Additive equations in dense variables via truncated restriction estimates
aa r X i v : . [ m a t h . C O ] M a r ADDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATEDRESTRICTION ESTIMATES
KEVIN HENRIOT
Abstract.
We study additive equations of the form P si =1 λ i P ( n i ) = 0 in variables n i ∈ Z d , where the λ i are nonzero integers summing up to zero and P = ( P , . . . , P r ) isa system of homogeneous polynomials making the equation is translation-invariant. Weinvestigate the solvability of this equation in subsets of density (log N ) − c ( P , λ ) of a largebox [ N ] d , via the energy increment method. We obtain positive results for roughly thenumber of variables currently needed to derive a count of solutions in the complete box[ N ] d , for the multidimensional systems of large degree studied by Parsell, Prendivilleand Wooley. Appealing to estimates from the decoupling theory of Bourgain, Demeterand Guth, we also treat the cases of the monomial curve P = ( x, . . . , x k ) and theparabola P = ( x , | x | ), for a number of variables close to or equal to the limit of thecircle method. Introduction
We are interested in solving additive diophantine equations in variables belonging toa thin subset of a box [ N ] d , for a large integer N >
2. More precisely, we consider asystem of r homogeneous integer polynomials P = ( P , . . . , P r ) in d variables, with each P i of degree k i >
1. Borrowing terminology from Parsell et al. [32], we call d = d ( P ) thedimension of the system P when each variable x i , 1 i d appears in a monomial withnonzero coefficient in at least one of the polynomials P , . . . , P r . We define the degree of P as k = k ( P ) = max i k i , and its weight as K = K ( P ) = P i k i . Furthermore, we saythat the system is reduced when the polynomials P i are linearly independent, in whichcase we call r = r ( P ) the rank of the system. We also fix coefficients λ , . . . , λ s ∈ Z r { } and study the system of r equations given by λ P ( x ) + · · · + λ s P ( x s ) = 0 , (1.1)with variables x , . . . , x s ∈ Z d . In order to solve this system in variables belonging tosubsets of Z d , we make the additional assumption that (1.1) is translation-invariant ,which imposes the condition λ + · · · + λ s = 0 that we assume from now on. Ourassumption of homogeneity also guarantees that (1.1) is dilation-invariant. Depending By this we mean that when ( x , . . . , x s ) is a solution of (1.1), so is ( x + u , . . . , x s + u ) for every u ∈ Z d . on the equation under study, one also typically defines a notion of non-trivial solutionwhich, at the very least, excludes the trivial diagonal solutions x = · · · = x s .Via Taylor expansions, one way to obtain translation-invariance in (1.1) is to pick alinearly independent subset P of the set of all partial derivatives of a given family ofpolynomials Q , . . . , Q h ∈ Z [ x , . . . , x d ], in which case we say that P is the seed systemgenerated by the seed polynomials Q , . . . , Q h . We also recall a more general definitionof Parsell et al. [32, Section 2]: we say that the system P is translation-dilation invariantif there exists a lower unitriangular matrix C ( ξ ) and a vector c ( ξ ) whose entries areinteger polynomials in ξ such that P ( x + ξ ) = c ( ξ ) + C ( ξ ) P ( x ) ( x , ξ ∈ Z d ) . It can be verified that this class of systems of polynomials contains the seed systems,and that it ensures again translation-dilation invariance in the equation (1.1).A classical question in additive combinatorics is to bound from below the lowest ad-missible density δ = δ ( N ) such that any subset A of [ N ] d of density at least δ containsa non-trivial solution to (1.1), as N tends to infinity. When specializing to the equa-tion x + x = 2 x detecting three-term arithmetic progressions, this covers the classicalsetting of Roth’s theorem [34], which says that the equation has a solution with all x i distinct in any subset of [ N ] of density at least (log log N ) − c . A subsequent argumentof Szemer´edi [38] and Heath-Brown [19] lowered the admissible density to (log N ) − c , fora small constant c >
0. A new framework was developed by Bourgain [10] to obtainthe exponent c = 1 / − ε , but in this work we only rely on the Heath-Brown-Szemer´edimachinery.The study of this question in cases of higher degree or dimension has generated afair amount of interest recently. The work of Smith [37] and Keil [27] concerned theone-dimensional quadratic case P = ( x, x ). Smith [36] has studied the degree- k case P = ( x, . . . , x k ), and Prendiville [33] has investigated the two-dimensional setting where P is given by a binary form and its derivatives. Prendiville’s result was later generalizedin work of Parsell et al. [32] to the class of all translation-dilation invariant systems ofpolynomials. In these references, doubly logarithmic bounds of the shape (log log N ) − c ( s ) were obtained via the method of Roth [34], for a number of variables sufficient to countthe number of solutions to (1.1) in [ N ] d by the circle method. In our previous work [20],we obtained logarithmic bounds of the shape (log N ) − c ( s, λ ) for the case P = ( x, x ), byadapting the Heath-Brown-Szemer´edi method [19, 38]. The purpose of this work is togeneralize this result to cases of larger degree or dimension.The discussion of our main theorem requires a little more context, but we can start bystating a representative result. Following Parsell et al. [32], we say that ( x , . . . , x s ) ∈ ( Z d ) s is a projected solution of (1.1) when all of the x i belong to a proper affine subspace DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 3 of Q d ; in dimension one this is equivalent to x = · · · = x s . We say that x is a subset-sum solution when there exists a partition [ s ] = E F · · · F E ℓ with ℓ > j ∈ [ ℓ ], P i ∈ E j λ i = 0 and P i ∈ E j λ i P ( x i ) = 0. This second definition is meantto exclude the obvious solutions obtained by setting the ( x i ) i ∈ E j to be equal for each j ∈ [ ℓ ]. Note that the space of projected solutions, and that of subset-sum solutions aretranslation-dilation invariant . Theorem 1.1 (Additive equations in subsets of monomial surfaces) . Let k > , d > , s > and λ , . . . , λ s ∈ Z r { } be such that λ + · · · + λ s = 0 . Suppose that P = ( x j · · · x j d d , j + · · · + j d k ) and let r denote the rank of P . Suppose also that the system of equations (1.1) possessesnonsingular real and p -adic solutions for every prime p . When s > r ( k + 1) + 1 ,there exists a constant c ( d, k, λ ) > such that every subset of [ N ] d of density at least N ) − c ( d,k, λ ) contains a solution to the system of equations (1.1) , which is neither aprojected nor a subset-sum solution. Note that the system of polynomials ( x j , | j | k ) is generated by