


∥∥∥∥2

L4t,x

1/2

. C(1, T, L)2

∑j∈Zd〈j1〉2s‖uBj1‖

2L4t,x

1/2C(1, T, L)∑`∈Zd‖uB`‖L4

t,x

2

. Lε0⟨ T

L2+θ

⟩1/2‖u‖3Zs ,

provided again that s > d/2.

5.5. The fixed point argument: proof of Theorem 5.1, Proposition 5.2 and Proposi-tion 5.3.

Proof of Theorem 5.1. Recall that

Φ(u) = eit∆βu0 + iλ2

tˆ

0

ei(t−s)∆β |u|2u(s) ds.


From the a priori ZsT estimate, we will deduce that Φ is a contraction on BZsT (0, 2A(TL−d)14 ) if

R := ρ(Aλ)2 ≤ 1

2.

Indeed, let us check that Φ is a contraction on BZsT (0, 2A(TL−d)14 ) under this condition: for u in

this ball,

‖Φ(u)− eit∆βu0‖ZsT =

∥∥∥∥∥∥λ2

tˆ

0

ei(t−s)∆β |u|2u(s) ds

∥∥∥∥∥∥ZsT

. λ2Lε0⟨ T

L2+θ

⟩ 12 ‖u‖3ZsT ≤ (Aλ)2ρ

(A(TL−d)

14

)≤ R

(A(TL−d)

14

)by (5.3). Also, if u, v ∈ BZsT (0, 2A(TL−d)

14 ),

‖Φ(u)− Φ(v)‖ZsT . λ2Lε0

⟨ T

L2+θ

⟩ 12(

2A(TL−d)14

)2‖u− v‖ZsT ≤ R‖u− v‖ZsT .

Proof of Proposition 5.2. From the above and the fact that ‖Φ(0)‖ZsT = A(TL−d)14 , we deduce that

‖Φ2(0)− Φ(0)‖ZsT ≤ R‖Φ(0)‖ZsT , and, by an obvious induction, that

‖Φj(0)− Φj−1(0)‖ZsT ≤ Rj−1‖Φ(0)‖ZsT . (5.6)

As for the energy estimate, it gives for u, v ∈ BZsT (0, 2A(TL−d)14 ),

‖Φ(u)− Φ(v)‖L∞T Hs ≤ R⟨ T

L2+θ

⟩− 14 ‖u− v‖ZsT . (5.7)

In order to bound u− ΦN (0) in ZsT , we write it

u− ΦN (0) =∞∑j=N

Φj+1(0)− Φj(0).

Therefore, using successively (5.7) and (5.6),

‖u− ΦN (0)‖L∞T Hs ≤∞∑j=N

∥∥Φj+1(0)− Φj(0)∥∥L∞T H

s

≤ R⟨ T

L2+θ

⟩− 14∞∑j=N

∥∥Φj(0)− Φj−1(0)∥∥Zs

≤⟨ T

L2+θ

⟩− 14∞∑j=N

Rj‖Φ(0)‖ZsT ≤ 2RN⟨ T

L2+θ

⟩− 14 ‖Φ(0)‖ZsT .

Proof of Proposition 5.3. By definition, Un,` can be obtained by the following iterative resolution

of the Schrodinger equation: Set vm0 = e2πit∆βu0 for 0 ≤ m ≤ 2n + 1 and for any 1 ≤ j ≤ n we

define vjm (0 ≤ m ≤ 2(n− j) + 1)

• vmj = vmj−1 if m < `j


• vmj = vm+2j−1 if m > `j

• v`jj is then given by the solution v of

i∂tv −1

2π∆v = −λ2v

`jj−1v

`j+1j−1 v

`j+2j−1 .

Finally, Un,` = v1n. We can then use the energy estimate to get

‖v1n‖L∞T Hs . λ2

⟨ T

L2+θ

⟩1/4Lε0‖v1

n−1‖Zs . . . ‖v3n−1‖Zs .

We can then estimate v`n−1

n−1 using the Zs estimate, and so on... This leads to the desired bound.

6. Improved integrability through randomization

Recall that

u0 =1

Ld

∑k∈ZdL

√φ(k)e2πik·xe2πiϑk(ω),

where the ϑk(ω) are independent random variables, uniformly distributed on [0, 2π].

For any t, s, ω,

‖eit∆u0‖Hs =1

Ld

∑k∈ZdL

〈k〉2sφ(k)

1/2

;

in other words, the randomization of the angles of the Fourier coefficients does not have any effecton L2 based norms. This is not the case for Lebesgue indices larger than 2.

Theorem 6.1. Assume that |φ(k)| . 〈k〉−s, with s > d2 . Then

(i) E∥∥eit∆βu0

∥∥pLpt,x([0,T ]×TdL)

. TLd−dp2 ‖u0‖pL2

x

(ii) (large deviation estimate)

P[∥∥eit∆βu0

∥∥4

L4t,x([0,T ]×TdL)

> λ]. exp

(−c(

λ

T 1/4L−d/4

)2)

Proof. (i) The proof is more or less standard. See [8] for instance.

(ii) We follow the argument in [8]. By Minkowski’s inequality (for p ≥ 4) and Khinchin’s inequality,∥∥eit∆βu0

∥∥Lpω(Ω,L4

t,x([0,T ]×TdL)).∥∥eit∆βu0

∥∥L4t,x([0,T ]×TdL,L

pω(Ω)))

.√p

Ld

∥∥∥∥(∑φ(k))1/2

∥∥∥∥L4t,x([0,T ]×TdL)

.√pT 1/4L−d/4.

By Chebychev’s inequality,

P[∥∥eit∆βu0

∥∥L4t,x([0,T ]×TdL)

> λ].√λ−p

(C0pT1/4L−d/4)p.

The desired inequality is then obvious if λ < 2eC0T1/4L−d/4; if not, it follows upon choosing

p =(

λC0T 1/4L−d/4e

)2.


As a consequence, we deduce the following proposition.

Proposition 6.2. Let ε0 > 0, ν > s+ d2 , and assume that |φ(k)| . 〈k〉−2ν . Then, for two constant

C, c > 0,

P[∥∥eit∆βu0

∥∥Zs< T 1/4Lε0−d/4

]> 1− Ce−cLε0 .

Proof. Applying Theorem 6.1 to (u0)Bj ,

P[∥∥eit∆βu0

∥∥L4t,x([0,T ]×TdL)

> 〈j〉−νT 1/4Lε02− d

4

]. exp(−c〈j〉2νLε0).

Therefore, for L sufficiently big,

P[∥∥eit∆βu0

∥∥Zs< T 1/4Lε0−d/4

]> 1−

∑j

P[∥∥eit∆β (u0)Bj

∥∥Zs> T 1/4L

ε02−d/4〈j〉−ν

]> 1− C

∑j

exp(−c〈j〉2νLε0)

> 1− Ce−cLε0 .

7. Proof of the main theorem

First choose ε0 > 0 sufficiently small.

Step 1: excluding exceptional data. Let Eε0,L be the event ∥∥eit∆βu0

∥∥Zs< T 1/4Lε0−d/4, and Fε0,L

its contrary: ∥∥eit∆βu0

∥∥Zs> T 1/4Lε0−d/4. By Proposition 6.2,

P(Fε0,L) . e−cLε0.

This is the set appearing in the statement of Theorem 2.5. In case when a global solution of (NLS)exists for all finite energy data (which is the case when d = 3, 4), we can use the conservation ofmass to conclude that

E(|ak(t)|2

)= E

(|ak(t)|2 | Eε0,L

)+O`∞(e−cL

ε0Ld).

Step 2: iterative resolution. We now apply Theorem (5.1), and choose A = Lε0 . Recall that

R = C0(λA)2⟨ T

L2+θ3

⟩1/2(TL−d)1/2Lε0 .

We will work under the assumption that RLε0 ≤ 1. This allows us to apply Theorem 5.1 andProposition 5.2 with f = u0, where ω ∈ Eε0,L, and A = Lε0 .

