On L^{12} square root cancellation for exponential sums associated with nondegenerate curves in {\mathbb R}^4
aa r X i v : . [ m a t h . C A ] J a n ON L SQUARE ROOT CANCELLATION FOR EXPONENTIAL SUMSASSOCIATED WITH NONDEGENERATE CURVES IN R CIPRIAN DEMETER
Abstract.
We prove sharp L estimates for exponential sums associated with nonde-generate curves in R . We place Bourgain’s seminal result [2] in a larger framework thatcontains a continuum of estimates of different flavor. We enlarge the spectrum of meth-ods by combining decoupling with quadratic Weyl sum estimates, to address new cases ofinterest. All results are proved in the general framework of real analytic curves. Introduction
Throughout this paper, φ , φ will be real analytic functions defined on some open intervalcontaining [ , k φ ′ k k C = X ≤ n ≤ max ≤ t ≤ | φ ( n ) k ( t ) | ≤ A , k ∈ { , } , (1) A ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det " φ (3)3 ( t ) φ (4)3 ( t ) φ (3)4 ( s ) φ (4)4 ( s ) ≤ A , t, s ∈ [ 12 , , (2) | φ (3)3 ( t ) | ≥ A , t ∈ [ 12 , . (3) A , . . . , A are positive numbers that will determine the implicit constants in various in-equalities. While φ , φ being C would suffice for our purposes, we choose to work with realanalytic functions for purely aesthetic reasons. The examples of most immediate interestare power functions φ ( t ) = t a , φ ( t ) = t b , with the real numbers a and b satisfying therestrictions a = b and a, b
6∈ { , , } .For a finite interval I ⊂ Z , we write (ignoring the dependence on φ k ) E I,N ( x ) = X n ∈ I e ( nx + n x + φ ( nN ) x + φ ( nN ) x ) . We make the following conjecture.
Conjecture 1.1.
Assume α ≥ β ≥ and α + β = 3 . Let ω = [0 , N α ] and ω = [0 , N β ] .Assume φ , φ are real analytic on (0 , and satisfy (1) , (2) and (3) . Then Z [0 , × [0 , × ω × ω |E [ N ,N ] ,N ( x ) | dx . ǫ N ǫ . (4) Key words and phrases. decoupling, Weyl sums, square root cancellation.The author is partially supported by the NSF grant DMS-1800305.AMS subject classification: Primary 42A45, Secondary 11L07 .
For virtually all conceivable applications of (4), both φ and φ will satisfy (3). Becauseof this symmetry, our restriction α ≥ β is essentially meaningless.Let us write A . B or A = O ( B ) if | A | ≤ CB for some, possibly large, but universalconstant C . The notation A ∼ B will mean that A . B and B . A at the same time. Wewill write A ≪ B or A = o ( B ) if | A | ≤ cB , for some small enough, universal constant c .The notation A / B will mean A . (log N ) O (1) B , where N will be the key scale parameter.Since |E [ N ,N ] ,N ( x ) | ∼ N if x ∈ [0 , o ( N )] × [0 , o ( N )] × [0 , o (1)] , the exponent 9 is optimal in(4). While the large value N is attained by |E [ N ,N ] ,N ( x ) | for a small subset of x , estimate (4)implies that the average value of the exponential sum on the given domain is O ( N + ǫ ), whenmeasured in L . We call this L square root cancellation. The relevance of the requirement α + β = 3 is that it guarantees | [0 , × [0 , × ω × ω | ( √ N ) = N . The case ( α, β ) = (2 ,
1) of the conjecture has been settled in [2]. This inequality wasproposed by Huxley [8], with φ ( t ) = t / , φ ( t ) = t / . In [2], it serves as the main ingredientin sharpening the record on the Lindel¨of hypothesis.The only other known case prior to our work was ( α, β ) = (3 , x andthe function φ play no role in this case, as φ ( nN ) x = O (1). We treat e ( φ ( nN ) x ) as acoefficient c n = O (1). In this case, (4) follows from the inequality Z [0 , × [0 ,N ] | X n ∈ I c n e ( nx + n x + φ ( nN ) x | dx dx dx . ǫ N ǫ k c n k l ∞ . Assuming φ satisfies (1) and (3), this was known as the Main Conjecture in Vinogradov’sMean Value Theorem, and was first solved in [10], then in [5]. This reduction fails for allother values α <
3, since the length of the interval ω becomes too small. The proofs in [2]and [5] for α = 2 and α = 3 are fairly different, in spite of both relying entirely on abstractdecoupling.Our main result here verifies Conjecture 1.1 in the range ≤ α < Theorem 1.2.
Assume that ≤ α ≤ . Assume φ , φ are real analytic on (0 , and satisfy (1) , (2) and (3) .Then Z [0 , × [0 , × ω × ω |E [ N ,N ] ,N ( x ) | dx . ǫ N ǫ . (5)If φ ( t ) = t a , φ ( t ) = t b with α ≤ a , β ≤ b , a + b ≤ a = b and a, b
6∈ { , , } , a verysimple rescaling argument for each member of the sum E [1 ,N ] ,N ( x ) = X ≤ M ≤ NM dyadic E [ M ,M ] ,N ( x )shows that (5) holds with E [ N ,N ] ,N replaced with E [1 ,N ] ,N ( x ). In particular, this is alwaysthe case for the moment curve φ ( t ) = t , φ ( t ) = t . Other cases such as ( α, β ) = (2 , a, b ) = ( , ) require slightly more sophisticated arguments similar to the one in [2], butwill not be pursued here. SQUARE ROOT CANCELLATION 3
Theorem 1.2 will follow from its bilinear analog. We prove this reduction in Section 7.
Theorem 1.3.
Let I , I be intervals of length ∼ N in [ N , N ] , with dist ( I , I ) ∼ N . Assume φ , φ are real analytic on (0 , and satisfy (1) , (2) and (3) . Assume that ≤ α ≤ .Then Z [0 , × [0 , × ω × ω |E I ,N ( x ) E I ,N ( x ) | dx . ǫ N ǫ . The implicit constant in this inequality is uniform over A , . . . , A ∼ . The reason we prove Theorem 1.2 using bilinear, as opposed to linear or trilinear methods,is rather delicate. As explained earlier, our results are sharp, the exponent 9 in (5) cannotbe lowered. Each of the decoupling inequalities relevant to this paper has a certain criticalexponent p c >
2. Experience shows that to achieve sharp results via decoupling, this toolmust be used at the critical exponent p c . When we apply decoupling on spatial balls ofradius M , we decouple the curve into arcs of length M − / . It seems very likely that thecritical exponent for such a decoupling in the linear setting is larger than 12. See [7] fora detailed discussion on this. Because of this, using linear L decoupling turns out to beinefficient. Bilinearizing instead, gives us access to the L decoupling of the parabola. Thisis an ideal scenario, since 6 is precisely the critical exponent in this setting.The fact that 12 = 6 × × L estimates forhypersurfaces in R . But the critical exponent here is 10 /
3, not 4.Most of the paper is devoted to proving Theorem 1.3. The proof will combine abstractdecoupling methods with quadratic Weyl sum estimates. The decoupling techniques areintroduced in Section 2. While these results are by now standard, the observation that thesuperficially stronger l ( L ) decoupling holds true for nondegenerate curves in the bilinearsetting appears to be new. One of the key features in our argument is the use of this inequalityin places where the l ( L ) decoupling used in [2] becomes inefficient. We combine this finerdecoupling with estimates from number theory. In short, here is how our approach works.The initial integral involves quartic Weyl sums, for which sharp estimates are totally out ofreach at the moment. Decoupling is applied once or twice in order to lower the complexity ofthe sums, to the level of manageable quadratic Weyl sums. These sums will appear in variouscombinations, and need to be tackled with extreme care, using various counting argumentssuch as Lemma 4.1 and Lemma 4.2.In Section 3, we start with a careful examination of Bourgain’s argument from [2], for α = 2. In many ways, this case turns out to be the easiest, as it works via just l L decoupling, and without any input from number theory. In Section 4 we introduce ournew methodology, addressing the symmetric case α = β = . This ends up being the mostdelicate case, since it captures the biggest region near the origin where constructive (andnear constructive) interference occurs. Also, it is in this case that the curve looks mostgenuinely four dimensional, as both ω and ω are large. For comparison, recall that when α = 3 the curve degenerates to a three dimensional one. Sections 5 and 6 extend our methodto the remaining cases, by successively building on each other. The case ≤ α < L p estimates for the moment curve CIPRIAN DEMETER on spatial domains smaller than the torus. In [6] only the moment curve in R is considered,and all estimates there rely solely on decoupling techniques.There remain a lot of interesting related questions. One of them has to do with provingConjecture 1.1 in the range 2 < α <
3. We may speculate that the solution would combinesome of the tools from our paper with those used to solve the case α = 3, see also Remark3.1. Second, L p moments are also worth investigating for smaller values p <
12, in particularfor p = 10. See for example [1] for some recent progress and some interesting applications.Section 8 of our paper contains an example that describes some of the enemies and limitationsof square root cancellation in this setting. It seems plausible that small cap decoupling forthe parabola (see [6]) will be the right tool to attack this problem. We hope to address someof these questions in future work. Acknowledgment.