the seed polyno-mials ( x j , | j | = k ). For that system, the estimates of Parsell et al. [32] for multidimen-sional Vinogradov mean values allow for a circle method treatment of the equation (1.1)in the same range s > r ( k + 1) + 1, and this is a substantial input in our proof. Animportant aspect of our approach, however, is that we need little number theoretic in-formation beyond mean value estimates to handle dense variables, and in the case ofthe above theorem the additional requirements consist only in simple bounds for localmultidimensional exponential sums.We now discuss in some depth the Fourier-analytic estimates involved in the treatmentequation (1.1) in dense variables, in order to motivate our main result. We define theweighted and unweighted exponential sums F ( P ) a ( α ) = X n ∈ [ N ] d a ( n ) e ( α · P ( n )) , F ( P ) ( α ) = X n ∈ [ N ] d e ( α · P ( n )) ( α ∈ T r ) . (1.2)The circle method expresses the number of solutions to (1.1) in a subset A of [ N ] d asa product of s weighted exponential sums of the above form, and therefore obtainingbounds on their s -th moments is of major importance. Restriction theory [16, 39, 42]provides a valuable framework to derive such bounds. When S is a finite subset of Z r equipped with a certain measure d σ S , the L q → L p extension problem is concerned with That is, they are invariant under translations ( x j ) j s ( x j + u ) j s , u ∈ Q d and dilations( x j ) j s γ ( x j ) j s , γ ∈ Q . Note that this forces N to be larger than a certain constant depending on P and λ . The constant c ( d, k, λ ) absorbs dependencies on s , considered as the dimension of the vector ( λ , . . . , λ s ). KEVIN HENRIOT establishing functional estimates of the form k ( g d σ S ) ∧ k L p ( T r ) k g k ℓ q ( S ) , and it is a dual version of the well-studied restriction problem. Bourgain [6–9] initiatedthe study of discrete restriction estimates for the squares, the sphere and the parabola.Recently, Wooley [45, 46] has given a formulation of the discrete restriction conjecturefor systems of homogeneous polynomials of dimension one, but the picture is less clearin higher dimensions. Short of guessing the right estimates, we put forward a conjecturewhich, when it does hold, provides us with exploitable estimates. We say that P satisfiesthe discrete restriction conjecture when it satisfies the estimate k F ( P ) a k pp . ε N ε k a k p (1.3)in the subcritical range p < K/d , the ε -full estimate k F ( P ) a k pp . ε N dp/ − K + ε k a k p (1.4)at the critical exponent p = 2 K/d , and the ε -free estimate k F ( P ) a k pp . p N dp/ − K k a k p (1.5)in the supercritical range p > K/d . In the case d = 1, it is believed that these estimatesall hold [45, 46]. Adding to the existing terminology, we say that P satisfies the weakdiscrete restriction conjecture when there exists θ > Z | F ( P ) a | > N d/ − θ k a k | F ( P ) a | q dm . q N dq/ − K k a k (1.6)for q > K/d . This weaker estimate is typically easier to obtain, and can be used [6,8] toobtain ε -free estimates for exponents q > p whenever an ε -full estimate of the form (1.4)is known.Only supercritical estimates are directly relevant to our problem, and therefore wequote the literature selectively. Bourgain established respectively in [6] and [8] that (1.5)holds in the full supercritical range p > P = ( x ) and p > P = ( x, x ). Keil [27]found an alternative proof of an L ∞ → L p estimate for p > P = ( x, x ).In the case of the d -dimensional parabola P = ( x , . . . , x d , x + · · · + x d ), which inour terminology is a system of dimension d and weight d + 2, Bourgain [8, Proposi-tions 3.82, 3.110, 3.114] proved the truncated estimate (1.6) in the whole supercriticalrange q > d + 2) /d , as well as estimates of the form (1.4) for d ∈ { , } , p > d > p > d + 4) /d . Eventually, the powerful decoupling theory of Bourgainand Demeter [11, Theorem 2.4] led to the conjectured estimates in all dimensions, thatis, (1.3) and (1.5) hold respectively for p = 2( d + 2) /d and p > d + 2) /d . DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 5
There have also been crucial developments for systems of polynomials of large degree.In that setting a natural object is the (multidimensional) Vinogradov mean value J s, P ( N ) = Z T r | F ( P ) ( α ) | s d α , which counts the number of solutions n i , m i ∈ [ N ] d to the sytem of equations P ( n ) + · · · + P ( n s ) = P ( m ) + · · · + P ( m s ) . A bound of the form J ℓ, P ( N ) . ε N dℓ − K + ε for an integer ℓ > K typically allows for asuccessful circle method treatment of the system of equations (1.1) in s > ℓ variables.Let us temporarily specialize to the case P = ( x, . . . , x k ) with k >
2, where K = k ( k + 1) and J s, P ( N ) = J s,k ( N ) is the usual Vinogradov mean value [41, Chapter 5].We introduce a new definition to facilitate the statement of later results. Definition 1.2.
For k > , we let s k denote the least integer s > K = k ( k + 1) suchthat J s,k ( N ) . ε N s − K + ε for every ε > . We restrict to s k > K since a simple averaging argument [41, Section 7] shows that J s,k ( N ) & N s + N s − K . The Vinogradov mean value conjecture, now a theorem, statesthat s k = K , and we discuss briefly the history leading to this result. The case k = 2is known to follow from simple divisor considerations. Classical work of Vinogradov [41]established an efficient asymptotic bound s k (3 + o k →∞ (1)) · k log k . In a majorachievement, Wooley [43,47,48] was able to settle the Vinogradov mean value conjecturefor k = 3 and to obtain the improved bound s k k − ∼ k →∞ K for k >
4, using hisefficient congruencing method. In a very recent breakthrough, Bourgain, Demeter andGuth [12] have settled the full Vinogradov mean value conjecture, that is s k = K , in theremaining cases k >
4, through a novel method rooted in multilinear harmonic analysis.Via the circle method [47, Section 9], it can be shown that R T k | F ( x,...,x k ) | p . N p − K for p > s k . Together with a well-known squaring argument for even moments , thisshows that an ε -free restriction estimate of the form (1.5) holds for p > s k + 2, andin fact it holds for p > s k via an observation of Hughes [23]. Up until the work ofBourgain-Demeter-Guth, the best available bounds on Vinogradov mean values wouldtherefore only produce an asymptotic range p > (1 + o k →∞ (1)) · K in such estimates.Wooley [44] was able to essentially halve this range , showing that (1.5) holds for p > k ( k + 1) ∼ k →∞ K , and his method extends to systems of polynomials. The stronger bound s k k ( k −
1) for k > By this we mean the bound k F ( P ) a k s s k F ( P ) k ss k a k s , which was used for instance by Bourgain [8,Proposition 2.36] and Mockenhaupt and Tao [29, Lemma 5.1]. The larger range p > k ( k −
1) was also announced in [46].