Assertion (iii) in Theorem 5.1 gives the inequality

‖u‖L∞Hs . 1 + λ2A3⟨ T

L2+θ3

⟩1/4(TL−d)3/4Lε0 .

We now split into two cases:

Case 1: 1 ≤ λ ≤ Ld−θ3−2

4−2ε0 . In this case, we take T := L

d+2+θ2

C0λ2L−4ε0 . Since T & L2+θ3 in this case,

we have (recall that 2 + θ3 < d):

‖u‖L∞Hs . 1, and R ≤ L−ε0 . (7.1)


Case 2: λ ≥ Ld−θ3−2

4−2ε0 . In this case, we take T := min(L2+θ3 , L

d−8ε0

C20λ

4 ), and one can check that the

above estimates in (7.1) still hold.

We now split

u = ΦN (0) + u− ΦN (0)

=N∑n=0

Un,` +∑

(n,`)∈SNUn,` + u− ΦN (0).

Here, SN can be defined as follows: write ΦN (0) as∑

(n,`)∈T NJn,`, for a finite set T N . Observe that

T N contains all the (n, `) for which 0 ≤ n ≤ N ; and some other pairs (n, `) for which n ≥ N + 1:they constitute SN . By Proposition 5.2 and Proposition 5.3, this implies that

u =

N∑n=1

∑`

Un,` +O`2,sL

(RN)

(where the implicit constant depends on N).

Switching to Fourier coefficients and squaring them, we obtain that

|ak(t)|2 =

∣∣∣∣∣N∑n=1

∑`

Jn,`

∣∣∣∣∣2

+O`1,2sL

(RN)

=

∣∣∣∣∣N∑n=1

∑`

Jn,`

∣∣∣∣∣2

+O`∞(LdRN

).

Step 3: pairing. By Proposition 4.2,∣∣∣∣∣N∑n=1

∑`

Jn,`

∣∣∣∣∣2

= E[|J1(k)|2 + J0(k)J2(k) + J0(k)J2(k)

]+O

(t

τ

t log t√τ

)

= φk +2λ4

L4d

∑k−k1+k2−k3

φkφk1φk2φk3

[1

φk− 1

φk1+

1

φk2− 1

φk3

] ∣∣∣∣sin(tπΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2+O`∞

(t

τ

t log t√τ

)

Step 4: big box limit L→∞ By definition of ν3, equidistribution of resonances holds on a scale

L−ν3−2, and therefore we find for t < Ld−ε

2λ4

L4d

∑k−k1+k2−k3=0

φkφk1φk2φk3

[1

φk− 1

φk1+

1

φk2− 1

φk3

] ∣∣∣∣sin(tπΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2=

2λ4

L2d


[1

φ(k)− 1

φ(k1)+

1

φ(k2)− 1

φ(k3)

] ∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2 dk1 dk2 dk3

+O`∞(t

τL−δ).

Step 5: large time limit t→∞ It is easy to check that, for f a smooth function,´ ∣∣∣ sin(πtx)

x

∣∣∣2 f(x) dx =

π2tf(0) +O(1). Therefore,


2λ4

L2d


[1

φ(k)− 1

φ(k1)+

1

φ(k2)− 1

φ(k3)

] ∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2 dk1 dk2 dk3

=t

τT (φ, φ, φ) +O

(1

τ

), τ =

L2d

2λ4

Step 6: Conclusion. Fix ε0 small enough. Recall that t ≤ T which is made precise in Step 2.

By picking, L ≥ L1(ε0) we can make the error term in Step 1 less than tτL−ε0 . Also, using the

fact the condition on T guarantees that R ≤ L−ε0 so that the error in Step 2 is also O( tτL−ε0)

by picking N = O(ε−10 ). Similarly for the error for steps 3 and 4 using the fact that λ ≥ 1. This

concludes the proof of Theorem 2.5.

8. Number theoretic results

In this section we will derive the asymptotic formula for lattice sums over the region,

RZdef= (p, q) ∈ Z2d ∩ [0, L]2d

∣∣ Q(p, q) ∈ [a, b], p 6= q,

in terms of integral over

Rdef= (x, y) ∈ R2d ∩ [0, L]2d

∣∣ Q(x, y) ∈ [a, b].

Furthermore, we will assume we are given generic β = (β1, . . . , βd) ∈ [1, 2]d .

8.1. Upper bounds. Upper bounds on lattice sums are much easier to obtain than finding anasymptotic formula for the sum. In fact, in our setting, if one is willing to allow the bounds tobe ε away from being optimal, then the bounds follow from well known facts in number theory, asexplained below.

Using the genericity of β = (β1, . . . , βd) ∈ [1, 2]d, a good upper bound for the linear form β ·n ∈ [a, b],where n = Zd is a consequence of the pigeonhole principle:

Lemma 8.1. The linear form β · n ∈ [a, b] satisfies the following bound

#n ∈ Zd ∩ [−M,M ]d∣∣ a ≤ β · n ≤ b =

∑a≤β·n≤b|n|≤M

1 .M (d−1)+(b− a) + 1 (8.1)

Proof. Since β = (β1, . . . , βd) are generic, then for 0 < |n| ≤M (see for example [9], Chapter VII)

|β · n| & 1

M (d−1)+.

For arbitrary n(1) 6= n(2) ∈ Zd satisfying a ≤ β · n(i) ≤ b and 0 <∣∣n(i)

∣∣ ≤M ,

1

M (d−1)+.∣∣∣β · (n(1) − n(2))

∣∣∣ ≤ b− a .By the pigeonhole principle we obtain (8.1).


For the quadratic form Q(p, q) defined by

Q(n) =d∑i=1

βin2i , n = (n1, . . . , nd),

Q(p, q) := Q(p)−Q(q) =

d∑i=1

βi(p2i − q2

i ),

(8.2)

an upper bound for the set (p, q) ∈ Z2d ∩ [0, L]2d∣∣ Q(p, q) ∈ [a, b], p 6= q, can be obtained in the

following manner. Define RZ` as

RZ` = (p, q) ∈ Z2d∩ [0, L]2d∣∣ a ≤ Q(p, q) ≤ b, pi 6= qi, for 1 ≤ i ≤ `, and pi = qi, for `+1 ≤ i ≤ d,

Since RZ consists of several copies of⋃d`=1RZ`, obtained by permuting βi(p2

i − q2i )d1, the bounds

we obtain for⋃d`=1RZ` are also bounds for RZ

A bound on RZ` can be obtained from Lemma 8.1 and the divisor bound d(k) .ε kε.

Lemma 8.2. For ` = 1, . . . d the cardinality of #RZ` satisfies the bound

#RZ` =∑RZ`

1 . L(d+`−2)+(b− a) + L(d−`)+ (8.3)

Proof. Define ki = (pi − qi)(pi + qi), for 1 ≤ i ≤ `. Since pi = qi, for `+ 1 ≤ i ≤ d, we conclude

#RZ` . Ld−`

∑a≤

∑i=1

βiki≤b

0<|k|.L2

∑(pi−qi)(pi+qi)=ki

1

By the divisor bound ∑(pi−qi)(pi+qi)=ki

1 . L0+ ,

and by (8.1), with M = L2, we obtain

#RZ` . L(d−`)+

(L2(`−1)+(b− a) + 1

),

and (8.3) follows.

Corollary 8.3. The number of elements in RZ, can be bounded by

#RZ . L2(d−1)+(b− a) + L(d−1)+ (8.4)

Moreover, if we further assume |a| , |b| ≤ 1, then we have the improved bound

#RZ . L2(d−1)+(b− a) + L(d−2)+ , (8.5)

Proof. It suffices to apply the Lemma 8.2, and to observe that ` ∈ 1, . . . , d since p = q isexcluded.

Remark 8.4. Note, that in terms of the first estimate (8.4), the second term may be treated as an

error so long as b− a ≥ L−(d−1)+ε0 for some ε0 > 0. Analogously, the second term of (8.5) may betreated as an error assuming b− a ≥ L−d+ε0.