The author is grateful to Hongki Jung and Zane Li for pointing out a fewtypos in the earlier version of the manuscript.2.
Decoupling for nondegenerate curves in R Let us start by recalling the decoupling for nondegenerate curves in R . Theorem 2.1 ([3]) . Let φ : [0 , → R be a C function, with k φ ′ k C = A < ∞ and min ≤ t ≤ | φ ′′ ( t ) | = A > . Then for each f : [0 , → C and each ball B N ⊂ R with radius N we have k Z [0 , f ( t ) e ( tu + φ ( t ) w ) dt k L u,w ( B N ) . ǫ N ǫ ( X J ⊂ [0 , | J | = N − / k Z J f ( t ) e ( tu + φ ( t ) w ) dt k L u,w ( B N ) ) / . The implicit constant is uniform over A ∼ , A ∼ . We use this result to illustrate in the simplest terms the reduction of cubic terms used inthe next section for α = 2. Namely, let us show that for 1 ≤ α < Z [0 , | X ≤ m ≤ M e ( mu + ( m + m M α ) w ) | dudw . ǫ M ǫ . (6)The term m M α w is not negligible (in the sense of Lemma 9.2), as it is not O (1). However,after a change of variables and using periodicity in u , we may rewrite the integral as1 M Z [0 ,M ] | X ≤ m ≤ M e ( mM u + φ ( mM ) w ) | dudw, where φ ( t ) = t + t M − α . Note that A , A ∼
1, uniformly over M . Inequality (6) is now astandard consequence of Theorem 2.1 with N = M . The cubic term becomes a perturbationof the quadratic term, and does not significantly affect the constant A . Theorem 2.4 willformalize this approach in four dimensions.Throughout this section, φ , φ are arbitrary functions satisfying (1) and (2). We denoteby E the extension operator associated with the curve ΦΦ( t ) = ( t, t , φ ( t ) , φ ( t )) , t ∈ [ 12 , . (7) SQUARE ROOT CANCELLATION 5
More precisely, for f : [ , → C and I ⊂ [ ,
1] we write E I f ( x ) = Z I f ( t ) e ( tx + t x + φ ( t ) x + φ ( t ) x ) dt. The following l ( L ) decoupling was proved in [2], see also [4]. It is in fact a bilinear versionof the l L decoupling for the curve (7). Theorem 2.2.
Let I , I be two intervals of length ∼ in [ , , with dist ( I , I ) ∼ . Letalso f i : [ , → C . Then for each ball B N of radius N in R we have k E I f E I f k L ( B N ) . ǫ N + ǫ ( X J ⊂ I X J ⊂ I k E J f E J f k L ( B N ) ) / . The sum on the right is over intervals J i of length N − / . In [2], this result is used in conjunction with the following estimate, an easy consequenceof transversality. k E J f E J f k L ( B N ) . N − k E J f k L ( B N ) k E J f k L ( B N ) . (8)This is inequality (13) in [4], and there is a detailed proof there. For reader’s convenience,we sketch a somewhat informal argument below.The Fourier transform of | E J i f i | is supported inside a rectangular box with dimensions ∼ N − × N − × N − × N − / . We have the following wavepacket representation on B N ,slightly simplified for exposition purposes | E J i f i ( x ) | ≈ X P i ∈P i a P i P i ( x ) . The coefficients a P i are nonnegative reals. The rectangular boxes P i have dimensions ∼ N × N × N × N / and tile B N . They can be thought of as N / -neighborhoods of cubeswith diameter ∼ N inside hyperplanes H i . Since dist ( J , J ) ∼
1, the angle between thenormal vectors of H i is ∼
1. Thus, we have that | P ∩ P | . N . We conclude by writing k E J f E J f k L ( B N ) ∼ X P ∈P X P ∈P a P a P | P ∩ P | . N − X P ∈P X P ∈P a P a P | P || P |≈ N − k E J f k L ( B N ) k E J f k L ( B N ) . It is worth observing that . in inequality (8) is essentially an (approximate) similarity ≈ , making (8) extremely efficient. Indeed, since P and P intersect B N , we have that | P ∩ P | ∼ N .To address new values of α in this paper, we will need the following l ( L ) decoupling.This implies the previous l ( L ) decoupling, and provides a critical improvement in the caseswhen the terms in the sum are of significantly different sizes. Theorem 2.3.
Let I , I be two intervals of length ∼ in [ , , with dist ( I , I ) ∼ . Letalso f i : [ , → C . Then for each ball B N of radius N in R we have k E I f E I f k L ( B N ) . ǫ N ǫ ( X J ⊂ I X J ⊂ I k E J f E J f k L ( B N ) ) / . CIPRIAN DEMETER
The sum on the right is over intervals J of length N − / .Proof. We briefly sketch the argument, that follows closely the one in [2]. Let b ( N ) be thebest constant such that k E I f E I f k L ( B N ) ≤ b ( N )( X J ⊂ I X J ⊂ I k E J f E J f k L ( B N ) ) / holds for all functions and balls as above. Fix B N and let B be a finitely overlapping coverof B N with balls ∆ of radius N / . It will soon become clear that the exponent 2 / k E I f E I f k L ( B N ) ≤ b ( N / )( X H ⊂ I X H ⊂ I k E H f E H f k L ( B N ) ) / . The intervals H have length N − / . We next analyze each term in the sum. Let l i be theleft endpoint of H i , and write a generic point t i ∈ H i as t i = l i + s i , with s i ∈ [0 , N − / ]. Weuse Taylor’s formula for k ∈ { , } φ k ( t i ) = φ k ( l i ) + φ ′ k ( l i ) s i + φ ′′ k ( l i )2 s i + ψ k,i ( s i ) , where, due to (1), k ψ k,i k L ∞ ([0 ,N − / ]) = O ( N ). Let us write y = x + 2 l x + φ ′ ( l ) x + φ ′ ( l ) x y = x + 2 l x + φ ′ ( l ) x + φ ′ ( l ) x y = x + φ ′′ ( l )2 x + φ ′′ ( l )2 x y = x + φ ′′ ( l )2 x + φ ′′ ( l )2 x . It follows that | E H f ( x ) E H f ( x ) | = | Z [0 ,N − / ] f ( l + s ) e ( s y + s y ) f ( l + s ) e ( s y + s y ) e ( L ( y, s , s )) ds ds | . Here L ( y, s , s ) = x ( ψ , ( s ) + ψ , ( s )) + x ( ψ , ( s ) + ψ , ( s )) . Lemma 9.1 shows that x , x depend linearly on y , y , y , y with coefficients O (1). Thisallows us to write e ( L ( y, s , s )) = e ( X i =1 y i ( g i ( s ) + h i ( s )))with k g i k ∞ , k h i k ∞ = O ( N ). Letting ¯ f i ( s i ) = f i ( l i + s i ) and η ( s , s ) = s + g ( s ) + h ( s ) η ( s , s ) = s + g ( s ) + h ( s ) η ( s , s ) = s + g ( s ) + h ( s ) η ( s , s ) = s + g ( s ) + h ( s )we write | E H f ( x ) E H f ( x ) | = SQUARE ROOT CANCELLATION 7 | Z [0 ,N − / ] ¯ f ( s ) e ( y η ( s , s ) + y η ( s , s )) ¯ f ( s ) e ( y η ( s , s ) + y η ( s , s )) ds ds | . For ¯ J i ⊂ [0 , N − / ] we write I ¯ J , ¯ J ( y ) = | Z ¯ J × ¯ J ¯ f ( s ) e ( y η ( s , s ) + y η ( s , s )) ¯ f ( s ) e ( y η ( s , s ) + y η ( s , s )) ds ds | . This is the extension operator associated with the surface ( η , . . . , η ), applied to the function f ⊗ f .We use Lemma 9.1 to write Z B N | E J f ( x ) E J f ( x ) | dx = Z ¯ B N I ¯ J , ¯ J ( y ) dy. Here ¯ B N is a ball of radius ∼ N and J i = ¯ J i + l i . Note that the surface ( η , . . . , η ) is within O ( N − ) from the surface ( s , s , s , s ) , s i ∈ [0 , N − / ] , so -for decoupling purposes- the two surfaces are indistinguishable when paired with spatialvariables y ranging through a ball of radius N . The latter surface admits an l ( L ) decou-pling, as can be easily seen by using Theorem 2.1 twice. The same remains true for thesurface ( η , . . . , η ), and thus kI [0 ,N − / ] , [0 ,N − / ] k L ( ¯ B N ) . ǫ N ǫ ( X ¯ J , ¯ J ⊂ [0 ,N − / ] kI ¯ J , ¯ J k L ( ¯ B N ) ) / . The sum on the right is over intervals of length N − / . If we undo the change of variableswe find k E H f E H f k L ( B N ) . ǫ N ǫ ( X J ⊂ H X J ⊂ H k E J f E J f k L ( B N ) ) / . Putting things together, we have proved the bootstrapping inequality b ( N ) . ǫ N ǫ b ( N / ) . We conclude that b ( N ) . ǫ N ǫ , as desired. (cid:3) We will also record the following close relative of Theorem 2.3, that will be needed in thenext sections.
Theorem 2.4.