KEVIN HENRIOT
We now return to the setting of a general system of polynomials P , and state ourmain abstract result. Given a translation-dilation invariant subset Z of ( Q d ) s , meant torepresent a space of trivial solutions to (1.1), we define the quantities N ( N, P , λ ) = { solutions ( x , . . . , x s ) ∈ [ N ] ds to (1.1) } , (1.7) N Z ( N, P , λ ) = { solutions ( x , . . . , x s ) ∈ [ N ] ds ∩ Z to (1.1) } . (1.8) Theorem 1.3.
Let s > and λ , . . . , λ s ∈ Z r { } be such that λ + · · · + λ s = 0 .Suppose that P is a system of r homogeneous polynomials of dimension d and weight K such that the system of equations (1.1) is translation-invariant, and Z is a translation-dilation invariant subset of ( Q d ) s . Suppose that, for a constant ω > depending on s and P , N ( N, P , λ ) & N ds − K and N Z ( N, P , λ ) . N ds − K − ω . (1.9) Suppose also that there exist real numbers < s ′′ < s ′ < s and θ > depending on s and P such that the following restriction estimates hold: Z T r | F ( P ) a | s ′′ d m . ε N ds ′′ − K + ε k a k s ′′ ∞ , (1.10) Z | F ( P ) a | > N d/ − θ k a k | F ( P ) a | s ′ d m . N ds ′ / − K k a k s ′ . (1.11) Then there exists a constant c ( P , λ ) > such that, for every subset A of [ N ] d of densityat least N ) − c ( P , λ ) , there exists a tuple ( x , . . . , x s ) ∈ A s r Z satisfying (1.1) . We first comment on the assumptions of this theorem. The bounds (1.9) essentiallymean that the circle method is successful in estimating the number of non-trivial solutionsto (1.1). The restriction estimates (1.10) and (1.11) are the main analytic informationneeded for the argument, and they are stronger than an L ∞ → L p estimate k F ( P ) a k pp . N dp − K k a k p ∞ with p < s , used in the method of Roth [34], but weaker than an L → L p estimate (1.5)with p < s , used in the Heath-Brown-Szemer´edi argument [19, 20, 38]. Note that ifwe have J ℓ ( N, P ) . ε N dℓ − K + ε for an integer ℓ > K , then an L ∞ → L ℓ estimate ofthe form (1.10) with s ′′ = 2 ℓ automatically holds . For this reason, assumption (1.10)is typically verified in practice when one is using Vinogradov mean value bounds toestimate the number of solutions N ( N, P , λ ), which is the case for systems of largedegree.Theorem 1.3 constitutes an abstract generalization of its predecessor [20, Theorem 2],and its proof is very similar in dimension one when a full L → L p restriction estimate This follows from the simple bound k F ( P ) a k s s k F ( P ) k s s k a k s ∞ for integers s > DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 7 of the form (1.5) is known. In the extension to the multidimensional setting, the onlysubstantial change to the original energy increment strategy occurs in the technicallinearization part of the argument [20, Section 9], and there we employ the frameworkof factors introduced to additive combinatorics by Green and Tao [17, 40] to handleeffectively the computations in higher dimensions. Finally, we need a new observation toexploit truncated restriction estimates of the form (1.11) instead of complete ones, whichis that for the kind of weight functions that arise in the energy increment iteration, onecan afford to ignore the moment tails of associated exponential sums.We now discuss several consequences of Theorem 1.3, starting with the one-dimensionalsetting. There the only translation-invariant system of equations of the form (1.1) up toequivalence is λ x j + · · · + λ s x js = 0 (1 j k ) , (1.12)corresponding to P = ( x, . . . , x k ). Using the optimal bound s k = 2 K to verify theassumptions (1.9) and (1.10) of Theorem 1.3, as well as a certain truncated restrictionestimate of our own, we obtain the following conclusion. Theorem 1.4 (Additive equations in subsets of monomial curves) . Let k > and K = k ( k +1) . Let s > and λ , . . . , λ s ∈ Zr { } be such that λ + · · · + λ s = 0 . Supposethat the system of equations (1.12) possesses nonsingular real and p -adic solutions forevery prime p . When s > K + 4 , there exists a constant c ( k, λ ) > such that everysubset of [ N ] of density at least N ) − c ( k, λ ) contains a solution to the system ofequations (1.12) , which is neither a projected nor a subset-sum solution. Note that, critically, our approach bypasses the need for complete L → L p restrictionestimates, which are at present only known [44] for p > k ( k + 1) ∼ k →∞ K . For thisreason, we are able to reach a number s of variables close to the limit of the circle method,which is s > K in this setting. Furthermore, this number of variables could be attainedif one only knew the truncated estimate (1.6) in the range p > K .For general systems of polynomials of large degree, the most general conclusion wecan obtain is the following, of which Theorem 1.1 is a special case. Theorem 1.5 (Additive equations in subsets of polynomial surfaces) . Let s > and λ , . . . , λ s ∈ Zr { } be such that λ + · · · + λ s = 0 . Suppose that P is a reduced translation-dilation invariant system of polynomials having dimension d , rank r , degree k and weight K . Suppose also that the system of equations (1.1) possesses nonsingular real and p -adicsolutions for every prime p . When k > and s > max(2 r ( k + 1) , K + d ) , there existsa constant c ( P , λ ) > such that every subset of [ N ] d of density at least N ) − c ( P , λ ) contains a solution to the system of equations (1.1) , which is neither a projected nor asubset-sum solution. KEVIN HENRIOT
To prove this result, one may choose to appeal to either the L → L p restriction esti-mates of Wooley [44], or to weaker truncated restriction estimates that we will provide.The assumptions (1.3) on the number of integer solutions are verified by quoting the as-ymptotic formulas of Parsell et al. [32], based on the efficient congruencing method. Asa parenthesis, we remark that in the special case where the coefficients ( λ i ) in (1.1) takea symmetric form ( µ , − µ , . . . , µ ℓ , − µ ℓ ), a simple Cauchy-Schwarz argument yields theconclusion of Theorems 1.1, 1.4 and 1.5 at power-like densities N − c ( P ) instead (see Propo-sition 5.3 below). It is expected [5] that the decoupling theory of Bourgain-Demeter-Guthcould also lead to to progress on bounds for multidimensional Vinogradov mean values,which could in turn improve the range of validity of Theorem 1.5.Finally, we consider the parabola system λ x + · · · + λ s x s = 0 ,λ | x | + · · · + λ s | x s | = 0(1.13)in variables x , . . . , x s ∈ Z d , which corresponds to the system of polynomials P = ( x , . . . , x d , x + · · · + x d ) , generated by the seed polynomial P ( x ) = | x | . When all the λ i but one have the samesign, say all but λ s , every solution x to (1.13) verifies λ | x − x s | + · · · + λ s − | x s − − x s | = 0by translation-invariance, and by definiteness we have x = · · · = x s . Barring thisunfortunate circumstance, which always occurs for s = 3, we can obtain a positive resultfor a number of dense variables exceeding the critical exponent p d = 2( d + 2) / Theorem 1.6 (Additive equations in subsets of the parabola) . Let d, s > and supposethat λ , . . . , λ s ∈ Z r { } are such that λ + · · · + λ s = 0 and at least two of the λ i arepositive, and at least two are negative. There exists a constant c ( d, λ ) > such thatevery subset of [ N ] d of density at least N ) − c ( d, λ ) contains a solution to the systemof equations (1.13) , which is neither a subset-sum solution nor a solution with two equalcoordinates, provided that (i) d = 1 and s > , or (ii) d = 2 and s > , or (iii) d > and s > . This result takes as input the aforementioned Strichartz estimates of Bourgain andDemeter [11] to verify the assumptions (1.10) and (1.11) of Theorem 1.3, while a lowerbound for the number of solutions to (1.13) can be obtained by reducing the system to
DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 9 a quadratic form of rank at least five. For dimensions d
6∈ { , } , or for d ∈ { , } and s > N ] d , underlocal solvability assumptions. Theorem 1.7.