Following this remark on identifying the leading order term, we can now identify subsets of RZ thatcontribute error terms only. The first such subsets are when |pi − qi| . L1−δ for some fixed δ > 0and some i that we may without loss of generality assume to be 1.


Lemma 8.5. For |a| , |b|, the number of elements in RZ satisfying |p1 − q1| . L1−δ satisfy thefollowing bound

#RZ ∩ (p, q) ∈ Z2d∣∣ |p1 − q1| . L1−δ . L2(d−1)+−δ(b− a) + L(d−2)+ .

Proof. If pi = qi for at least one i, then by Corollary 8.3 with d replaced by d− 1, we have

#RZ ∩ (p, q) ∈ Z2d∣∣ pi = qi . L

(L2(d−2)+(b− a) + L(d−3)+

),

which is lower order. Moreover, if pi 6= qi for all i, and |p1−q1| . L1−δ, then the sum over 2 ≤ i ≤ dcan be bounded by L2(d−2)+(b− a) + L0+ , using Lemma 8.2, while the sum over p1 and q1 can be

by L2−δ. This gives a bound of L2−δ(L2(d−2)+(b− a) + L0+

), which is lower order if d ≥ 3.

Next we show that if one pi or qi is less than L1−δ, where we may again assume i = 1, then thecontribution to the number of elements in RZ is lower order.

Lemma 8.6. For |a| , |b| ≤ 1, we have the following estimate

#RZ ∩ (p, q) ∈ Z2d∣∣ |p1| . L1−δ . L2(d−1)+−δ(b− a) + L(d−2)+

Proof. If both |p1| . L1−δ and |q1| . L1−δ or pi = qi for at least one i, then by Lemma 8.5 we have

the stated bound. Otherwise, the sum over 2 ≤ i ≤ d contributes L2(d−2)+(b− a) + L0+ , while thesum over p1 and q1 contributes L2−δ.

From Lemma 8.5 and Lemma 8.6, we have

Corollary 8.7. Setting

RZδ = RZ \d⋃i=1

((p, q) ∈ Z2d

∣∣ where, |pi| . L1−δ, |qi|, or |pi − qi| . L1−δ, for at least one i).

Then, for |a| , |b| ≤ 1, we have the following cardinality bound on the set difference RZ \RZδ

#RZ \RZδ . L2(d−1)+−δ(b− a) + L(d−2)+

8.2. Equidistribution. In [5] Bourgain considered a generic positive definite diagonal quadraticform in 3 variables, and proved that the pair correlation problem is equidistributed at a scale of1Lρ , for 1 < ρ < 2: i.e., for d = 3, provided that a < b, |a|, |b| < O(1) and L−ρ < b− a < 1, then∑

RZ

1 = L2(d−1)(b− a)H2d−1((x, y) ∈ [−1, 1]2d

∣∣ Q(x, y) = 0)

+O(Ld−2−δ(b− a)

),

where H2d−1 is the 2d-1 dimensional Hausdorff measure.

Our aim here is to generalize the theorem of Bourgain to obtain the following asymptotic formula


Theorem 8.8. For any ε > 0, there exists a sufficiently small δ such that we have the following:Suppose 0 < t ≤ Ld−ε,∑

ki∈ZdLk−k1+k2−k3=0

φkφk1φk2φk3

[1

φk− 1

φk1+

1

φk2− 1

φk3

] ∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2 =

L2d

ˆ (δ(Σ)φ(k)φ(k1)φ(k2)φ(k3)

[1

φ(k)− 1

φ(k1)+

1

φ(k2)− 1

φ(k3)

] ∣∣∣∣sin(πtΩ(k, k1, k2, k3))

πΩ(k, k1, k2, k3)

∣∣∣∣2)dk1 dk2 dk3

+O(tL2d−δ

)where we recall Σ(k, k1, k2, k3) = k − k1 + k2 − k3.

If we rescale time µ := tL−2, let Ki = Lki, and denote g(x) =(

sin(πx)πx

)2and

W0

(K

L,K1

L,K2

L,K3

L

)= φkφk1φk2φk3

[1

φk− 1

φk1+

1

φk2− 1

φk3

],

then the sum can be expressed as

t2∑

K,K1,K2,K3∈ZdK−K1+K2−K3=0

W0

(K

L,K1

L,K2

L,K3

L

)g(µΩ(K,K1,K2,K3)) .

By defining

u′ = K1 −K ∈ Zd, u′′ = K3 −K ∈ Zd, and u = (u′, u′′) ∈ Z2d

thenΩ(K,K1,K2,K3) = Q0(u)

whereQ0(u) := −2β1u

′1u′′1 − 2β2u

′2u′′2 − · · · − 2βdu

′du′′d , (8.6)

and the sum can be expressed as

t2∑u∈Z2d

W0

(K

L,u′ +K

L,u′ + u′′ +K

L,u′′ +K

L

)g(µQ0(u)). (8.7)

The quadratic form Q0 can be diagonalized by making the change of coordinates

pi = u′i + u′′i , qi = u′i − u′′iwhere pi and qi are either both even or both odd, i.e.∑

ui∈Z2

=∑

pi,qi∈2Z+

∑pi,qi∈(2Z+1)

=∑

pi,qi∈Z−

∑pi∈2Z,qi∈Z

−∑

pi∈Z,qi∈2Z+2

∑pi,qi∈2Z

.

Consequently, the sum (8.7), can be written as four different sums of the form,

t2∑

(p,q)∈Z2d

W( pL,q

L

)g(µQ(p, q)),

where Q(p, q) is given by3 (8.2), and where we suppressed the dependence on k for convenience. Upto an admissible error, we can replace W with a compactly supported function, in particular due tothe rapid decay of W , the contribution of pairs p, q ∈ Zd ∩ ([−L1+δ, L1+δ])c is of order O(L−N ) forany N ≥ 1. Next we note that we can ignore all pairs (p, q) such that |pj | = |qj | for each j. The sum

such pairs such that |p| , |q| ≤ L1+δ is of order O(t2Ld(1+δ)) and hence contributes to an admissible

3There are factors of 2 missing due to sums over even terms. However this has no impact since β is generic.


error, where here we used the restriction t ≤ Ld−ε. Due to symmetry, it is sufficient to restrictourselves to the positive sector p, q ∈ Zd+ ∩ [0, L1+δ] for p 6= q. Here we are using that the subset of(p, q) such that pj = 0 or qj = 0 for some j is an admissible error. This follows as a consequenceof Lemma 8.6. To rigorously carry out such an estimate, one must split the contributions when|Q(p, q)| ≤ µ−1 and |Q(p, q)| > µ−1. Assuming without of loss of generality that p1 = 0, thensplitting up the later part dyadically in the size of |Q(p, q)| and using |g(x)| . 1

|x|2 one obtains the

estimate

t2∑

(p,q)∈Z2d+

p1=0, p 6=q

∣∣∣W ( pL,q

L

)g(µQ(p, q))

∣∣∣ . t2(L2(d−1)+−2δ

µ+ L(d−2)+

). tL2d−δ .

With all these reductions in mind, proving Theorem 8.8 will follow as a consequence of the followingtheorem.

Theorem 8.9 (Equidistribution). Fix ε > 0 and let δ > 0 be sufficiently small. Then for genericβ ∈ [1, 2]d, we have that for any smooth function W : R2d → R, compactly supported in a ball ofradius Lδ, the following holds,∑

(p,q)∈Z2d+

p 6=q

W( pL,q

L

)g(µQ(p, q)) = L2d

¨

R2d

W (x, y)g(L2µQ(x, y)) dxdy +O

(L2(d−1)−δ

µ

)

where 0 < µ ≤ Ld−1−ε.

We remark that the above theorem is actually stronger than required: in view of the restriction oft in the hypothesis of Theorem 8.8, we need only consider µ within the range 0 < µ ≤ Ld−2−ε.

The difficulty in proving this theorem is mainly due to the fact that we are trying to prove equidis-tribution down to a scale of 1

µ ∼ L−d+1+ε. In fact, if we want to prove equidistribution on a coarser

scale, e.g. 1µ = L

43 , then the theorem is trivial, as can be seen below.