Assume ψ , . . . , ψ : [ − , → R have C norm O (1) , and in addition satisfy | ψ ′′ ( t ) | , | ψ ′′ ( t ) | ≪ , ∀ | t | ≤ and | ψ ′′ ( t ) | , | ψ ′′ ( t ) | ∼ , ∀ | t | ≤ . Let E be the extension operator associated with the surface Ψ( ξ , ξ ) = ( ξ , ξ , ψ ( ξ ) + ψ ( ξ ) , ψ ( ξ ) + ψ ( ξ )) , | ξ | , | ξ | ≤ . CIPRIAN DEMETER
More precisely, for F : [ − , → C , R ⊂ [ − , and x ∈ R we write E R F ( x ) = Z R F ( ξ , ξ ) e ( x · Ψ( ξ , ξ )) dξ dξ . Then for each ball B N ⊂ R with radius N we have k E [ − , F k L ( B N ) . ǫ N ǫ ( X H ,H ⊂ [ − , k E H × H F k L ( B N ) ) / , (9) where the sum is taken over intervals of length N − / .In particular, for each constant coefficients c m ,m ∈ C we have k X m ≤ N / X m ≤ N / c m ,m e ( x · Ψ( m N / , m N / )) k L ( B N ) . ǫ N ǫ k c m ,m k l | B N | / , (10) while if M ≥ N / we have k X m ≤ M X m ≤ M c m ,m e ( x · Ψ( m M , m M )) k L ( B N ) (11) . ǫ N ǫ ( X J ,J k X m ∈ J X m ∈ J c m ,m e ( x · Ψ( m M , m M )) k L ( B N ) ) / , with J i intervals of length ∼ M N − / partitioning [1 , M ] .The implicit constants in both inequalities are independent of N , M and of ψ i .Proof. The exponential sum estimates (10), (11) are standard consequences of (9), so wewill focus on proving the latter. WhenΨ( ξ , ξ ) = ( ξ , ξ , C ξ + C ξ , C ξ + C ξ )with | C | , | C | ∼ | C | , | C | ≪
1, the result follows by applying l ( L ) Theorem 2.1twice (after an initial affine change of variables that reduces it to the case C = C = 1, C = C = 0).We use a bootstrapping argument similar to the one in Theorem 2.3. Let d ( N ) be thesmallest constant in (9). We need to prove that d ( N ) . ǫ N ǫ .We first note that k E [ − , F k L ( B N ) ≤ d ( N / )( X U ,U ⊂ [ − , k E U × U F k L ( B N ) ) / , where U i are intervals of length N − / centered at l i . When | t | < N − / we have ψ i ( t ) = ψ i ( l i ) + ψ ′ i ( l i ) t + C i t + O ( 1 N ) , where | C | , | C | ∼ | C | , | C | ≪
1. It follows that, after an affine change of variables,the restriction of Ψ to U × U may be parametrized as( ξ , ξ , C ξ + C ξ + O ( 1 N ) , C ξ + C ξ + O ( 1 N )) , | ξ | , | ξ | = O ( N − / ) . We decouple this on B N , using the observation at the beginning of the proof. We find k E U × U F k L ( B N ) . ǫ N ǫ ( X H ⊂ U ,H ⊂ U k E H × H F k L ( B N ) ) / . SQUARE ROOT CANCELLATION 9
It follows that d ( N ) . ǫ N ǫ d ( N / ), which forces d ( N ) . ǫ N ǫ . (cid:3) Bourgain’s argument for the case α = 2For the remainder of the paper, we will use the following notation E I ( x ) = X n ∈ I e ( N Φ( nN ) x ) = X n ∈ I e ( nx + n N x + φ ( nN ) N x + φ ( nN ) N x ) . Note that, compared to E I,N , we dropped the subscript N and renormalized the variables x , x and x . Letting Ω = [0 , N ] × [0 , x , we need to prove that Z Ω |E I E I | . ǫ N ǫ . Bourgain’s argument from [2] for the case α = 2 of Theorem 1.3 involves three successivedecouplings. We simplify it slightly it and reduce it to only two decouplings.Step 1. Cover Ω with cubes B of side length 1, apply l ( L ) decoupling (Theorem 2.2)on each B (or rather N B , after rescaling), then sum these estimates to get Z Ω |E I E I | . ǫ N ǫ X J ⊂ I X J ⊂ I X B ⊂ Ω Z B |E J E J | . Here J , J are intervals of length N / .The remaining part of the argument will show the uniform estimate O ( N ǫ ) for eachterm P B ⊂ Ω R B |E J E J | . Fix J i = [ h i , h i + N / ].Step 2. Note that when x ∈ Ω |E J i ( x ) | = | X m ≤ N / c m ( x ) e ( mu i + m w i + η i ( m ) x ) | where u i = x + h i N x + φ ′ ( h i N ) x w i = x N + φ ′′ ( h i N ) x N η i ( m ) = m φ ′′′ ( hiN )3! N + m φ ′′′′ ( hiN )4! N + . . . . (12)We hide the whole contribution from x into the coefficient c m ( x ). Indeed, since φ ( h i + mN ) N x = φ ( h i N ) N x + mφ ′ ( h i N ) x + O (1) ,x does not contribute significantly with quadratic or higher order terms, so it produces nocancellations. We will only use that c m ( x ) = O (1).At this point, we seek a change of variables. We want the new domain of integration tobe a rectangular box, to allow us to separate the four-variable integral R Ω |E J E J | into theproduct of two-variable integrals. Note that the ranges of x , x , x are the same, [0 , N ],but x is restricted to the smaller interval [0 , range of x to [0 , N ], because the individual waves e ( mφ ′ ( h i N ) x ) have different periods withrespect to the variable x . Because of this, the variable x is practically useless from thispoint on, it will not generate oscillations. To generate a fourth variable with range [0 , N ] forthe purpose of a change of coordinates, Bourgain produces a piece of magic.First, he applies (8) on each cube N B Z B |E J E J | = N − Z NB |E J ( · N ) E J ( · N ) | . N − Z NB |E J ( · N ) | Z NB |E J ( · N ) | = Z B |E J | Z B |E J | . Second, he uses the following abstract inequality, that only relies on the positivity of |E J i | X B ⊂ Ω Z B |E J | Z B |E J | . Z Ω dx Z ( y,z ) ∈ [ − , × [ − , |E J ( x + y ) E J ( x + z ) | dydz. Using periodicity in the y , z variables, this is . N Z x ∈ [0 ,N ] x ,y ,y ,y ,z ,z ,z ∈ [ − , dx . . . dz Z y ,z ,x ,x ∈ [0 ,N ] |E J ( x + y ) E J ( x + z ) | dy dz dx dx . In short, the variable x is now replaced with the new variables y and z . It remains toprove that Z y ,z ,x ,x ∈ [0 ,N ] |E J ( x + y ) E J ( x + z ) | dy dz dx dx . ǫ N ǫ , (13)uniformly over x , x , y , y , y , z , z , z . With these variables fixed, we make the affinechange of variables ( y , z , x , x ) ( u , u , w , w ) u = ( y + x ) + h N ( x + y ) + φ ′ ( h N )( x + y ) u = ( z + x ) + h N ( x + z ) + φ ′ ( h N )( x + z ) w = x + y N + φ ′′ ( h N ) x + y N w = x + z N + φ ′′ ( h N ) x + z N . (14)The Jacobian is ∼ N , due to (3). Note that x + y = A ( w − w ), where A depends juston h , h , and | A | ∼ N . Using (12) we may write the last integral as N × Z | ui | . N | wi | . | X m i ≤ N c m ,m e ( m u + m w + m u + m w + ( η ( m ) + η ( m )) A ( w − w )) | . (15)The coefficient c m ,m depends only on m , m , x , y , z , but not on the variables of integra-tion u i , w i . The argument of each exponential may be rewritten as m N / u N / + ( ψ ( m N / ) + ψ ( m N / )) w N + m N / u N / + ( ψ ( m N / ) + ψ ( m N / )) w N SQUARE ROOT CANCELLATION 11 where ψ ( ξ ) = ξ + ξ Aφ ′′′ ( h N )3! N / + ξ Aφ ′′′′ ( h N )4! N + . . .ψ ( ξ ) = ξ Aφ ′′′ ( h N )3! N / + ξ Aφ ′′′′ ( h N )4! N + . . .ψ ( ξ ) = − ξ Aφ ′′′ ( h N )3! N / − ξ Aφ ′′′′ ( h N )4! N − . . .ψ ( ξ ) = ξ − ξ Aφ ′′′ ( h N )3! N / − ξ Aφ ′′′′ ( h N )4! N − . . . . These functions satisfy the requirements in Theorem 2.4. The integral (15) is the same as N − Z ui = O ( N / wi = O ( N ) | X m ≤ N / X m ≤ N / c m ,m e (( u , u , w , w ) · Ψ( m N / , m N / )) | du du dw dw . If we cover the domain of integration with ∼ N balls B N and apply (10) on each of them,we may dominate the above expression by N − N ( N + ǫ N / ) = N ǫ . This proves (13) and ends the proof. Note that this argument treats the cubic and higherorder terms as perturbations of quadratic factors, as explained in the proof of (6).In summary, what is special about the case α = 2 is that the range of x in our initialintegral over Ω is [0 , N ]. This was needed in producing the large spatial range w i = O ( N ) forour final variables, crucial for the application of (10). This inequality provides decouplinginto point masses, reducing the initial exponential sum to individual waves. In Section 6we will see that when α is slightly smaller than 2, inequality (11) will have to replace (10),leading to quadratic Weyl sums whose handling demands number theory. Remark . It is not clear whether a version of Bourgain’s method could be made to workin the range 2 < α <