Let d, s > and λ , . . . , λ s ∈ Z r { } . Suppose that the system ofequations (1.13) has a nonsingular real solution in (0 , + ∞ ) ds and nonsingular p -adicsolutions for every prime p . Let N ( N, d, λ ) denote the number of solutions to (1.13) in [ N ] d . For s > d , we have N ( N, d, λ ) ∼ S · J · N ds − ( d +2) as N → ∞ , where S , J > . The factors S and J are defined in (6.12) and (6.13) below (with T = ∞ ), and throughfurther analysis they be given the traditional interpretation in terms of products of localdensities associated to the system of equations (1.13), though we do not provide thedetails here. When counting solutions to (1.13) in [ − N, N ] d ∩ Z d instead, one needs onlyassume the existence of a nonzero real solution to (1.13), as we explain in Section 6.The approach by reduction to a quadratic form is also likely to produce an asymptoticformula, but it is not clear that one would recover the same expression for local densities.We close this already lengthy introduction by discussing certain limitations of theprevious results. First, an annoying feature of Theorem 1.3 is the dependency of thelogarithm exponent on the coefficients ( λ i ) and the system of polynomials P . This is aseemingly irreducible feature of the Heath-Brown-Szemer´edi argument [19, 38] which isnot present in other methods such as Roth’s [34]. Secondly, our approach does not yieldthe expected density of solutions c ( δ ) N ds − K to the equations (1.1) in a subset of density δ of a box [ N ] d , and it would be very desirable to find a density increment strategythat addresses this shortcoming . For systems given by one quadratic form which is ina sense far from being diagonal (that is, with large off-rank), Keil [25, 26] has devisedsuch a strategy, which relies on finding a uniform majorant of weighted exponential sumsby Weyl differencing. However, it seems difficult to obtain such bounds in the diagonalsituation, where the weights are not easily eliminated, and we anticipate that a set oftechniques involving Bohr sets might be required instead. Remark.
A prior version of this article was publicized before the announcement ofBourgain, Demeter and Guth [12]. This new version records the consequences of thisnew development for some of our estimates. This question was raised to the author by ´Akos Magyar, whom we thank here.
Acknowledgements.
We thank Lilian Matthiesen for an interesting remark whichinspired Proposition 5.3. We thank Trevor Wooley for communicating us an advancedcopy of his forthcoming manuscript [44]. This work was supported by NSERC Discorerygrants 22R80520 and 22R82900. 2.
Notation
For x ∈ R and q ∈ N , we write e ( x ) = e iπx and e q ( x ) = e ( xq ). For functions f : T d → C and g : Z d → C , we define b f ( k ) = R T d f ( α ) e ( − k · α )d α and b g ( α ) = P n ∈ Z d g ( n ) e ( α · n ).For a function f defined on abelian group G and x, t ∈ G , we let τ t f ( x ) = f ( x + t ).When k > a ∈ Z k and q ∈ N , we write ( a , q ) = gcd( a , . . . , a k , q ), and we let q | a denote the fact that q | a , . . . , q | a k . For q > Z q as a shorthand forthe group Z /q Z . We write k x k or sometimes k x k T for the distance of a real x to Z .We let d m denote the Lebesgue measure on R d , or on T d identified with any cube ofthe form [ − θ, − θ ) d , and we let dΣ denote the counting measure on Z d .When Ω is a finite set and f : Ω → C is a function, we write E Ω f = E x ∈ Ω f ( x ) = | Ω | − P x ∈ Ω f ( x ). When P is a property, we let 1 P or 1[ P ] denote the boolean whichequals 1 when P is true, and 0 otherwise. When n is an integer we write [ n ] = { , . . . , n } ,and we let N = N ∪ { } . We let A F B denote the disjoint union of sets A and B .3. Additive equations in dense variables
In this section, we prove Theorem 1.3. We employ the arithmetic energy-incrementmethod from our previous work [20], with several simplifications to make the high-dimensional framework more bearable, and with a more significant modification to usetruncated restriction estimates.We start by introducing the relevant objects. We fix a system of r homogeneouspolynomials P = ( P , . . . , P r ), where each P i ∈ Z [ x , . . . , x d ] has degree k i >
1, and werecall that k = max i r k i is the degree of P and K = k + · · · + k r is its weight. Wealso fix coefficients λ , . . . , λ s ∈ Z r { } such that λ + · · · + λ s = 0. We fix an integer N > λ P ( n ) + · · · + λ s P ( n s ) = 0(3.1)in variables n , . . . , n d ∈ [ N ] d . We also fix a translation-dilation invariant subset Z of( Q d ) s , to be thought of as a set of trivial solutions to (3.1), and we define the quantities N ( N, P , λ ) and N Z ( N, P , λ ) as in (1.7) and (1.8). From now on, we place ourselvesunder the assumptions of Theorem 1.3, which in particular imply that N can be takenlarger than any fixed constant depending on P and λ . Unless otherwise specified, allexplicit and implicit constants throughout the section may depend on P and λ . DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 11
Next, we fix a prime number M ∼ DN , where D = D ( P , λ ) > M, λ i ) = 1 for all i and so that, for n , . . . , n s ∈ [ N ] d , the system ofequations (3.1) is equivalent to λ P j ( n ) + · · · + λ s P j ( n s ) ≡ M k j (1 j r ) . (3.2)Accordingly we define Z M = Q rj =1 Z /M k j Z ; note that | Z M | = M K ≍ N K . When f : Z d → C is a function, we also define F f : T r → C and H f : Z M → C by F f ( α ) = X n ∈ [ N ] d f ( n ) e (cid:18) r X j =1 α j P j ( n ) (cid:19) , H f ( ξ ) = E n ∈ [ N ] d f ( n ) e (cid:18) r X j =1 ξ j P j ( n ) M k j (cid:19) , (3.3)so that H f ( ξ ) = N − d F f ( ξ /M k , . . . , ξ r /M k r ) and F f = F ( P ) f in the notation of theintroduction. We write respectively F and H for the unweighted versions of F f and H f where one takes f ≡
1. For p >
0, we define the ℓ p norm of a function G : Z M → C by k G k p = ( P ξ ∈ Z M | G ( ξ ) | p ) /p .Next, we define the multilinear operator T acting on functions f i : Z d → C by T ( f , . . . , f s ) = D K N ds − K X n ,..., n s ∈ [ N ] d f ( n ) · · · f s ( n s )1 (cid:20) s X i =1 λ i P ( n i ) = 0 (cid:21) . (3.4)The normalizing constant D is unimportant and will be eventually absorbed in big O notation. Note that T (1 [ N ] d , . . . , [ N ] d ) = D K N − ( ds − K ) N ( N, P , λ ). As mentioned in theintroduction, a fact of key importance to us is that the operator T is controlled by s -thmoments of the exponential sums H f . Proposition 3.1.