Proposition 8.10. Fix δ > 0 sufficiently small, then if L1+4δ ≤ b−a ≤ Lδ, we have the asymptoticformula

#

(p, q) ∈ Zd ∩ [0, L]2d∣∣ Q(p, q) ∈ [a, b]

= L2(d−1)(b− a)

¨

R2d

1[0,1]2d(x, y)δdirac(Q(x, y)) dxdy

+O(L2(d−1)−δ(b− a)

).

Proof. First we will smooth the characteristic functions by extending the region to a slightly biggerregion with a controlled error term. This is done as follows. Let wL ∈ C∞c ([L−δ, L + Lδ]) be abump function satisfying wL(x) = 1 for x ∈ [0, L] and

‖wL‖CN . L−Nδ ,

then ∑p,q∈Zd

WL

( pL,q

L

)− 1[0,L]2d (p, q) = O

(L2d−1+δ

).

Moreover, if we let hL ∈ C∞c ([a− L1+2δ, b+ L1+2δ]) to be a bump function hL(x) = 1 for x ∈ [a, b]and

‖hL‖CN . L−N(1+2δ) .


then by Corollary 8.3, we have∑p,q∈Zd

WL

( pL,q

L

)hL (Q(p, q))− 1[0,L]2d (p, q)1[a,b](Q(p, q)) = O

(L2d−1+δ

)+O

(L(2d−1+2δ)+

)= O

(L2(d−1)−δ(b− a)

).

assuming that b− a ≥ L1+4δ. Thus, it is sufficient to obtain the asymptotic formula for

S :=∑p,q∈Zd

WL

( pL,q

L

)hL (Q(p, q)) .

Using Fourier transform, we express S as

S =

ˆ ∞−∞

hL(s)∑p,q

WL

( pL,q

L

)e(Q(p, q)s) ds :=

ˆ ∞−∞

hL(s)S(s) ds (8.8)

Applying Poisson summation we may rewrite S(s) as

S(s) =∑`

ˆWL

(xL,y

L

)e(Q(x, y)s−m · x− n · y) dx dy (8.9)

=L2d∑`

ˆWL (z) e(L2Q(z)s− L` · z) dz (8.10)

where z = (x, y), and ` = (m,n).

The term ` = 0 contributes the asymptotic formula

L2d

ˆWL(z)hL(L2Q(z))dz = L2(d−1)(b−a)

¨

R2d

1[0,1]2d(x, y)δdirac(Q(x, y)) dxdy+O(L2(d−1)−δ(b− a)

)where we used (b − a) < L−δ in replacing hL(L2Q)) by δdirac(Q). So it remains to show that thesum for ` 6= 0 can be treated as error. First we estimate the sum for s ≤ 1

L1+δ . In this case we write

Φ(z, `, s) = L2Q(z)s−L` · z, and note that since |s| ≤ 1L1+δ and |z| . 1, then |∇zΦ(z,m, s)| ≥ L|`|

2 ,where

∇zΦ(z, `, s) = L2∇Q(z)s− L` , (8.11)

and thus upon integrating (8.10) by parts, we obtain

S(s) =∑6=0

L2d

ˆ∇j(

WL(z)

2πi∇jΦ(z, `, s)

)e(Φ(z, `, s)) dz. (8.12)

Since each derivative of WL contributes L1−δ, then each integration by parts contributes a factor of1

Lδ|`| . Applying a sufficient number of integrations by parts, and using the fact that |hL(s)| . b−a,

we may ensure that the contribution for ` 6= 0 and |s| ≤ 1L1+δ is arbitrarily small.

For |s| ≥ 1L1+δ we note that

|hL(s)| . (b− a)1

(L1+2δ|s|)N,

for all N , and consequently, we can treat this term as an error as well. Thus we conclude the statedresult.

Before we present the proof of Theorem 8.9, we will present Bourgain’s theorem.


Theorem 8.11. Fix ε > 0, then for δ > 0 sufficiently small the following statement is true:Suppose Ij , Jj ⊂ [0, L], j = 1, . . . , d for d ≥ 3 are intervals with length satisfying

L1−δ ≤ |Ij | , |Jj | ≤ L (8.13)

Then for a, b satisfying |a| , |b| ≤ 1 and L−d+ε < b− a < L−ε we have∑a≤Q(p,q)≤bpj∈Ij ,qj∈Jj

p 6=q

1 =

Î1×···×Id

ˆJ1×···×Jd

1a≤Q(x,y)≤b dxdy +O(L2(d−1−dδ)(b− a)) . (8.14)

In order to prove Theorem 8.11, we first make a series of reductions.

Step 1: Ignore intervals that contribute lower order sums. Set δ = 4dδ, then by Corollary 8.7 we

have for δ sufficiently small,∑RZ

1 =∑RZδ

1 +O(L2(d−1)+−δ(b− a))

)+O(L(d−2)+ =

∑RZδ

1 +O(L2(d−1)+−δ(b− a)

)(8.15)

where we have used the restriction of a− b and assumed δ to be sufficiently small compared to ε.

Thus we restrict our attention to the case where

(a) ∀pi ∈ Ei, and ∀qi ∈ Fi, we have |pi| > L1−δ, |qi| > L1−δ,

(b) distance(Ei, Fi) > L1−δ.

With this reduction at hand, we divide each interval into at most L3δ intervals, Ei = ∪αIαi andFi = ∪αJαi each satisfying

(c) 12L

1−3δ ≤ |Iαi | , |Jαi | ≤ L1−3δ,

and prove that for intervals Iαi and Jαi , satisfying Conditions (a), (b), and (c) we have∑a≤Q(p,q)≤bpj∈Iαj ,qj∈Jαj

p 6=q

1 =

Îα1 ×···×Iαd

ˆJα1 ×···×Jαd

1a≤Q(x,y)≤b dxdy +O(L2(d−1)−(3d+1)δ(b− a)) . (8.16)

Summing in α and using (8.15) we have∑a≤Q(p,q)≤bpj∈Ij ,qj∈Jj

p 6=q

1 =∑α

(Îα1 ×···×Iαd


1a≤Q(x,y)≤b dxdy +O(L2(d−1−(3d+1)δ)(b− a))

)

+O(L2(d−1)+−4dδ(b− a)

)=∑α



1a≤Q(x,y)≤b dxdy +O(L2(d−1)+−δ(b− a)

).

Using that δ = 4dδ and∣∣∣∣∣Î1×···×Id

ˆJ1×···×Jd

1a≤Q(x,y)≤b dxdy −∑α



1a≤Q(x,y)≤b dxdy

∣∣∣∣∣. L2(d−1)+−δ(b− a)

we conclude (8.14).


Summarizing, if by abuse of notation, we drop the index α and replace δ with δ, we have reducedthe proof of Theorem 8.11 to proving the following proposition.

Proposition 8.12. Fix ε > 0, then for δ > 0 sufficiently small the following statement is true:Suppose Ij , Jj ⊂ [−L,L], j = 1, . . . , d for d ≥ 3 are intervals satisfying

(1) ∀pi ∈ Ii, and ∀qi ∈ Ji, we have |pi| > L1−δ, |qi| > L1−δ.

(2) distance(Ii, Ji) > L1−δ.

(3) 12L

1−3δ ≤ |Ii| , |Ji| ≤ L1−3δ

Then for a, b satisfying |a| , |b| ≤ 1 and L−d+ε < b− a < L−ε we have∑a≤Q(p,q)≤bpj∈Ij ,qj∈Jj

p 6=q

1 =

ˆI1×···×Id

ˆJ1×···×Jd

1a≤Q(x,y)≤b dxdy +O(L2(d−1)−(3d+1)δ(b− a)) . (8.17)

Let us now suppose Ij and Jj satisfy the hypothesis of Proposition 8.12.