3. If successful, this would potentially provide a new argument forVinogradov’s Mean Value Theorem in R . Decoupling on cubes with size N β − and using(8) on balls B N β leads to variables y , z with associated period equal to 1, much bigger thantheir range N β − . The change of variables (14) is no longer efficient in this case.4. Proof of Theorem 1.3 in the case α = This time we let Ω = [0 , × [0 , N ] × [0 , N / ] × [0 , N / ]. Recall that E I ( x ) = X n ∈ I e ( nx + n N x + φ ( nN ) N x + φ ( nN ) N x ) . We need to prove Z Ω |E I E I | dx . ǫ N ǫ . (16)In this case, we will only need assumptions (1) and (2), but not (3). We start by presentinga general principle that will explain the subtleties of our argument. See also Remark 4.3.Consider two partitions of Ω, one into cubes B with side length l and another one intocubes ∆ with side length L ≥ l . The intervals J i have length q Nl and partition I i . The intervals U i have length q NL and partition I i . The following holds, via two applications ofTheorem 2.3 (on cubes B and ∆, combined with Minkowski’s inequality) kE I E I k L (Ω) . ǫ N ǫ ( X J ,J kE J E J k L (Ω) ) / . ǫ N ǫ ( X U ,U kE U E U k L (Ω) ) / . (17)Also, combining the above inequalities with H¨older shows that kE I E I k L (Ω) . ǫ N ǫ ( ♯ ( J , J )) ( X J ,J kE J E J k L (Ω) ) / . ǫ N ǫ ( ♯ ( U , U )) ( X U ,U kE U E U k L (Ω) ) / . (18)Invoking periodicity in x , x and the invariance of (1), (2), (3) under the change of sign φ k
7→ − φ k , (16) is equivalent with proving that Z [ − N / ,N / ] × [ − N,N ] × [ − N / ,N / ] × [ − N / ,N / ] |E I ( x ) E I ( x ) | dx . ǫ N + ǫ . (19)We first demonstrate the inefficiency of l L decoupling for this case, by working with thesmaller domain S = [ − o ( N / ) , o ( N / )] . We cover S with unit cubes and apply decoupling into intervals J , J of length N / as inthe previous section, to dominate Z S |E I ( x ) E I ( x ) | dx . ǫ N ǫ N − ) X J ,J Z S |E J ( x ) E J ( x ) | dx. (20)We will next show that the right hand side is too big, thus leading to an overestimate forour initial integral. When J = [ h, h + N / ] and n = h + m ∈ J we write φ k ( nN ) = φ k ( hN ) + φ ′ k ( hN ) mN + 12 φ ′′ k ( hN )( mN ) + O ( mN ) . If | x | , | x | ≪ N / and m ≤ N / , we guarantee that the contribution from higher orderterms is small O ( mN ) N ( | x | + | x | ) ≪ . If we collect the contributions from linear and quadratic terms we find |E J ( x ) | = | X m ≤ N / e ( mu + m w + o (1)) | where ( u = x + hN x + φ ′ ( hN ) x + φ ′ ( hN ) x w = x N + φ ′′ ( hN ) x N + φ ′′ ( hN ) x N . Using Lemma 9.1 we write Z S |E J ( x ) E J ( x ) | dx & N Z ( u ,u ∈ [0 ,o ( N / w ,w ∈ [0 ,o ( N − / | X m ≤ N / e ( mu + m w + o (1)) | | X m ≤ N / e ( mu + m w + o (1)) | . SQUARE ROOT CANCELLATION 13
We now use the fact that we have constructive interference | X m ≤ N / e ( mu + m w + o (1)) | ∼ N / on the set of measure ∼ N ( u, w ) ∈ ( [ l ∈{ , ,...,o ( √ N ) } [ l, l + 1 √ N ]) × [0 , N ] . It follows that Z S |E J ( x ) E J ( x ) | dx & N N N − = N . It is not hard to prove that this lower bound is sharp, but this has no relevance to us here.The point of working with the symmetric domain S was to make sure that w , w ∼ N arein the new domain of integration. Going back to (20), the l ( L ) decoupling method leadsto the upper bound Z S |E I ( x ) E I ( x ) | dx . ǫ N ǫ . This falls short by the factor N / from proving (19).The second inequality in (4) shows that using l ( L ) decoupling on cubes ∆ that are largerthan N will only worsen the upper bounds we get. On the other hand, working with smallercubes will render decoupling inefficient. The resulting exponential sums will be very difficultto handle using number theory, since the cubic terms are no longer O (1) in this case.Let us now describe the correct approach, that will critically rely on l , rather than l decoupling. The following level set estimate will play a key role in various countingarguments. The main strength of the lemma is in the case when | l | ∼ | l | .Throughout the remainder of the paper, the letter l will be used to denote integers, andtheir relative proximity to powers of 2 will be denoted using the symbol ∼ . We make theharmless convention to write 0 ∼ . Lemma 4.1.
Assume φ , φ satisfy (1) and (2) . Let l , l with max {| l | , | l |} ∼ j , j ≥ ,and let f ( t ) = l φ ′′ ( t ) + l φ ′′ ( t ) . Then we can partition the range of f into sets R s with ≤ s ≤ j , each of which is the unionof at most two intervals of length ∼ s , such that for each v ∈ R s we have | f − ( v + [ − O (1) , O (1)]) ∩ [ 12 , | . √ j + s . All implicit constants are universal over all pairs of such φ , φ and over l , l , s .Proof. The result is trivial if l = l = 0, so we will next assume that max {| l | , | l |} ≥ f to the interval [ , (cid:20) f ′ ( t ) f ′′ ( t ) (cid:21) = " φ (3)3 ( t ) φ (3)4 ( t ) φ (4)3 ( t ) φ (4)4 ( t ) l l (cid:21) , (2) implies that for each t ∈ [ ,
1] we havemax {| f ′ ( t ) | , | f ′′ ( t ) |} ∼ j . (21) We let t be a point in [ ,
1] where | f ′ | attains its minimum. If | f ′ ( t ) | ∼ j , then we maytake R j to be the whole range of f , and all other R s to be empty. Indeed, the Mean ValueTheorem shows that | f ( t ) − f ( t ) | & | t − t | & − j . It is worth observing that if | l | ≫ | l | , then (3) would immediatelyguarantee that | f ′ ( t ) | ∼ j .We now assume that | f ′ ( t ) | ≪ j . Due to (21), we must have that | f ′′ ( t ) | ∼ j . Wewrite for t ∈ [ , f ( t ) = f ( t ) + f ′ ( t )( t − t ) + f ′′ ( t )( t − t ) O (2 j ( t − t ) ) . (22)Case 1. Consider s with 2 j ≥ s > C max { | f ′ ( t ) | j , } , for some large enough C independentof j . Using this and (22), we see that | f ( t ) − f ( t ) | ≪ s whenever | t − t | ≪ s − j . (23)Define R s = { v : | v − f ( t ) | ∼ s } . Let v ∈ R s and let w = v + O (1). Thus, we also have | w − f ( t ) | ∼ s . Let t , t be such that f ( t ) = v , f ( t ) = w . Using (23) it follows that | t − t | , | t − t | & s − j . Our assumptionshows that 2 s − j ≫ | f ′ ( t ) | j . Thus, | t − t | , | t − t | ≫ | f ′ ( t ) | j , and using (22) again we concludethat | f ( t i ) − f ( t ) | ∼ j | t i − t | . Thus, | t i − t | ∼ s − j . Using again (22) we find that if t , t are on the same side of t then | f ( t ) − f ( t ) | ∼ s + j | t − t | . We conclude that | t − t | . √ j + s , as desired.Next, we define R s for smaller values of s . We distinguish two cases.Case 2a. Assume now that | f ′ ( t ) | ≤ j/ . For s such that 2 s is the largest dyadic power ≤ C max { | f ′ ( t ) | j , } = C we define R s = { v : | v − f ( t ) | . s } . We also let R s ′ = ∅ for smaller values of s ′ . Let v ∈ R s and w = v + O (1). Let t , t be suchthat f ( t ) = v , f ( t ) = w . Since in fact | f ( t i ) − f ( t ) | .