For functions f , . . . , f s : Z d → C , we have | T ( f , . . . , f s ) | k H f k s · · · k H f s k s . (3.5) Proof.
For convenience we define the bilinear form h x , y i = P rj =1 x j y j M − k j on Z M . Byequivalence of (3.1) and (3.2) for n i ∈ [ N ] d and by orthogonality, we have T ( f , . . . , f s ) = D K N ds − K X n ,..., n d ∈ [ N ] d f ( n ) · · · f s ( n s ) 1 M K X ξ ∈ Z M e ( h ξ , λ P ( n ) + · · · + λ s P ( n s ) i )Interchanging summations, and renormalizing, we obtain T ( f , . . . , f s ) = X ξ ∈ Z M E n ,..., n s ∈ [ N ] d f ( n ) e ( h λ ξ , P ( n ) i ) · · · f s ( n s ) e ( h λ s ξ , P ( n s ) i )= X ξ ∈ Z M H f ( λ ξ ) · · · H f s ( λ s ξ ) . By H¨older’s inequality, we deduce that | T ( f , . . . , f s ) | s Y i =1 k H f i ( λ i · ) k s . For every i ∈ [ s ], we have k H f i ( λ i · ) k s = k H f i k s , since the M k j , j ∈ [ r ] are all coprimeto λ i , and this concludes the proof. (cid:3) The exponential sums H f , being discretized versions of F f , behave exactly the sameinsofar as moments are concerned. Lemma 3.2.
Uniformly for functions f : [ N ] d → C , we have, for every p > , k H f k pp . p N K − dp k F f k pp . Proof.
Define g : Z r → C by g ( m ) = X n ∈ [ N ] d : P ( n )= m f ( n ) , so that F f = b g by (3.3). By [20, Proposition 6.1], we have therefore k H f k pp = N − dp X ξ ∈ Z /M k Z · · · X ξ r ∈ Z /M kr Z (cid:12)(cid:12)(cid:12)b g (cid:16) ξ M k , . . . , ξ r M k r (cid:17)(cid:12)(cid:12)(cid:12) p . p N K − dp Z T r | b g ( θ , . . . , θ r ) | p d θ . . . d θ r = N K − dp k F f k pp . (cid:3) We also need a technical lemma to transform the assumptions of Theorem 1.3 intouseful restriction estimates. It is more natural at this point to work with scaled averages,and thus for a function f : [ N ] d → C and p > k f k L p [ N ] = ( E n ∈ [ N ] d | f ( n ) | p ) /p . Lemma 3.3.
Let d, r > , θ > and < q < p . Suppose that T : ℓ ( Z d ) → L ∞ ( T r ) isan operator such that, for every ε > , Z T r | T f | q d m . ε N dq − K + ε k f k q ∞ , (3.6) Z | T f | > N d − θ k f k L N ] | T f | p d m . N dp − K k f k pL [ N ] . (3.7) Then, uniformly for functions f : [ N ] d → C , we have k T f k pp . p,q,θ N dp − K k f k p − qL [ N ] k f k q ∞ . DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 13
Furthermore, for < ν < ( pq − θ we have, uniformly for functions f : [ N ] d → C suchthat k f k ∞ / k f k L [ N ] N ν , k T f k pp . p,q,θ,ν N dp − K k f k pL [ N ] . Proof.
Since k f k L [ N ] k f k ∞ and we have the estimate (3.7), it suffices in both casesto bound the tail I = Z | T f | N d − θ k f k L N ] | T f | p d m. To obtain the first estimate, observe that by (3.6) we have I N ( p − q )( d − θ ) k f k p − qL [ N ] Z T r | T f | q d m . ε N ε − ( p − q ) θ N dp − K k f k p − qL [ N ] k f k q ∞ . For ε small enough, we obtain the first estimate. To obtain the second estimate, notethat when k f k ∞ N ν k f k L [ N ] , we have I N ε + qν − ( p − q ) θ N p − K k f k pL [ N ] . For ν < ( pq − θ and ε small enough, we obtain the second estimate. (cid:3) Using the previous lemmas, we can translate these assumptions into a simple L → L p estimate for the operator f H f acting on functions of small L ∞ / L ratio, and into aninhomogeneous “mixed norms” estimate for general functions. Proposition 3.4.
Uniformly for functions f : [ N ] d → C , we have k H f k p . k f k − ( s ′′ /s ′ ) L [ N ] k f k s ′′ /s ′ ∞ k f k ∞ for p > s ′ . (3.8) There exists a constant ν ∈ (0 , depending at most on s ′ , s ′′ , θ such that, uniformly forfunctions f : [ N ] d → C such that k f k ∞ and k f k L [ N ] > N − ν , we have k H f k p . k f k L [ N ] for p > s ′ . (3.9) Proof.