Step 2: Transform the region of summation. The sum can be written as,∑pi∈Ij ,qj∈Jj

1[a,b](Q(p, q)) =∑

pi∈Ij ,qj∈Jj

1[0, b−a2 ]

(Q(p, q)− a+ b

2

)(8.18)

By writing Id = [u−∆u, u+∆u], and Jd = [v−∆v, v+∆v], and utilizing the fact that |u−v| > L1−δ,we express the region RZ as,∣∣∣∣∣

∑d−1j=1 βj(p

2j − q2

j )− b+a2

βd(p2d − q2

d)+ 1

∣∣∣∣∣ ≤ b− a2βd

∣∣p2d − q2

d

∣∣ ,≤ b− a

2βd |u2 − v2|+O

((b− a)L−δ

|u2 − v2|

)since ∣∣p2

d − q2d − u2 + v2

∣∣ . L(∆u+ ∆v) . L2−3δ and∣∣u2 − v2

∣∣ ≥ L2−2δ .

Setting ξ = b+a2 and η = b−a

2 , then by taking logarithms and Taylor expanding ln(x) around x = 1we obtain∣∣∣∣∣∣ln

d−1∑j=1

βj(p2j − q2

j )− ξ

− ln(p2d − q2

d

)− lnβd

∣∣∣∣∣∣ ≤ η

βd |u2 − v2|+O

(ηL−δ

|u2 − v2|

), (8.19)

here we assumed, without loss of generality,∑d−1

j=1 βj(p2j − q2

j )− ξ > 0 and p2d − q2

d > 0.

Step 3: Replace the sum with an analogous sum.

Instead of considering the sum over the region RZ, we will consider the sum over the region SZ,defined as

SZ =

(p, q) ∈d∏j=1

Ij ×d∏

k=1

Jk :

∣∣∣∣∣∣lnd−1∑j=1

βj(p2j − q2

j )− ξ

− ln(p2d − q2

d

)− lnβd

∣∣∣∣∣∣ ≤ η

βd |u2 − v2|

(8.20)


In order to make this reduction, we need a bound on cardinality of (p, q) satisfying∣∣∣∣∣∣lnd−1∑j=1

βj(p2j − q2

j )− ξ

− ln(p2d − q2

d

)− lnβd

∣∣∣∣∣∣ =η

βd |u2 − v2|+O

(ηL−δ

|u2 − v2|

),

Such a bound would follow as a consequence of a version of a weaker version of Proposition 8.12with the asymptotic formula (8.21) replaced with a sharp upper bounded, i.e.

Proposition 8.13. Fix ε > 0, then for δ > 0 sufficiently small the following statement is true:Suppose Ij and Jj satisfy the hypothesis of Proposition 8.12, then for a, b satisfying |a| , |b| ≤ 1 and

L−d+ε < b− a < L−ε we have ∑a≤Q(p,q)≤bpj∈Ij ,qj∈Jj

p 6=q

1 = O(L2(d−1)−3dδ(b− a)) . (8.21)

We note that for Proposition 8.12 compared with Proposition 8.13 we may require a stricter small-ness criteria on δ relative to the choice of ε. With this in mind, applying Proposition 8.13, thedifference in summing in p and q satisfying (8.19) and computing the cardinality of SZ is of order

O(L2(d−1)−(3d+1)δ(b− a)) and hence can be treated as an error.

By the arguments above, the sum in Proposition 8.13 may be estimated from above by the cardi-nality of SZ with η replaced by 2η in the set’s definition. Hence up to a factor of 2 in the definitionof SZ, to prove both Proposition 8.13 and Proposition 8.12, it suffices to obtain an asymptoticformula for SZ.

Step 4: Expressing the sum using Fourier Transform. The number #SZ can be expressed usingFourier transform as follows. Let

F (p, q) = ln

d−1∑j=1

βj(p2j − q2

j )− ξ

− ln(p2d − q2

d

)− lnβd, A =

η

|u2 − v2|,

and write ∑SZ

1 =∑

(pj ,qj)∈Ii×Jj

1[−A,A](F (p, q)) =∑

(pj ,qj)∈Ij×Jj

ˆeiF (p,q)t 1[−A,A](t)dt

=

ˆS1(t)S2(t)e−it lnβd 1[−A,A] dt,

where,

S1(t) =∑

pi∈Ii,qi∈Jii=1,...,d−1

d−1∑j=1

βj(p2j − q2

j ) + ξ

it

(8.22)

S2(t) =∑

pd∈Id,qd∈Jd

(p2d − q2

d)it. (8.23)

Step 5: A scaling argument. As mentioned earlier, if A is large compared to L−1, then comparingthe sum over SZ and the area of S is relatively simple. For this reason we split our sum by scaling

with a factor AA0

, where A0 = L4/3

|u2−v2| , i.e., split the integral into two terms,

A

A0

ˆS1(t)S2(t)e−it lnβd 1[−A0,A0] dt+

ˆS1(t)S2(t)e−it lnβd

(1[−A,A] −

A

A0

1[−A0,A0]

)dt = I + II .


The first integral is counting p, q such that∣∣∣∣∣∣lnd−1∑j=1

βj(p2j − q2

j )− ξ

− ln(p2d − q2

d

)− lnβd

∣∣∣∣∣∣ ≤ A0β−1d .

This amounts to counting ∣∣∣∣∣∣d∑j=1

βj(p2j − q2

j )− ξ

∣∣∣∣∣∣ ≤ L 43 +O(L

43−δ) .

Applying a similar upper/lower bounding argument to that used in Step 3 with the use of Propo-sition 8.13 replaced by the use of Proposition 8.10, we obtain

I = L2(d−1)

ˆI1×···×Id

ˆJ1×···×Jd

1a≤Q(x,y)≤b dxdy +O(L2(d−1)−(3d+1)δ(b− a)) .

For the purpose of proving Proposition 8.13, one simply observes that the first term is of orderO(L2(d−1)−3dδ(b − a)). Thus in order to complete the proof of Proposition 8.12, Proposition 8.13,and by implication Theorem 8.11, it suffices to estimate II.

Step 6: II is an error. Now consider II, we aim to show that

|II| . L2(d−1)−3dδη (8.24)

for a set of (β2, βd) of full measure. By Chebyshev’s inequality, it suffices to show

‖II‖L2β2,βd

. L2(d−1)−(3d+1)δη.

To see this, define

ΩL = β ∈ [1, 2]d∣∣ |II| > L2(d−1)−3dδη .

By Chebyshev’s inequality we have

|ΩL| .1

L4(d−1)−6dδ‖II‖2L2

β2,βd

. L−2δ .

Then since∞⋂j=N

|Ω2j | ≥ 1− C∞∑j=N

2−2jδ = 1− C 4δ(1−N)

4δ − 1→ 1 as N →∞ .

then we obtain (8.24) for a set of (β2, βd) of full measure, where implicit the constant depends(β2, βd).

The Fourier Transform in the integrand of II can be computed explicitly,∣∣∣∣1[−A,A](t)−A

A0

1[−A0,A0](t)

∣∣∣∣ = A

∣∣∣∣sin(At)

At− sin(A0t)

A0t

∣∣∣∣ . min

(AA2

0 |t|2 ,

A

1 +A |t|

),

by Borell-Cantelli we have


Averaging in β2 and βd, and using Plancherel’s theorem for the integral in βd, we have from the

bounds A = ηL−2+2δ and A0 = L−23

+2δ

‖II‖2L2β2,βd

. A2

(A4

0

ˆ|t|≤L

13

t4 ‖S1‖2L2β2

|S2|2 dt+

ˆ|t|≥L

13

1

1 +A2t2‖S1‖2L2

β2

|S2|2 dt

)

. η2L−203

+12δ

ˆ|t|≤L

13

t4 ‖S1‖2L2β2

|S2|2 dt+ η2L−4+4δ

ˆ|t|≥L

13

1

1 + η2L−4t2‖S1‖2L2

β2

|S2|2 dt

. η2L−83

+6δ

ˆ|t|≤L

13

t4 ‖S1‖2L2β2

dt︸︷︷︸III

+ η2L−4+4δ

ˆ|t|≥L

13

1

1 + η2L−4t2‖S1‖2L2

β2

|S2|2 dt︸︷︷︸IV

where we have used the trivial bound #S2 ≤ #Id#Jd ≤ L2−6δ.