1, (22) forces | t i − t | . − j/ ∼ √ j + s ,as desired.Case 2b. Assume now that | f ′ ( t ) | > j/ . For s such that 2 s is the largest dyadic power ≤ C max { | f ′ ( t ) | j , } = C | f ′ ( t ) | j we define R s = { v : | v − f ( t ) | . s } . SQUARE ROOT CANCELLATION 15
We also let R s ′ = ∅ for smaller values of s ′ . Let v ∈ R s and w = v + O (1). Let t , t be suchthat f ( t ) = v , f ( t ) = w . Using that | f ′ ( t ) | ≥ | f ′ ( t ) | for all t , we find that | f ( t ) − f ( t ) | ≥ | t − t || f ′ ( t ) | . We conclude that | t − t | . | f ′ ( t ) | ∼ √ j + s , as desired. (cid:3) From now on, we will implicitly assume that all Weyl sums are smooth, as in Lemma9.2. This can be easily arranged using partitions of unity, namely working with smooth γ satisfying X l ∈ Z γ ( · + l ) = 1 R . To simplify notation, these weights will be ignored.Cover Ω with unit cubes B = B p,l ,l = [0 , × [ p, p + 1] × [ l , l + 1] × [ l , l + 1] with p ≤ N, l , l ≤ N / . We first write Z Ω |E I E I | ∼ X B ⊂ Ω Z B |E I E I | . We use l decoupling (Theorem 2.3) on each B Z B |E I E I | . ǫ N ǫ ( X J ⊂ I X J ⊂ I ( Z B |E J E J | ) / ) where J i is of the form [ h i , h i + N / ]. When x ∈ B and J = [ h, h + N / ] |E J ( x ) | = | X m ≤ N / e ( mu + m w + m v + O ( N − / )) | where u = x + hN x + φ ′ ( hN ) x + φ ′ ( hN ) x w = x N + φ ′′ ( hN ) x N + φ ′′ ( hN ) x N v = φ ′′′ ( hN ) x + φ ′′′ ( hN ) x N . (24)The term O ( N − / ) can be dismissed as it produces tiny errors consistent with square rootcancellation. Note that since v = O ( N − / ), we have | X m ≤ N / e ( mu + m w + m v ) | ≈ | X m ≤ N / e ( mu + m w ) | . See Lemma 9.2 for a rigorous argument. The key point is that we may dismiss the cubicterms.Write I ( h , h , B ) = Z ( u ,u ,w ,w ) ∈ [0 , × [ a − O (1) N , a O (1) N ] × [ a − O (1) N , a O (1) N ] | Y i =1 X m i ≤ N / e ( m i u i + m i w i ) | du du dw dw , where ( a = p + l φ ′′ ( h N ) + l φ ′′ ( h N ) a = p + l φ ′′ ( h N ) + l φ ′′ ( h N ) . (25)Via the change of variables with Jacobian ∼ N (Lemma 9.1) u = x + h N x + φ ′ ( h N ) x + φ ′ ( h N ) x w = x N + φ ′′ ( h N ) x N + φ ′′ ( h N ) x N u = x + h N x + φ ′ ( h N ) x + φ ′ ( h N ) x w = x N + φ ′′ ( h N ) x N + φ ′′ ( h N ) x N we see that Z B |E J E J | . N I ( h , h , B ) . Writing I a = Z [0 , × [ a − O (1) N , a + O (1) N ] | X m ≤ N / e ( mu + m w ) | dudw we find that Z B |E J E J | . N I a I a . Let us analyze (25). The question is, for fixed B , what are the values of a , a that arise(modulo O (1) error terms), and what is their multiplicity, when h , h range through themultiples of N / in [1 , N ].Assume l ∼ j , l ∼ j , with 2 j , j ≤ N / . We may assume j ≤ j , the othercase is completely similar. We apply Lemma 4.1 to f ( t ) = ( l φ ′′ ( t ) + l φ ′′ ( t )). For each0 ≤ s , s ≤ j and each p we have O (2 s + s ) pairs ( a , a ) of integers with a − p ∈ R s ( l , l )and a − p ∈ R s ( l , l ). Note that we index the intervals R s i from Lemma 4.1 by l , l . Foreach such pair ( a , a ), (25) has O ( N j s s ) solutions ( h , h ). When we count solutions, wetolerate error terms of size O (1).Thus X B ⊂ Ω Z B |E I E I | . N X p ≤ N X j . N / X j . j X l ∼ j X l ∼ j X s ,s ≤ j ( N j + s s ) ( X a ∈ p + R s ( l ,l ) X a ∈ p + R s ( l ,l ) I / a I / a ) . N X p ≤ N X j . N / X j . j j + j ( N j ) X s ,s ≤ j ( X a ∈ p + R s ( l ,l ) I / a ) / ( X a ∈ p + R s ( l ,l ) I / a ) / . The last inequality follows from Cauchy–Schwarz. Next, we observe that p + R s i ( l , l ) ⊂ [ p − O (2 j ) , p + O (2 j )]. These intervals are roughly the same for roughly 2 j values of p . We SQUARE ROOT CANCELLATION 17 can thus dominate the above by / N X j . N / X j . j j + j ( N j ) j X H ⊂ [0 ,N ] | H | =2 j ( X a ∈ H I / a ) ∼ N X j . N / X H ⊂ [0 ,N ] | H | =2 j ( X a ∈ H I / a ) . (26)The sum runs over pairwise disjoint intervals H . It is easily seen to be O ( N ), by usingthe following lemma with M = N / . Lemma 4.2.
Let I a = Z [0 , × [ a − O (1) M , a + O (1) M ] | X m ≤ M e ( mu + m w ) | dudw For each j ≤ M we have X H ⊂ [0 ,M | H | =2 j ( X a ∈ H I / a ) / M j + M . Proof.
The arcs { x ∈ [0 ,
1) : dist ( x − bq , Z ) ≤ qM } , with 1 ≤ b ≤ q ≤ M and ( b, q ) = 1,cover [0 , I a with aM in some arc with q ∼ Q . Here Q isdyadic and Q . M . We separate the proof into two cases. Note that H/M ⊂ [0 ,
1] and haslength 2 j /M . Also, | b/q − b ′ /q ′ | ≥ /qq ′ .Case 1. Q > M j . Each H/M intersects . j Q M arcs with q ∼ Q . For each such b/q ,and each 1 ≤ k ≤ Mq there are ∼ M k Q values of a with | aM − bq | ∼ qM k . Call A ( Q, k ) the collection of all these a . For each a ∈ A ( Q, k ), Lemma 9.2 gives I a . k M ( M / k/ ) = 2 k M. The contribution from a ∈ A ( Q, k ) is X H ⊂ [0 ,M | H | =2 j ( X a ∈ H ∩ A ( Q,k ) I / a ) . M j ( 2 j Q M M k Q ) ( M k ) = M k j Q . This is easily seen to be O (2 j M ), since 2 k = O ( M Q − ) and Q = O ( M ). The contributionto the full sum is acceptable, since there are / Q and k .Case 2. Q < M j . There are . Q arcs with q ∼ Q . Essentially, each H is either dis-joint from all these (so not contributing at this stage) or (essentially) contained inside oneof them. We distinguish two subcases. (a) If j M < QM k (this is stronger than Q < M j ), there are QM k M j intervals H/M contained in [ bq − QM k , bq + QM k ]. Their contribution is X b QM k , for each b/q with q ∼ Q there is only one H/M that intersects | t − bq | ∼ qM k with at most M qM k values of a contributing from H . The contribution from the O ( Q )arcs with denominator ∼ Q is . Q ( MQ k ) (2 k M ) = 2 k M Q .