By reverse nesting of ℓ p ( Z M ) norms, it suffices to prove both estimates at theendpoint s ′ . We rewrite the assumptions (1.10) and (1.11) as Z T r | F f | s ′′ d m . ε N ds ′′ − K + ε k f k s ′′ ∞ , Z | F f | > N d − θ k f k L N ] | F f | s ′ d m . N ds ′ − K k f k s ′ L [ N ] , where 0 < s ′′ < s ′ < s and θ >
0. The proof follows by applying Lemma 3.3 to
T f = F f with ( q, p ) = ( s ′′ , s ′ ) and ν = ( s ′′ s ′ − θ , and then invoking Lemma 3.2. (cid:3) With the previous analytical tools in place, we can carry out the first step of the usualdensity increment strategy, which is to extract a large moment of the exponential sum H f . When A is a subset of [ N ] d of density δ , we write f A = 1 A − δ [ N ] d for its balancedindicator function, here and throughout the section. Proposition 3.5.
There exists a constant c > such that the following holds. If A isa subset of [ N ] d of density δ such that T (1 A , . . . , A ) c δ s , then . k H f A /δ k s . Proof.
We expand 1 A = f A + δ [ N ] d by multilinearity in O ( c δ s ) = T (1 A , . . . , A )= δ s T (1 [ N ] d , . . . , [ N ] d ) + P T ( ∗ , . . . , f A , . . . , ∗ )= δ s D K N − ( ds − K ) N ( N, P , λ ) + P T ( ∗ , . . . , f A , . . . , ∗ ) , where the sum is over 2 s − f A or δ [ N ] d . Recalling the assumption (1.9), we assume that c is small enough and use thepigeonhole principle to obtain a lower bound of the form δ s . | T ( f , . . . , f s ) | , where a number ℓ > f i are equal to f A , and others are equal to δ [ N ] d .Therefore, by (3.5) and (3.8), we have δ s . k H f A k ℓs · δ s − ℓ k H k s − ℓs . δ s − ℓ k H f A k ℓs . After some rearranging we find that δ . k H f A k s , which finishes the proof. (cid:3) The next step is identical to that in the one-dimensional case [20, Section 8]: weextract a large restricted moment involving few frequencies.
Proposition 3.6.
There exist positive constants c , c , C such that the following holds.If A is a subset of [ N ] d of density δ such that T (1 A , . . . , A ) c δ s , then there exists R ( δ/ − C and distinct frequencies ξ , . . . , ξ R ∈ Z M such that R c . R X i =1 | H f A /δ ( ξ i ) | s ′ . Proof.
By Proposition 3.5 and (3.8), we have1 . X ξ | H f A /δ ( ξ ) | s , X ξ | H f A /δ ( ξ ) | s ′ . δ − s ′ . The proposition then follows at once from [20, Lemma 8.1] upon reordering the | H f A /δ ( ξ ) | by size. (cid:3) DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 15
The next stage of the arithmetic Heath-Brown-Szemer´edi method requires an estimateof simultaneous diophantine approximation essentially due to Schmidt [4, Chapter 7] andrefined by Green and Tao [18, Proposition A.2]. Here we use the more general version ofLyall and Magyar [28, Proposition B.2], which applies to monomials of arbitrary degree.
Proposition 3.7.
Let
L, T ∈ N and k > . There exist constants c, C > depending atmost on k such that, for any θ , . . . , θ T ∈ R and for L > (2 T ) CT , there exist q L such that k q k θ i k T L − cT − for all i T . We define a cube progression as a set of the form u + q [ L ] d with u ∈ Z d and q, L > φ : Z d → T simply as a map φ ( x ) = G ( x ) mod 1,for a polynomial G ∈ R [ x , . . . , x d ], and we define the degree of φ to be that of G . When Q is a subset of Z d and φ : Z d → T is a polynomial phase function, we letdiam Q ( φ ) = sup x , y ∈ Q k φ ( x ) − φ ( y ) k T . With this vocabulary in place, we now carry out a familiar linearization procedure.
Proposition 3.8 (Simultaneous linearization of polynomial phases) . Let k > and d > . There exist constants c, C > depending at most on k and d such that thefollowing holds. Let R > and suppose that φ , . . . , φ R : Z d → T are polynomial phasefunctions such that φ j (0) = 0 and deg φ j k for all j ∈ [ R ] . Assume that N > (2 R ) CR k .Then there exists a partition of the form [ N ] d = ( F i Q i ) F Ξ , where each Q i is a cubeprogression of size | Q i | > N cR − k such that diam Q i ( φ j ) N − cR − k for every j ∈ [ R ] , andwhere | Ξ | N d − cR − k .Proof. We induct on k > k = 0 all the polynomials are zero and we can take Q = [ N ] d and Ξ = ∅ . We now assume that k >
1, and throughout the proof we letimplicit or explicit constants depend at most on k and d . The letters c and C denotepositive such constants whose value may change from line to line.Let L > q > N ] d into congruence classes and then into subcubes, it is easy to find a partition of the form[ N ] d = F v ∈ V ( v + q [ L ] d ) F Ξ with | Ξ | . N d − / , as long as qL N / . Consider an index j ∈ [ R ] and the Taylor expansion of φ j at v ∈ V given by φ j ( v + q x ) = X | α | k ∂ α φ j ( v ) α ! q | α | x α = X | α | = k q k θ α ,j x α + ψ v ,q,j ( x ) , where x ∈ Z d , θ α ,j ∈ R and every ψ v ,q,j ∈ R [ x , . . . , x d ] has degree less than k andzero constant coefficient (since φ j has degree at most k , its derivatives of order k are This is a slight abuse of notation, since G is not uniquely defined from φ , but in practice we considerpolynomial phase functions as formal couples ( φ, G ). constant). Consequently we have, for every j ∈ [ R ], v ∈ V , x , y ∈ Z d , φ j ( v + q x ) − φ j ( v + q y ) = X | α | = k q k θ α ,j ( x α − y α ) + ψ v ,q,j ( x ) − ψ v ,q,j ( y ) . When x , y ∈ [ L ] d , by the triangle inequality for the distance on T , this implies that k φ j ( v + q x ) − φ j ( v + q y ) k T . L k max | α | = k k q k θ α ,j k T + k ψ v ,q,j ( x ) − ψ v ,q,j ( y ) k T . (3.10)At this point we use Proposition 3.7 to pick 1 q N / such that k q k θ α ,j k T N − cR − for every j ∈ [ R ] and every | α | = k , which is possible for N > (2 R ) CR . Foreach fixed v ∈ V , we assume that L > (2 R ) CR k − and use the induction hypothesisto obtain a partition [ L ] d = ( F w ∈ W Q v , w ) F Ξ v , where each Q v , w is a cube progressionsuch that | Q v , w | > L cR − k − and diam Q v , w ( ψ v ,q,j ) L − cR − k − for every j ∈ [ R ], andwith | Ξ v | L d − cR − k − . Inserting these diophantine and diameter bounds into (3.10),we obtain k φ j ( v + q x ) − φ j ( v + q y ) k T . L k N − cR − + L − cR − k − , (3.11)uniformly for j ∈ [ R ], v ∈ V and x , y ∈ [ L ] d .We choose finally L = N c ′ R − with c ′ small enough so that L N / and the right-hand side of (3.11) is O ( N − cR − k ). Working back through the conditions on L , wefind that this requires N > (2 R ) CR k , and when C is large enough we have thereforediam v + qQ v , w φ j N − cR − k for all j, v , w . We obtain a partition[ N ] d = F v ∈ V w ∈ W ( v + qQ v , w ) F F v ∈ V ( v + q Ξ v ) F Ξ . Since each set v + q Ξ v has density at most N − cR − k in its ambient box v + q [ L ] d , thedisjoint union Ξ ′ = F v ∈ V ( v + q Ξ v ) contained in [ N ] d has size at most N d − cR − k , andΞ ′′ = Ξ ′ F Ξ has size at most N d − c ′ R − k . (cid:3) To proceed further we need to recall the language of factors [40, Section 6], a special-ization of the theory of conditional expectations [15, Chapter 7] to the finite setting. Wecall factor a σ -algebra of the finite set [ N ] d . It can be verified that the factors of [ N ] d are in one-to-one correspondence with its partitions via( B i ) i ∈ [ ℓ ] such that [ N ] d = ℓ F i =1 B i
7→ B = (cid:8) F i ∈ J B i , J ⊂ [ ℓ ] (cid:9) . (3.12)We define an atom of a factor B as a minimal non-empty element of B , and those arethe sets B i under the correspondence (3.12). It can be verified that f : [ N ] d → C is B -measurable if and only if it is constant on every atom of B . We define the full factor B full as the factor whose atoms are all the singletons of [ N ] d , so that every function DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 17 f : [ N ] → C is B full -measurable, and has a well-defined conditional expectation E (cid:2) f |B (cid:3) for any factor B of [ N ] d . One can check that E (cid:2) f |B (cid:3) = P i ∈ [ ℓ ] ( E B i f )1 B i under thecorrespondence (3.12). All the usual properties of conditional expectation can be verifieddirectly in the finite setting, and we encourage the reader to do so as needed.In our situation, the language of factors will serve to simplify the step [20, Section 9]of the energy-increment strategy where the balanced function is replaced by an averagedversion of itself over a family of arithmetic progressions, which we now interpret asa conditional expectation. The function g below corresponds to the function f A /δ ofProposition 3.6, and when Ξ is a subset of [ N ] d we write Ξ c = [ N ] d r Ξ. Proposition 3.9 (Conditioning the balanced function) . Let δ ∈ (0 , and suppose that g : [ N ] d → C is such that k g k ∞ δ − . Suppose that, for certain constants c , C > ,there exist R ( δ/ − C and distinct frequencies ξ . . . , ξ R ∈ Z M such that R c . R X i =1 | H g ( ξ i ) | s ′ . (3.13) Then there exists C > such that, when N > e ( δ/ − C , the following holds. Considerthe polynomial phase functions φ , . . . , φ R : Z d → T such that H g ( ξ i ) = E n ∈ [ N ] d g ( n ) e ( φ i ( n )) (1 i R ) , and consider the partition [ N ] d = ( F i Q i ) F Ξ given by Proposition 3.8. Let B be thefactor of [ N ] d corresponding to this partition, and write e g = E (cid:2) g Ξ c |B (cid:3) . Then R c . k H e g k s ′ s ′ . Proof.
Consider an index i ∈ [ R ]. We first neglect the error set Ξ via H g ( ξ i ) = E (cid:2) ge ( φ i ) (cid:3) = E (cid:2) g Ξ c e ( φ i ) (cid:3) + O ( δ − N − cR − k ) . (3.14)Since 1 Ξ c e ( φ i ) is almost constant on each cube progression Q j and zero on Ξ, we have E (cid:2) g Ξ c e ( φ i ) |B (cid:3) = E (cid:2) g Ξ c |B (cid:3) e ( φ i ) + O ( δ − N − cR − k ) . Returning to (3.14), we can exploit this fact by conditioning on B in H g ( ξ i ) = E h E (cid:2) g Ξ c e ( φ i ) |B (cid:3)i + O ( δ − N − cR − k )= E h E (cid:2) g Ξ c |B (cid:3) e ( φ i ) i + O ( δ − N − cR − k )= H e g ( ξ i ) + O ( δ − N − cR − k ) . We can insert this estimate in (3.13) to obtain R c . R X i =1 | H e g ( ξ i ) | s ′ + O (cid:0) R ( δ − N − cR − k ) s ′ (cid:1) . Recalling the size condition on R , and completing the sum, we obtain the desired state-ment when N > e ( δ/ − C with C > (cid:3) Using the previous proposition and restriction estimates, we aim to obtain a lowerbound on the energy of the conditioned balanced function. If we succeed in doing so,the following proposition then yields a density increment.
Proposition 3.10 ( L density increment) . Let κ ∈ [ c , + ∞ ) for a constant c > .Suppose that B is a factor of [ N ] d with atoms ( Q i ) , Ξ such that | Ξ | N d − ( δ/ C for aconstant C > . Suppose also that A is a subset of [ N ] d of density δ such that κδ k E (cid:2) f A Ξ c |B (cid:3) k L [ N ] . Then there exists C > such that, for N > e − ( δ/ − C , there exists an atom Q i with (1 + κ ) δ | A ∩ Q i || Q i | . Proof.
First note that E (cid:2) [ N ] d Ξ c |B (cid:3) = 1 [ N ] d r Ξ . We write k · k = k · k L [ N ] throughoutthis proof. Expanding the square, we obtain κ δ k E (cid:2) A r Ξ |B (cid:3) − δ [ N ] d r Ξ k k E (cid:2) A r Ξ |B (cid:3) k − δ h E (cid:2) A r Ξ |B (cid:3) , [ N ] d r Ξ i + δ k [ N ] d r Ξ k . Let A ′ = A r Ξ. Since the conditional expectation operator is self-adjoint, we have then κ δ k E (cid:2) A ′ |B (cid:3) k − δ h A ′ , E (cid:2) [ N ] d r Ξ |B (cid:3) i + δ + O ( N − ( δ/ C )= k E (cid:2) A ′ |B (cid:3) k − δ + O ( N − ( δ/ C ) . Assuming that N > e − ( δ/ − C with C > κ ) δ k E (cid:2) A ′ |B (cid:3) k k E (cid:2) A ′ |B (cid:3) k ∞ · E h E (cid:2) A ′ |B (cid:3)i max i ( E Q i A ) · δ, where we have ignored the Ξ-average since E Ξ A ′ = 0. This gives the desired conclusionupon dividing by δ . (cid:3) We are finally ready to derive our main iterative proposition. It is at this point thatwe genuinely exploit the two types of restriction estimates of Proposition 3.4, in order
DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 19 to first obtain a lower bound on the energy of the conditioned balanced function, andthen apply a complete L → L p estimate. At this stage we may also reduce our workinghypothesis to A not containing any non-trivial solutions, by our assumption (1.9) andthe fact that N is already assumed to be quite large with respect to the density δ . Proposition 3.11.