To bound ‖S1‖L2β2

, we rewrite

|S1(t)|2 =∑


∑rj∈Ij ,sj∈Jjj=1,...,d−1

d−1∑j=1

βj(p2j − q2

j )− ξ

itd−1∑j=1

βj(r2j − s2

j )− ξ

−it

=∑



(p21 − q2

1 + β2(p22 − q2

2) + ψ1)it(r21 − s2

1 + β2(r22 − s2

2) + ψ2)−it

=∑



e(t(ln(p2

1 − q21 + β2(p2

2 − q22) + ψ1

)− ln

(r2

1 − s21 + β2(r2

2 − s22) + ψ2

)))

where

ψ1 :=

d−1∑j=3

βj(p2j − q2

j )− ξ and ψ2 =

d−1∑j=3

βj(r2j − s2

j )− ξ

for d > 3 or ψ1 = ψ2 = ξ for the case d = 3. Setting

φ := ln(p2

1 − q21 + β2(p2

2 − q22) + ψ1

)− ln

(r2

1 − s21 + β2(r2

2 − s22) + ψ2

)we have

|∂β2φ| =∣∣∣∣ p2

2 − q22

p21 − q2

1 + β2(p22 − q2

2) + ψ1− r2

2 − s22

r21 − s2

1 + β2(r22 − s2

2) + ψ2

∣∣∣∣≥∣∣∣∣(p2

2 − q22)(r2

1 − s21 + ψ2)− (r2

2 − s22)(p2

1 − q21 + ψ1)

L4

∣∣∣∣ ,then for t ≤ L4, and by taking the sup over indices 3 ≤ i ≤ d− 1, we have

ˆ|S1(t)|2 dβ2 . sup

ψ1,ψ2

L2(d−3)∑

pi∈Ii,qi∈Jiri∈Ii,si∈Jii=1,2

(1 + |t| inf

β2|∂β2Ψ|

)−1

.

Here we a using the trivial bound for the case 1 ≥ |t| infβ2 |∂β2Ψ|, otherwise we use an oscilla-tory estimate (see for example [37] Chapter 8, Proposition 2). For the former case, to apply theproposition, we split the integral into regions for which ∂β2Φ is monotonic in β2.


Set (pi− qi)(pi + qi) = wi and (ri− si)(ri + si) = zi, and sum over fixed wi and zi using the divisorbound d(k) .ε |k|ε, we obtain


ψ1,ψ2

L2(d−3)+∑

L2−2δ≤|wi|,|zi|≤L2

(1 +|t|L4|w2(z1 + ψ2)− z2(w1 + ψ1)|

)−1

The above sum can rearranged by summing first over the set,

A (k,w2, z2) = L2−2δ ≤ |w1| , |z1| ≤ L2∣∣ ⌊|w2(z1 + ψ2)− z2(w1 + ψ1)|

⌋= k,

and then over (k,w2, z2) to obtain,


ψ1,ψ2

L2(d−3)+∑

0≤k.L2

L2−2δ≤|w2|,|z2|≤L2

#A (k,w2, z2)L4

L4 + |t| k

. supψ1,ψ2

L2(d−3)+∑

L2−2δ≤|w2|,|z2|≤L2

maxk

#A (k,w2, z2)

(1 +

L4+

|t|

)

Now we estimate #A (k,w2, z2) for a fixed (k,w2, z2). Assume A (k,w2, z2) 6= ∅, then there existsw0 and z0, such that, L2−2δ ≤ |w0 − ψ2| ≤ L2 and L2−2δ ≤ |z0 − ψ1| ≤ L2 and

[|w2(z0)− z2(w0)|] = k .

Thus

#A . #w2z1 = z2w1

∣∣ |w1 − w0| , |z1 − z0| ≤ L2 = #w1 =w2z1

z2

∣∣ |w1 − w0| , |z1 − z0| ≤ L2

Since w1 ∈ Z then #A . 1 + L2 gcd(w2,z2)z2

, and consequently

ˆ|S1(t)|2 dβ2 . L

2(d−3)+

(1 +

L4+

|t|

) ∑L2−2δ≤|w2|,|z2|≤L2

(1 +

L2 gcd(w2, z2)

z2

)

. L2(d−3)+

(1 +

L4+

|t|

) ∑L2−2δ≤|w2|,|z2|≤L2

1 +∑

L2−2δ≤|w2|,|z2|≤L2

gcd(w2,z2) 6=1

L2+

. L2(d−1)+δ

(1 +

L4

|t|

)

Hence, applying this bound to III yields

III . η2L−83

+6δ

ˆ|t|≤L

13

|t|3 L2(d+1+δ) dt . η2L2d+1 (8.25)


Now consider IV , we have

IV . η2L2(d−3)+5δ

ˆ|t|≥L

13

|t|+ L4

1 + η2L−4t21

|t||S2|2 dt

. η2L2(d−3)+5δ sup

κ≥L13

κ1+ + L4κ0+

1 + η2L−4κ2

ˆ|t|≥L

13

1

|t|1+ |S2|2 dt

. η2L2(d−3)+5δ

(L2+

η+ L4+

)supkL−

13 2−k

ˆ L13 2k+1

L13 2k

|S2|2 dt

. η2L2(d−3)+5δ

(L2+

η+ L4+

)supkL−

13 2−k

ˆ L13 2k+1

L13 2k

|S2|2 dt

(8.26)

It is now convenient to rewrite S2 in terms of the coordinates m = pd− qd, n = pd + qd and the set

K := pd − qd)∣∣ (pd, qd) ∈ Id × Jd .

The sum S2 may then be rewritten as

S2 =∑pd,qd

1Id(pd)1Jd(qd)(p2d − q2

d)it

=∑m,n

1[−1,1]

(m+ n− 2u

2∆u

)1[−1,1]

(m− n− 2v

2∆v

)mitnit

=∑m∈K

mit∑n

1[−1,1]

(m+ n− 2u

2∆u

)1[−1,1]

(m− n− 2v

2∆v

)nit .

Without loss of generality, we may assume ∆u ≤ ∆v. Let us cover K by disjoint intervals Mj of

length L1−100dδ and define wj to be the center of Mj . It is not difficult to show that that may be

achieved such that #Mj . L100dδ we have the following bound on the set difference

#

(⋃k

Mj

)\K . L2−100dδ .

Thus we have∣∣∣∣∣∣S2 −∑j

∑m∈Mj

mit∑n

1[−1,1]

(m+ n− 2u

2∆u

)1[−1,1]

(m− n− 2v

2∆v

)nit

∣∣∣∣∣∣ . L2−100dδ .

Up to a controllable error, we may also replace m+ n with wj . Specifically, we have the estimate∣∣∣∣∣∑n

(1[−1,1]

(m+ n− 2u

2∆u

)1[−1,1]

(m− n− 2v

2∆v

)− 1[−1,1]

(wj + n− 2u

2∆u

)1[−1,1]

(wj − n− 2v

2∆v

))∣∣∣∣∣ . L1−100dδ ,

and hence

S2 =∑j

∑m∈Mj

mit∑n

1[−1,1]

(wj + n− 2u

2∆u

)1[−1,1]

(wj − n− 2v

2∆v

)nit +O(L2−100dδ) .

Again, up to an allowable error we may also replace the sharp cut-off cutoff functions with a smoothcut-off ψ ≡ 1 on [−1 + L−100d, 1− L−100d] and supported on the interval [−1, 1], i.e.

S2 =∑j

∑m∈Mj

mit∑n

ψ

(wj + n− 2u

2∆u

)ψ

(wj − n− 2v

2∆v

)nit︸︷︷︸

Sj

+O(L2−100dδ) (8.27)


We proceed to estimate |Sj |2. Defining

χ(z) = ψ

(wj − 2u

2∆u+ z

)ψ

(wj − 2v

2∆u− z)

and letting χ denote Mellin transform of χ, then

S2j =

(1

2π

ˆ<s=2

χ(s)(2∆u)sζ(s− it)ds)2

.