Since 2 k . M , this term is O ( M ). (cid:3) Remark . One may wonder whether there is a clever way to estimate the sum X B ⊂ Ω ( X J ⊂ I X J ⊂ I ( Z B |E J E J | ) / ) , without using number theory. To this end, the most natural thing to try is to use Minkowski’sinequality and to bound this expression by( X J ⊂ I X J ⊂ I ( Z Ω |E J E J | ) / ) . (27)However, a change of variables as before shows that for each J , J Z Ω |E J E J | ≥ N − / Z [0 ,N / ] |E J E J | ∼ N − / N [ Z ( u,w ) ∈ [0 ,N / ] × [0 ,N − / ] | N / X m =1 e ( mu + m w ) | dudw ] ∼ N / . Using this approach, the upper bound we get for (27) is N . As in our earlier attempt touse l rather than l decoupling, this estimate falls short by a factor of N / from the sharpupper bound N .Also, due to (17), the expression (27) is only getting larger if J i are replaced with smallerintervals. Thus, decoupling on cubes larger than B (such as N / -cubes) only worsens ourupper bound. SQUARE ROOT CANCELLATION 19
A similar computation shows that the only case of Conjecture 1.1 that can be approachedwith l L decoupling is the case α = 2 discussed in the previous section.5. The case < α ≤ Let Ω = [0 , N α − ] × [0 , N ] × [ N , N α − ] × [0 , N β − ]. Using 1-periodicity in x , we needto prove that Z Ω |E I E I | . ǫ N α + ǫ . We cover Ω with cubes B of side length N α − and write Z Ω |E I E I | ∼ X B ⊂ Ω Z B |E I E I | . The size of these cubes is the smallest that will make cubic terms negligible after the decou-pling. Since we need α − ≤ β − α ≤ . Note also that the x coordinate is ≥ N / . We can afford this omission because ofthe α = case discussed in the previous section. Since we are about to decouple on cubes B with size larger than 1, Remark 4.3 tells us that applying this method for x near the originleads to losses. Our next argument will make explicit use of the fact that x is away fromthe origin.We use l decoupling (Theorem 2.3) on each B (or rather N B , after rescaling) Z B |E I E I | . ǫ N ǫ ( X J ⊂ I X J ⊂ I ( Z B |E J E J | ) / ) where J i is of the form [ h i , h i + M ], with M = N − α . We write B as[0 , N α − ] × [ pN α − , ( p + 1) N α − ] × [ l N α − , ( l + 1) N α − ] × [ l N α − , ( l + 1) N α − ]with the integers 0 ≤ p ≤ M , N − α ≤ l ≤ N α and 0 ≤ l ≤ N − α . Since α > , wehave that l ≫ l .We let as before I a = Z [0 , × [ a − O (1) M , a + O (1) M ] | X m ≤ M e ( mu + m w ) | dudw With a change of variables as in the previous section, we have Z B |E J E J | ∼ N ( N α − ) I a I a where ( a = p + l φ ′′ ( h N ) + l φ ′′ ( h N ) a = p + l φ ′′ ( h N ) + l φ ′′ ( h N ) . (28)It is crucial that the cubic (and also the higher order) term is O (1), cf. (24) m φ ′′′ ( hN ) x + φ ′′′ ( hN ) x N = O (1) , ∀ x ∈ Ω , so it may be neglected according to Lemma 9.2. If l ∼ j , it is immediate that | a − p | . j , | a − p | . j . Also, for fixed a , a , p, l , l ,(28) has O (( NM j ) ) solutions ( h , h ), modulo O (1). We do not need Lemma 4.1 here, sincethis time l is much larger than l . We now dominate R Ω |E I E I | as before by . ǫ N ǫ X j : N − α ≤ j ≤ N α X l ∼ j X j : 2 j ≤ N − α X l ∼ j ( NM j ) N ( N α − ) X p ≤ M ( X a : | a − p | . j I / a ) . We use Cauchy–Schwarz for the last expression to dominate the above by N ǫ X j : N − α ≤ j ≤ N α j N − α − j N α N ( N α − ) j j X | H |∼ j ( X a ∈ H I / a ) = N α + ǫ X j : N − α ≤ j ≤ N α − j X | H |∼ j ( X a ∈ H I / a ) . Using Lemma 4.2, this is dominated by N α + ǫ X j : N − α ≤ j ≤ N α ( M j + M − j ) . N ǫ ( N α + N + α ) . This is O ( N α + ǫ ), as desired, since α > .6. The case ≤ α < , N δ ] × [0 , N ] × [ N , N α − ] × [0 , N β − ] . Because of the case addressed in the previous section, we may assume x ≥ N . This willbuy us some extra flexibility in choosing δ . In fact, we can work with any δ satisfying2 − β ≤ δ ≤ − β. (29)We need to prove that Z Ω |E I E I | . ǫ N δ + ǫ . We will first decouple on cubes B with side length N β − . This is the largest size that isavailable to us, due to the range in the x variable. Unlike the case from the previous section,the resulting intervals are not small enough to make the cubic terms negligible, to allow usto use estimates for quadratic Weyl sums. We will accomplish that by means of a furtherdecoupling, on cubes of side length N δ , similar to the case α = 2 described earlier.To get started, we use l decoupling (Theorem 2.3) on each cube B of side length N β − Z B |E I E I | . ǫ N ǫ ( X J ⊂ I X J ⊂ I ( Z B |E J E J | ) / ) , where J i = [ h i , h i + M ] has length M = N − β .Next, we cover Ω with boxes∆ = [0 , N δ ] × [ pN δ , ( p + 1) N δ ] × [ lN δ , ( l + 1) N δ ] × [0 , N β − ] , SQUARE ROOT CANCELLATION 21 with p ≤ N − δ , N − δ ≤ l ≤ N α − − δ . If we sum up the above inequality over cubes B ⊂ ∆and use Minkowski’s inequality, we find Z ∆ |E I E I | . ǫ N ǫ ( X J ⊂ I X J ⊂ I ( X B ⊂ ∆ Z B |E J E J | ) / ) . (30)Next, we fix J , J and perform a second decoupling for the term R ∆ |E J E J | . We proceedas in Section 3 Z B |E J E J | = N − Z NB |E J ( · N ) E J ( · N ) | . N − − β Z NB |E J ( · N ) | Z NB |E J ( · N ) | = N − β Z B |E J | Z B |E J | . Then X B ⊂ ∆ Z B |E J | Z B |E J | . N − β Z ∆ dx Z ( y,z ) ∈ [0 ,N β − ] × [0 ,N β − ] |E J ( x + y ) E J ( x + z ) | dydz. Combining these two and using periodicity in the y , z variables we get Z ∆ |E J E J | . N − β − δ × Z ( x ,x ,y ,y ,y ,z ,z ,z ) ∈ S dx . . . dz Z y ,z ∈ [0 ,Nδ ] x ∈ [ pNδ, ( p +1) Nδ ] x ∈ [ lNδ, ( l +1) Nδ ] |E J ( x + y ) E J ( x + z ) | dy dz dx dx , where S is characterized by0 ≤ x ∈ [0 , N δ ] , x , y , y , y , z , z , z ∈ [0 , N β − ] . We seek to estimate the second integral uniformly over x , x , y , y , y , z , z , z . With thesevariables fixed, we make the affine change of variables ( y , z , x , x ) ( u , u , w , w ) u = ( y + x ) + h N ( x + y ) + φ ′ ( h N )( x + y ) + φ ′ ( h N )( x + y ) u = ( z + x ) + h N ( x + z ) + φ ′ ( h N )( x + z ) + φ ′ ( h N )( x + y ) w = x N + φ ′′ ( h N ) x N w = x N + φ ′′ ( h N ) x N . The Jacobian is ∼ N , due to (3). The second integral is comparable to N Z ( u i ,w i ) ∈ [0 ,N δ ] × [ ai − O (1) M ∗ , ai + O (1) M ∗ ] 2 Y i =1 | X m i ≤ M e ( m i u i + m i w i + η i ( m i ) x ) | du dw du dw . (31)Here M ∗ = N − δ , a i = p + l φ ′′ ( hiN )2 and η i ( m ) = m φ ′′′ ( hiN )3! N + m φ ′′′′ ( hiN )4! N + . . . . Note that since vN = O ( M − ) for v equal to any of the variables x , y , y , y , z , z , z , we have dismissed thecontribution of these variables associated with quadratic (as well as the higher order) terms.See Lemma 9.2. We may apply again Theorem 2.4, using that x = A ( w − w ) with A = O ( N ). Notehowever that this time we cannot decouple into point masses (as in (10)), since M ∗ issignificantly larger than 1. Instead, applying (11) with N = ( MM ∗ ) we dominate (31) by N ǫ × (32)( X J ′ ,J ′ [ Z ( u i ,w i ) ∈ [0 ,N δ ] × [ ai − O (1) M ∗ , ai + O (1) M ∗ ] 2 Y i =1 | X m i ∈ J ′ i e ( m i u i + m i w i + η i ( m i ) x ) | du dw du dw ] ) . The intervals J ′ i partitioning [1 , M ] have length M ∗ . What we have gained by doing thisdecoupling is that, when m i is confined to a small interval J ′ i = [ h ′ i , h ′ i + M ∗ ], the contributionof the term η i ( m i ) = η i ( h ′ i + m ′ i ) = η i ( h ′ i ) + η ′ i ( h ′ i ) m ′ i + η ′′ i ( h ′ i ) ( m ′ i ) O ( ( m ′ i ) N ) (33)can be neglected. To see this, note first that η ′′ i ( h ′ i ) = X n ≥ φ ( n )3 ( h i N ) ( h ′ i ) n − N n − ( n − . Making another linear change of variables such that w ′ i = w i + η ′′ i ( h ′ i )2 A ( w − w ) , we write, using that | x | ≤ N α − Y i =1 | X m i ∈ J ′ i e ( m i u i + m i w i + η i ( m i ) x ) | = Y i =1 | X m ′ i ∈ [1 ,M ∗ ] e ( m ′ i u ′ i + ( m ′ i ) w ′ i + O (( m ′ i ) N α − N )) | . The range of w ′ i is (a subset of) [ a ′ i − O (1) M ∗ , a ′ i + O (1) M ∗ ], where a ′ i = p + l X n ≥ φ ( n )3 ( h i N ) ( h ′ i ) n − N n − ( n − p + l φ ′′ ( h i + h ′ i N )2 . Since α − − β ≤ δ by (29), we have that a i − a ′ i = O (1). Thus, the quadratic termin (33) will not affect the domain of integration. Moreover, the contribution of the higherorder terms in (33) is negligible (cf. Lemma 9.2), as long as we can guarantee that we have N α − N = O ( M − ∗ ). This is equivalent to δ ≥ − β , and follows from (29) and the fact that β ≤ . Under this assumption, we dominate (32) by N ǫ X J ′ ,J ′ Y i =1 Z [0 ,N δ ] × [ ai − O (1) M ∗ , ai + O (1) M ∗ ] | X m i ∈ J ′ i e ( m i u i + m i w i ) | du i dw i . This is ∼ N δ + ǫ ( MM ∗ ) I a I a , where I a = Z [0 , × [ a − O (1) M ∗ , a + O (1) M ∗ ] | X m ≤ M ∗ e ( mu + m w ) | dudw SQUARE ROOT CANCELLATION 23 is independent of J ′ , J ′ . Recall that a = p + l φ ′′ ( h N )2 a = p + l φ ′′ ( h N )2 . (34)Assume now that l ∼ j , with N − δ . j . N α − − δ . For fixed p, l , and fixed ( a , a ) (within a factor of O (1)), the system (34) has . ( N j M ) solutions ( h , h ). Getting back to (30), summing over ∆ ⊂ Ω we find that Z Ω |E I E I | . ǫ N − β − δ | S | N δ + ǫ N δ X p ≤ M ∗ X N − δ . j . N α − − δ − j X l ∼ j ( X | a − p | . j I / a ) . We use Cauchy–Schwarz to dominate this by N δ + β + ǫ X N − δ . j . N α − − δ X H ⊂ [1 ,M ∗ ] | H | =2 j − j ( X a ∈ H I / a ) . Using Lemma 4.2, it remains to check that X N − δ . j . N α − − δ M ∗ j + X N − δ . j . N α − − δ M ∗ − j . N − β − δ . The first sum is in order, since α + β = 3. So is the second sum, as long as δ ≤ − β , whichis guaranteed by (29). 7. Proof of Theorem 1.2
This section shows that Theorem 1.3 implies Theorem 1.2. The argument is inspired by[2].The parameter K will be very large and universal, independent of N , φ k . The larger the K we choose to work with, the smaller the ǫ from the N ǫ loss will be at the end of thesection. Proposition 7.1.
Assume α + β = 3 and ≤ α ≤ . Assume φ , φ : (0 , → R are realanalytic and satisfy (1) , (2) and (3) . Let as before ω = [0 , N α ] , ω = [0 , N β ] and E I,N ( x ) = X n ∈ I e ( nx + n x + φ ( nN ) x + φ ( nN ) x ) . We consider arbitrary integers N , M satisfying ≤ M ≤ N K and N + [ M, M ] ⊂ [ N , N ] .Let H , H be intervals of length MK inside N + [ M, M ] such that dist ( H , H ) ≥ MK .Then Z [0 , × [0 , × ω × ω |E H ,N ( x ) E H ,N ( x ) | . ǫ N M ǫ . Proof.
Write H = N + I , H = N + I with I , I intervals of length MK inside [ M, M ]and with separation ≥ MK . We use the following expansion, certainly valid for all m in I i . φ ( N + mN ) = Q ( m ) + X n ≥ φ ( n )3 ( N N ) n ! ( mN ) n = Q ( m ) + ( MN ) X n ≥ φ ( n )3 ( N N )( MN ) n − n ! ( mM ) n . Here Q ( m ) = A + Bm + Cm with B = O ( N ), C = O ( N ). We introduce the analogue ˜ φ of φ at scale M ˜ φ ( t ) = X n ≥ φ ( n )3 ( N N )( MN ) n − n ! t n . This series is convergent as long as N N + t ∈ (0 , , N ≤ N .Let δ > K large enough we can make MN as small aswe wish, so we may guarantee that for each t ∈ [ ,
1] we have | ˜ φ (3)3 ( t ) − φ (3)3 ( N N ) | ≤ δ. (35)Thus, we can guarantee (3) for ˜ φ , with a slightly smaller, but uniform value of A . Thesame will work with (1) and (2), as it will soon become clear. To this end, we may alsoenforce | ˜ φ (4)3 ( t ) | ≤ δ. (36)We also define, with Q ( m ) = D + Em + F m satisfying E = O ( N ), F = O ( N ) φ ( N + mN ) = Q ( m ) + X n ≥ φ ( n )4 ( N N ) n ! ( mN ) n = Q ( m ) + φ (3)4 ( N N )3! ( mN ) + ( MN ) X n ≥ φ ( n )4 ( N N )( MN ) n − n ! ( mM ) n . The last two terms are equal to φ (3)4 ( N N ) φ (3)3 ( N N ) ( MN ) ˜ φ ( mM ) + ( MN ) X n ≥ φ ( n )4 ( N N ) φ (3)3 ( N N ) − φ (3)4 ( N N ) φ ( n )3 ( N N ) φ (3)3 ( N N ) n ! ( MN ) n − ( mM ) n . Let ˜ φ be the analogue of φ at scale M defined by˜ φ ( t ) = X n ≥ φ ( n )4 ( N N ) φ (3)3 ( N N ) − φ (3)4 ( N N ) φ ( n )3 ( N N ) φ (3)3 ( N N ) n ! ( MN ) n − t n . As before, by choosing K large enough, we can arrange that for all t ∈ [ , | ˜ φ (4)4 ( t ) − φ (4)4 ( N N ) φ (3)3 ( N N ) − φ (3)4 ( N N ) φ (4)3 ( N N ) φ (3)3 ( N N ) | ≤ δ. SQUARE ROOT CANCELLATION 25
Combining this with (35) and (36) we may arrange thatdet " ˜ φ (3)3 ( t ) ˜ φ (4)3 ( t )˜ φ (3)4 ( s ) ˜ φ (4)4 ( s ) − det " φ (3)3 ( N N ) φ (4)3 ( N N ) φ (3)4 ( N N ) φ (4)4 ( N N ) is as small in absolute value as we wish, uniformly over t, s ∈ [ , φ , ˜ φ ), with slightly modified, but uniform values of A , A .Similar comments apply regarding (1).Now |E H k ,N ( x ) | = | X m ∈ I k e ( mx + ( m + 2 N m ) x + φ ( N + mN ) x + φ ( N + mN ) x ) | = | X m ∈ I i e ( m ( x + 2 N x + Bx + Ex ) + m ( x + Cx + F x )+ ( MN ) ˜ φ ( mM )( x + φ (3)4 ( N N ) φ (3)3 ( N N ) x )+ ( MN ) ˜ φ ( mM ) x ) | . Recall N ∼ N , B, E = O (1 /N ), C, F = O (1 /N ). We make the change of variables y = x + 2 N x + Bx + Ex y = x + Cx + F x y = ( MN ) ( x + φ (3)4 ( N N ) φ (3)3 ( N N ) x ) y = ( MN ) x . Due to periodicity, we may extend the range of x to [0 , N ]. This linear transformationmaps [0 , N ] × [0 , × ω × ω to a subset of a box ˜ ω × ˜ ω × ˜ ω × ˜ ω centered at the origin,with dimensions roughly N , , M N α − , M N β − .Thus |E H k ,N ( x ) | = |E I k ,M ( y ) | where E I k ,M ( y ) = X m ∈ I k e ( my + m y + ˜ φ ( mM ) y + ˜ φ ( mM ) y ) . We may write, using again periodicity in y and y Z [0 , × [0 , × ω × ω |E H ,N ( x ) E H ,N ( x ) | = 1 N Z [0 ,N ] × [0 , × ω × ω |E H ,N ( x ) E H ,N ( x ) | ≤ ( NM ) Z [0 , × [0 , × ˜ ω × ˜ ω |E I ,M ( y ) E I ,M ( y ) | . Finally, we use Theorem 1.3 with N = M , noting that ˜ ω ⊂ [ − M α , M α ] and ˜ ω ⊂ [ − M β , M β ], to estimate the last expression by( NM ) M ǫ = N M ǫ . (cid:3) We can now prove Theorem 1.2.Choose K large enough, depending on ǫ . Write H n ( I ) for the collection of dyadic intervalsin I with length N K n . We write H H to imply that H , H are not neighbors. Then |E I,N ( x ) | ≤ H ∈H ( I ) |E H,N ( x ) | + K max H H ∈H ( I ) |E H ,N ( x ) E H ,N ( x ) | / . We repeat this inequality until we reach intervals in H l of length ∼
1, that is K l ∼ N . Wehave |E I,N ( x ) | . l + l l K max ≤ n ≤ l max H ∈H n ( I ) max H H ∈H n +1 ( H ) |E H ,N ( x ) E H ,N ( x ) | / . (log N ) N log K max ≤ n ≤ l max H ∈H n ( I ) max H H ∈H n +1 ( H ) |E H ,N ( x ) E H ,N ( x ) | / . Using Corollary 7.1 we finish the proof Z [0 , × [0 , × ω × ω |E I,N | . K N log K X n X H ∈H n ( I ) max H H ∈H n +1 ( H ) Z [0 , × [0 , × ω × ω |E H ,N ( x ) E H ,N ( x ) | . K,ǫ N ǫ +log K X n K n N ( NK n ) . K,ǫ N ǫ +log K N . Choosing K large enough, we may make log K Other values of p The reason Theorem 1.2 was accessible via the bilinear result in Theorem 1.3 has to dowith the fact that 6 is the critical exponent for the decoupling for the parabola (at thecanonical scale). Thus, our arguments rely fundamentally on this dimensional reduction. In[6], the small cap decoupling for the parabola is settled, and the associated critical exponentslie between 4 and 6. In principle, this new tool can be used to determine L p moments forcurves in R , in the range 8 ≤ p ≤ Conjecture 8.1 (Square root cancellation in L p ) . Let ≤ p ≤ . Assume α ≥ β ≥ satisfy α + β = p − . Let φ , φ , ω , ω be as in Theorem 1.2. Then Z [0 , × [0 , × ω × ω |E [ N ,N ] ,N | p . ǫ N p − ǫ . The case β = 0 was proved in [6] in the larger range 9 ≤ p <
12. As mentioned earlier,when β = 0, the curve collapses to a three dimensional curve. However, the next resultshows that the restriction p ≥
11 is needed if β > . The new obstruction can be describedas constructive interference on spatially disjoint blocks. SQUARE ROOT CANCELLATION 27
Theorem 8.2.