Shifting the contour to <s = 12 we pick up the residue

χ(1 + it)(2∆u)1+it ,

which for |t| ≥ L13 is order O(L−N ) for any N due to the decay of ψ. Then using ∆u ∼ L1−3δ

S2j =

(1

2π

ˆ<s= 1

2

χ(s)(2∆u)sζ(s− it)ds

)2

+O(1)

. L1−3δ

(ˆ<s= 1

2

χ(s)ζ(s− it)ds

)2

+O(1) = L1−3δ

(χ

(1

2+ ·)∗ ζ(

1

2− i(·)

)(t)

)2

+O(1) .

Again, using the rapid decay of ψ, we have

S2j . L

1−3δ

((1[−L100dδ,L100dδ](·)χ

(1

2+ ·))∗ ζ(

1

2− i(·)

)(t)

)2

+O(1) .

We now utilize following classical L4 bound of the zeta function in the critical strip (Heath-Brown’The fourth power moment of the Riemann zeta function’ 1979)

1

T

ˆ T

0

∣∣∣∣ζ (1

2− it

)∣∣∣∣4 dt . T 0+ .

Using the above bound yields∥∥∥∥(1[−L100dδ,L100dδ](·)χ(

1

2+ ·))∗ ζ(

1

2− i(·)

)∥∥∥∥4

L4([L13 2k,L

13 2k+1])

. ‖χ‖L∞∥∥∥∥ζ (1

2+ i(·)

)∥∥∥∥4

L4([L13 2k−L100dδ,L

13 2k+1+L100dδ])

. L2k

where we have bounded δ powers of L by L23 . Thus we obtain

‖Sj‖4L4([L

13 2k,L

13 2k+1])

. L32k .

A analogous argument also yields∥∥∥∥∥∥∑

mj∈Mj

mit

∥∥∥∥∥∥4

L4([L13 2k,L

13 2k+1])

. L32k .

Using the decomposition (8.27) and the bound #Mj . L100dδ, we have

‖S2‖2L2([L

13 2k,L

13 2k+1])

. L3+200dδ2k + L133−200dδ2k . L

133−200dδ2k .


Thus, combining the above estimate on S2 with (8.26), we obtain

IV . η2L2(d−3)+5δ

(L2+

η+ L4+

)(supkL−

13 2−kL

133−200dδ2k

). η2L2(d−1)+(5−200d)δ

(L2+

η+ L4+

). η2L4(d−1)−2(3d+1)

as desired, using that η > L−d.

Before we prove Theorem 8.9, we will need a couple of auxiliary lemmas. The following lemma ishelpful in bounding errors to the asymptotic formula.

Lemma 8.14. Let ε > 0. Given a generic quadratic from Q(p, q) as defined in (8.2), we have thefollowing estimate ∑

(p,q)∈Z2d∩[0,L]2d

p 6=q,|Q(p,q)|≥a

1

Q(p, q)2.L(2d−2)+

a. (8.28)

for a ≥ L−d+ε

Proof. We begin by dyadically subdividing the interval [a,CL2], for some large C, we define

RZ(m)

def= (p, q) ∈ Z2d ∩ [0, L]2d

∣∣ |Q(p, q)| ∈ [2m, 2m+1], p 6= q,mmin = blog2 ac, and mmax = dlog2CL

2e .

Applying Lemma 8.2 yields ∑(p,q)∈Z2d∩[0,L]2d

p 6=q,|Q(p,q)|≥a

1

Q(p, q)2.

mmax∑m=mmin

2−2m#RZ(m)

.mmax∑

m=mmin

2−2mL(2d−2)+2m

.mmax∑

m=mmin

L(2d−2)+

a.

The following lemma will be useful localizing the sum in Theorem 8.9.

Lemma 8.15. Fix ε > 0, then for δ > 0 sufficiently small the following statement is true: SupposeIj , Jj ⊂ [0, L] for j = 1, . . . , n are intervals with length satisfying

L1−δ ≤ |Ij | , |Jj | (8.29)

and define

S(I,J)def= (p, q) ∈ Z2d

∣∣ pj ∈ Ij , qj ∈ Jj , p 6= q .Then for µ satisfying Lε ≤ µ ≤ Ld−ε we have∑

(p,q)∈S(I,J)

g(µQ(p, q)) =

ˆI1×···×Id

ˆJ1×···×Jd

g(µQ(x, y)) dxdy +O

(L2(d−1−dδ)

µ

). (8.30)


Proof. First note that by Lemma 8.14∣∣∣∣∣∣∣∣∣∑

(p,q)∈S(I,J)

g(µQ(p, q))−∑

(p,q)∈S(I,J)

|Q(p,q)|≤µ−1L4dδ

g(µQ(p, q))

∣∣∣∣∣∣∣∣∣ .L2(d−1)+(1−4d)δ

µ

.L2(d−1−dδ)

µ.

Define the sum A(y) and the integral A as follows

A(y) =∑

(p,q)∈S(I,J)

|Q(p,q)|≤y

1 and A(y) =

ˆI1×···×Id

ˆJ1×···×Jd

1[−y,y](Q(u, v)) dudv

Then in the sense of distributions∑(p,q)∈S(I,J)


=

ˆ µ−1L4dδ

0g(µy)A′(y) dy

= −µˆ µ−1L4dδ

0g′(µy)A(y) dy + g(L4dδ)A(µ−1L4dδ)


0g′(µy)A(y) dy +O

(L2(d−2)−4dδ

µ

)where in the last inequality we applied Lemma 8.2 and the bound Lε ≤ µ ≤ Ld−ε. WritingA = A+ (A− A), we have∑

(p,q)∈S(I,J)



0g′(µy)A(y) dy + µ

ˆ µ−1L4dδ

0g′(µy)(A(y)− A(y)) dy +O

(L2(d−2)−4dδ

µ

)

By Theorem 8.11 (by choosing δ smaller than the δ used in the theorem) it follows that assumingy ≥ L−d+ε then ∣∣∣A(y)− A(y)

∣∣∣ . L2(d−1)−10dδy .

For y ≤ L−d+ε by the trivial bound∣∣∣A(y)− A(y)∣∣∣ . A(y) + A(y) . Ld−2+ε+δ .

Using the trivial bound g′(z) . 1 we have

µ

ˆ µ−1L4dδ

0

∣∣∣g′(µy)(A(y)− A(y))∣∣∣ dy

. µˆ L−d+ε

0

∣∣∣g′(µy)(A(y)− A(y))∣∣∣ dy +

ˆ µ−1L4dδ

L−d+ε

∣∣∣g′(µy)(A(y)− A(y))∣∣∣ dy

. µL−2+2ε+δ + µL2(d−1)−10dδ

ˆ µ−1L4dδ

L−d+εydy

. µ−1L2(d−1)−ε+δ + µ−1L2(d−2)−2dδ


where in the last inequality we used µ ≤ Ld−ε. Choosing δ sufficiently small in relation to ε, thisconstitutes an allowable error. The proof concludes by noting that by integration by parts

−µˆ µ−1L4dδ

0g′(µy)A(y) dy =

ˆI1×···×Id

ˆJ1×···×Jd

1[−µ−1L4dδ,µ−1L4dδ](Q(x, y))(g(µQ(x, y)) + g(L4dδ)

)dxdy

Proof of Theorem 8.9. We first note that by symmetry, it is sufficient to restrict ourselves to thepositive sector p, q ∈ Zd+. Note that Lemma 8.6 implies the subset of (p, q) such that pj = 0 orqj = 0 may be treated as an admissible error. Thus, it suffices to show

∑(p,q)∈Z2d

+p 6=q

W( pL,q

L

)g(µQ(p, q)) = L2d

¨

R2d+

W (x, y)g(L2µQ(x, y)) dxdy +O

(L2(d−1)−δ

µ

)

=L2(d−1)

µ

¨

R2d+

W (x, y)δdirac(Q(x, y)) dxdy +O

(L2(d−1)−δ

µ

)

Divide [0, Lδ]d × [0, Lδ]d into products of cubes Mj , Nk ⊂ R3+ of length L−10dδ. Define Wj,k to be

the average of W over Mj ×Nk:

Wj,k :=

Mj

Nk

W (x, y) dxdy

Note that if (x, y) ∈Mj ×Nk then from the smoothness of W

|W (x, y)−Wj,k| . L−10dδ .