Assume p < . Let ω = [ − N α , N α ] , ω = [ − N β , N β ] , α ≥ β and α + β = p − . Assume also that β > .Then, for some δ > and φ ( t ) = t , φ ( t ) = t we have Z [ − , × [ − , × ω × ω |E [ N ,N ] ,N | p & N p − δ . Proof.
Lemma 8.3 shows that the integral is greater than X J ⊂ I Z [ − , × [ − , × ω × ω |E J,N | p , where the sum runs over intervals J of length M < N , partitioning [ N , N ]. The parameter M will be determined later. In some sense, the components E J,N behave as if they werespatially supported on pairwise disjoint sets.By periodicity Z [ − , × [ − , × ω × ω |E J,N | p = N − Z [ − N ,N ] × [ − N,N ] × ω × ω |E J,N | p . Write H = [ h + 1 , h + M ]. Note that |E J,N ( x ) | = | X ≤ m ≤ M e ( my + m y + m y + m y ) | where y = x + 2 hx + h N x + h N x y = x + hN x + h N x y = x N + hN x y = x N . This change of variables maps [ − N , N ] × [ − N, N ] × ω × ω to a set containing S = [ − o ( N ) , o ( N )] × [ − o ( N ) , o ( N )] × [ − o ( N α − ) , o ( N α − )] × [ − o ( N β − ) , o ( N β − )] . We have used that 3 ≥ α ≥ β −
1. Thus Z [ − , × [ − , × ω × ω |E J,N ( x ) | p dx ≥ N Z S | X ≤ m ≤ M e ( my + m y + m y + m y ) | p dy. Let M = max { N − α , N − β } . Note that since β > M = N − ǫ for some ǫ >
0. Note also that[0 , o ( M − )] × [0 , o ( M − )] ⊂ [0 , o ( N α − )] × [0 , o ( N β − )] . Using constructive interference we get Z S | X ≤ m ≤ M e ( my + m y + m y + m y ) | p dy & M p − N . Putting things together we conclude that Z [ − , × [ − , × ω × ω |E [ N ,N ] ,N | p & NM N M p − N . Note that this is ≥ N p − δ , for some δ > (cid:3) Lemma 8.3.
Let R be a collections of rectangular boxes R in R n , with pairwise disjointdoubles R . Let F be a Schwartz function in R n that can be written as F = X R ∈R F R , with the spectrum of F R inside R . Then for each ≤ p ≤ ∞ we have ( k F R k p ) l p ( R ) . k F k p . The implicit constant is independent of F and R .Proof. Interpolate between 2 and ∞ . (cid:3) Auxiliary results
This section records two auxiliary results that are used repeatedly throughout the paper.
Lemma 9.1.
Assume t, s ∈ [ , satisfy | t − s | ∼ . The Jacobian of the transformation y = ψ ( x ) y = x + 2 tx + φ ′ ( t ) x + φ ′ ( t ) x y = x + 2 sx + φ ′ ( s ) x + φ ′ ( s ) x y = x N + φ ′′ ( t ) x N + φ ′′ ( t ) x N y = x N + φ ′′ ( s ) x N + φ ′′ ( s ) x N is ∼ N . Moreover, ψ maps cubes Q with side length L to subsets of rectangular boxes of dimensionsroughly L × L × LN × LN .If the cube Q is centered at the origin, ψ ( Q ) contains the rectangular box [ − o ( L ) , o ( L )] × [ − o ( LN ) , o ( LN )] .Proof. Let φ ( u ) = u , φ ( u ) = u . Then the Jacobian is1 N det φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) φ ′′ ( t ) φ ′′ ( t ) φ ′′ ( t ) φ ′′ ( t ) φ ′′ ( s ) φ ′′ ( s ) φ ′′ ( s ) φ ′′ ( s ) . (37)Note thatdet φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) φ ′′ ( t ) φ ′′ ( t ) φ ′′ ( t ) φ ′′ ( t ) φ ′′ ( s ) φ ′′ ( s ) φ ′′ ( s ) φ ′′ ( s ) = lim ǫ → ǫ det φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) φ ′ ( t + ǫ ) φ ′ ( t + ǫ ) φ ′ ( t + ǫ ) φ ′ ( t + ǫ ) φ ′ ( s + ǫ ) φ ′ ( s + ǫ ) φ ′ ( s + ǫ ) φ ′ ( s + ǫ ) . SQUARE ROOT CANCELLATION 29
A generalization of the Mean-Value Theorem (see [9], Voll II, part V, Chap 1, No. 95)guarantees that det φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) φ ′ ( t + ǫ ) φ ′ ( t + ǫ ) φ ′ ( t + ǫ ) φ ′ ( t + ǫ ) φ ′ ( s + ǫ ) φ ′ ( s + ǫ ) φ ′ ( s + ǫ ) φ ′ ( s + ǫ ) == ǫ ( t − s ) ( t + ǫ − s )( s + ǫ − t ) det φ ′ ( τ ) φ ′ ( τ ) φ ′ ( τ ) φ ′ ( τ ) φ ′′ ( τ ) φ ′′ ( τ ) φ ′′ ( τ ) φ ′′ ( τ ) φ ′′′ ( τ ) φ ′′′ ( τ ) φ ′′′ ( τ ) φ ′′′ ( τ ) φ ′′′′ ( τ ) φ ′′′′ ( τ ) φ ′′′′ ( τ ) φ ′′′′ ( τ ) = ǫ ( t − s ) ( t + ǫ − s )( s + ǫ − t ) det (cid:20) φ ′′′ ( τ ) φ ′′′ ( τ ) φ ′′′′ ( τ ) φ ′′′′ ( τ ) (cid:21) for some τ i ∈ [ ,
1] depending on t, s, ǫ . The conclusion follows by letting ǫ → y ∈ [ − cL, cL ] × [ − c LN , c LN ] , for some small enough c , independent of N . We need to prove that y = ψ ( x ) for some x ∈ Q .This can be seen by solving for x . For example, x ∼ N det y φ ′ ( t ) φ ′ ( t ) φ ′ ( t ) y φ ′ ( s ) φ ′ ( s ) φ ′ ( s ) y φ ′′ ( t ) N φ ′′ ( t ) N φ ′′ ( t ) N y φ ′′ ( s ) N φ ′′ ( s ) N φ ′′ ( s ) N . This and (1) show that | x | . N ( | y | + | y | N + | y | + | y | N ) . The same inequality holds for all x i , which proves the desired statement. (cid:3) Lemma 9.2.
Let γ be a Schwartz function supported on [ − , . Define the smooth Weylsums for u, w, v ∈ R G ( u, w, v ) = X k ∈ Z γ ( k/M ) e ( ku + k w + k v ) . Let ≤ b ≤ q ≤ M with ( b, q ) = 1 . Assume that dist ( w − bq , Z ) := ϕ ≤ qM and that | v | . M . Then for each ǫ > we have | G ( u, w, v ) | . ǫ M ǫ q / min { M, ϕ / } if u ∈ M = [ m ∈ Z [ mq − ϕM ǫ , mq + ϕM ǫ ] and | G ( u, w, v ) | . ǫ M − if u
6∈ M . Proof.
Invoking periodicity, we may assume that w = bq + ϕ with | ϕ | ≤ Mq . Using therepresentation k = rq + k , 0 ≤ k ≤ q − G ( u, w, v ) = q − X k =0 e ( k b/q ) X r ∈ Z γ ( k + rqM ) e (( rq + k ) u + ( rq + k ) ϕ + ( rq + k ) v )= X m ∈ Z " q q − X k =0 e ( k b/q − k m/q ) R γ ( y/M ) e (( u + mq ) y + ϕy + vy ) dy (cid:21) = X m ∈ Z S ( b, m, q ) J ( u, v, ϕ, m, q ) (38)where S ( b, m, q ) = 1 q q − X k =0 e ( k b/q − km/q ) J ( u, v, ϕ, m, q ) = Z R γ ( y/M ) e (( u + mq ) y + ϕy + vy ) dy = M Z R γ ( z ) e ( M ( u + mq ) z + ϕM z + vM z ) dz. If | ϕ | . M , we are content with the bound | J ( u, v, ϕ, m, q ) | . M .Assume now that | ϕ | ≫ M . The classical van der Corput estimate (second derivativetest) reads | Z R γ ( z ) e ( Az + Bz + Cz ) dz | . | B | − / , if | B | ≫ | C | . In our case | B | = | ϕ | M ≫ & | vM | = | C | . In either case we get | J ( u, v, ϕ, m, q ) | . min { M, | ϕ | − / } . On the other hand, repeated integration by parts (first derivative test) shows that for each α > | J ( u, v, ϕ, m, q ) | . α A α when | A | = M | u + mq | ≥ M ǫ ϕM . Thus, when u ∈ M , only O ( M ǫ ) values of m will havea non-negligible contribution to the sum, while if u
6∈ M then the contribution from all m will be negligible.Combining these with the classical estimate | S ( b, m, q ) | . √ q finishes the argument. (cid:3) SQUARE ROOT CANCELLATION 31
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