Hence using Lemma 8.14∣∣∣∣∣∣∣∣∣∑

(p,q)∈Z2d+

p 6=q

W( pL,q

L

)g(µQ(p, q))−

∑j,k

∑p∈Mj ,q∈Nk

p 6=q

Wj,kg(µQ(p, q))

∣∣∣∣∣∣∣∣∣ .L2(d−1)(1+δ)+δ−10dδ

µ

.L2(d−1)−δ

µ

Applying Lemma 8.15 (taking δ to be sufficiently small) we obtain

∑j,k

∑p∈Mj ,q∈Nk

p 6=q

Wj,kg(µQ(p, q)) =∑j,k

ˆMj

ˆMk

Wj,kg(µQ(x, y)) dxdy +O

(L2(d−1)−δ

µ

)

=

ˆR3+

ˆR3+

W (p, q)g(µQ(x, y)) dxdy +O

(L2(d−1)−δ

µ

)


References

[1] D. Benedetto, F. Castella, R. Esposito, and M. Pulvirenti. Some considerations on the derivation of the nonlinearquantum Boltzmann equation. J. Statist. Phys., 116(1-4):381–410, 2004.

[2] D. Benedetto, F. Castella, R. Esposito, and M. Pulvirenti. Some considerations on the derivation of the nonlinearquantum Boltzmann equation. II. The low density regime. J. Stat. Phys., 124(2-4):951–996, 2006.

[3] D. Benedetto, F. Castella, R. Esposito, and M. Pulvirenti. From the N -body Schrodinger equation to thequantum Boltzmann equation: a term-by-term convergence result in the weak coupling regime. Comm. Math.Phys., 277(1):1–44, 2008.

[4] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinearevolution equations. I. Schrodinger equations. Geometric and Functional Analysis, 3(2):107–156, 1993. bibtex:Bourgain.

[5] J. Bourgain. On pair correlation for generic diagonal forms. arXiv e-prints, June 2016.[6] J. Bourgain and C. Demeter. The proof of the l2 decoupling conjecture. Ann. of Math. (2), 182(1):351–389, 2015.[7] T. Buckmaster, P. Germain, Z. Hani, and J. Shatah. Effective dynamics of the nonlinear Schrodinger equation

on large domains. Comm. Pure Appl. Math., 71(7):1407–1460, 2018.[8] N. Burq and N. Tzvetkov. Random data Cauchy theory for supercritical wave equations. I. Local theory. Invent.

Math., 173(3):449–475, 2008.[9] J. W. S. Cassels. An introduction to Diophantine approximation. Hafner Publishing Co., New York, 1972. Fac-

simile reprint of the 1957 edition, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45.[10] A. de Suzzoni and N. Tzvetkov. On the propagation of weakly nonlinear random dispersive waves. Arch. Ration.

Mech. Anal., 212(3):849–874, 2014.[11] Y. Deng, P. Germain, and L. Guth. Strichartz estimates for the Schrodinger equation on irrational tori. J. Funct.

Anal., 273(9):2846–2869, 2017.[12] Y. Deng, P. Germain, L. Guth, and S. Myerson. Strichartz estimates for the Schroedinger equation on non-

rectangular two-dimensional tori. arXiv e-prints, Oct. 2018.[13] L. Erdos. Lecture notes on quantum brownian motion. In Quantum Theory from Small to Large Scales, pages

3–98. Oxford University Press, may 2012.[14] L. Erdos, M. Salmhofer, and H. Yau. On the quantum Boltzmann equation. J. Statist. Phys., 116(1-4):367–380,

2004.[15] L. Erdos, M. Salmhofer, and H. Yau. Quantum diffusion of the random Schrodinger evolution in the scaling

limit. II. The recollision diagrams. Comm. Math. Phys., 271(1):1–53, 2007.[16] L. Erdos, M. Salmhofer, and H. Yau. Quantum diffusion of the random Schrodinger evolution in the scaling

limit. Acta Math., 200(2):211–277, 2008.[17] M. Escobedo and J. J. L. Velazquez. On the theory of weak turbulence for the nonlinear Schrodinger equation.

Mem. Amer. Math. Soc., 238(1124):v+107, 2015.[18] E. Faou. Linearized wave turbulence convergence results for three-wave systems. arXiv e-prints, May 2018.[19] E. Faou, P. Germain, and H. Hani. The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrodinger

equation. Journal of the American Mathematical Society, 2015.[20] I. Gallagher, L. Saint-Raymond, and B. Texier. From Newton to Boltzmann: hard spheres and short-range

potentials. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zurich, 2013.[21] P. Germain, H. Hani, and L. Thomann. On the continuous resonant equation for NLS, II: Statistical study. Anal.

PDE, 8(7):1733–1756, 2015.[22] P. Germain, H. Hani, and L. Thomann. On the continuous resonant equation for NLS. I. Deterministic analysis.

J. Math. Pures Appl. (9), 105(1):131–163, 2016.[23] P. Germain, A. D. Ionescu, and M.-B. Tran. Optimal local well-posedness theory for the kinetic wave equation.

arXiv e-prints, Nov. 2017.[24] K. Hasselmann. On the non-linear energy transfer in a gravity-wave spectrum Part 1. General theory. Journal

of Fluid Mechanics, 12(04):481–500, Apr. 1962.[25] K. Hasselmann. On the non-linear energy transfer in a gravity wave spectrum. II. Conservation theorems; wave-

particle analogy; irreversibility. J. Fluid Mech., 15:273–281, 1963.[26] A. D. Ionescu and B. Pausader. The energy-critical defocusing NLS on T3. Duke Math. J., 161(8):1581–1612,

2012.[27] O. E. Lanford, III. Time evolution of large classical systems. Springer, Berlin, 1975.[28] J. Lukkarinen and H. Spohn. Not to normal order—notes on the kinetic limit for weakly interacting quantum

fluids. Journal of Statistical Physics, 134(5):1133–1172, Mar 2009.[29] J. Lukkarinen and H. Spohn. Weakly nonlinear Schrodinger equation with random initial data. Invent. Math.,

183(1):79–188, 2011.


[30] A. J. Majda, D. W. McLaughlin, and E. G. Tabak. A one-dimensional model for dispersive wave turbulence. J.Nonlinear Sci., 7(1):9–44, 1997.

[31] S. Nazarenko. Wave turbulence, volume 825 of Lecture Notes in Physics. Springer, Heidelberg, 2011.[32] A. C. Newell and B. Rumpf. Wave turbulence: a story far from over. In Advances in wave turbulence, volume 83

of World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, pages 1–51. World Sci. Publ., Hackensack, NJ, 2013.[33] R. Peierls. Zur kinetischen theorie der warmeleitung in kristallen. Annalen der Physik, 395(8):1055–1101, 1929.[34] A. Soffer and M.-B. Tran. On the energy cascade of acoustic wave turbulence: Beyond Kolmogorov-Zakharov

solutions. arXiv e-prints, Nov. 2018.[35] H. Spohn. Derivation of the transport equation for electrons moving through random impurities. J. Statist. Phys.,

17(6):385–412, 1977.[36] H. Spohn. The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat.

Phys., 124(2-4):1041–1104, 2006.[37] E. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Prince-

ton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S.Murphy, Monographs in Harmonic Analysis, III.

[38] V. Zakharov, V. L’vov, and G. Falkovich. Kolmogorov spectra of turbulence I: Wave turbulence. Springer Science& Business Media, 2012.

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