There exist positive constants c, C such that the following holds.Suppose that A is a subset of [ N ] d of density δ such that all solutions ( n i ) ∈ A s to (3.1) lie in Z , and that N > e − ( δ/ − C . Then there exists R ( δ/ − C and a cubeprogression Q ⊂ [ N ] d of size N ′ such that, writing δ ′ = | A ∩ Q | / | Q | , we have δ ′ > (1 + cR c ) · δ, N ′ > N cR − k . Proof.
In the context of this proof, we let c, C denote positive constants whose valuemay change from line to line, and which may depend on P and λ as usual. Since allsolutions ( n i ) ∈ A s to (3.1) lie in Z , it follows from (3.4) and (1.9) that T (1 A , . . . , A ) CN − ω c δ s , for N > Cδ − s/ω , where c is the constant in Proposition 3.6. Assuming furthermore that N > e ( δ/ − C for a large enough C >
0, we can then combine Propositions 3.6 and 3.9 toobtain 1 R ( δ/ − C such that δR c . k H e f A k s ′ , (3.15)where e f A = E (cid:2) f A Ξ c |B (cid:3) and B is a factor of [ N ] d generated by atoms ( Q i ) , Ξ, with each Q i being a cube progression with | Q i | > N cR − C and with | Ξ | N d − ( δ/ C . From (3.15)and (3.8), noting also that k e f A k ∞ k f A k ∞
1, we deduce that for some
C > δ C . k e f A k L [ N ] . By assuming that δ > N − c with c > N large, we can ensure that N − ν k e f A k L [ N ] , where ν is the constant from Proposition 3.4, and on the other handwe have k e f A k ∞
1. We may therefore apply (3.9) in (3.15) to obtain δR c . k E (cid:2) f A Ξ c |B (cid:3) k L [ N ] . At this stage we can simply apply Proposition 3.10 to obtain the coveted density incre-ment. (cid:3)
The proof of Theorem 1.3 now follows by an iteration entirely similar to the one inthe one-dimensional setting [20, Section 4].
Proof of Theorem 1.3.
It suffices to follow the proof of [20, Theorem 2] in [20, Sec-tion 4], mutadis mutandis , replacing [20, Proposition 4.1] by Proposition 3.11, arithmeticprogressions by cube progressions, and trivial solutions by the set Z . The powers of R differ in the two cases but this does not affect the final bound. Since the constants inthe statement of Proposition 3.11 were allowed to depend on P , λ , the final logarithmexponent now depends on these parameters as well. When the algorithm stops, oneobtains a cube progression Q = v + q [ L ] d with v ∈ Z d and q > A ∩ Q = v + qA ′ , there exists ( n i ) ∈ ( A ′ ) s r Z satisfying (3.1). By translation-dilationinvariance of Z and of (3.1), it follows that ( v + q n i ) ∈ A s r Z also satisfies (3.1), andthe proof is complete. (cid:3) On epsilon-removal
We fix an integer N > k >
3. We writeΓ = { ( n, . . . , n k ) , n N } , d σ Γ = 1 Γ dΣ . We define the corresponding Weyl sum F ( α ) = X n N e ( α n + · · · + α k n k ) ( α ∈ T k ) . Given a weight function g : Z k → C , we also define F g ( α ) = X n ∈ Γ g ( n ) e ( α · n ) = ( g d σ Γ ) ∧ ( α ) ( α ∈ T k ) . (4.1)so that F = (d σ Γ ) ∧ in the unweighted case g ≡
1. The goal of this section is to provean estimate of the form (1.6) for P = ( x, . . . , x k ), by a modification of the argument ofBourgain [6] for squares. Hughes was the first to obtain results in this direction in unpub-lished work from 2013. We include our alternative argument for two main reasons: toillustrate the philosophy that truncated restriction estimates are simpler to obtain thanfull ones, requiring as they do only major arc information on unweighted exponentialsums, and also to show how these estimates naturally extend to the multidimensionalsetting. Proposition 4.1 (Truncated restriction estimate for monomial curves) . Let k > andwrite K = k ( k + 1) . Let θ = 1 / if k = 3 , and θ = max(2 − k , / s k − ) else. Then, forevery ε > , Z | F g | > N − θ + ε +1 / k g k | F g ( α ) | p d α . p,ε N p − K k g k p for p > K + 4 . We refer to Definition 1.2 for the meaning of s k . We pay attention to the quality ofthe exponent θ above, although this is not necessary for our applications, and the proof Note that F g = F ( x,...,x k ) a with a ( n ) = g ( n, . . . , n k ) in the notation of the introduction, but this newdefinition is more natural from a Fourier-analytic point of view. Very recently, Wooley [44] has independently obtained a similar estimate.
DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 21 could be simplified slightly by ignoring this aspect. The previous proposition has thefollowing more familiar consequence, which again is not strictly required for our laterargument.
Corollary 4.2 ( ε -removal for monomial curves) . Let k > and write K = k ( k + 1) .Suppose that, for some q > , Z T k | F g | q d m . ε N q − K + ε k g k q for every ε > . Then, for p > max (2 K + 4 , q ) , Z T k | F g | p d m . N p − K k g k p . Proof.
Without loss of generality we may assume that k g k = 1. By Proposition 4.1, itsuffices to bound the tail Z | F g | N − θ + ε +1 / | F g | p d m N − ( p − q )( θ − ε ) N ( p − q ) / Z T k | F g | q d m . ε N ε − ( p − q )( θ − ε ) N p/ − K . N p/ − K . (cid:3) We start by recalling the basics of the discrete Tomas-Stein argument [6, 8]. We fix afunction g : Z d → C , and for a parameter η > E η = {| F g | > ηN / } , f = 1 E η F g | F g | , f = 1 E η . We assume that k g k = 1 throughout, so that | F g | N / by Cauchy-Schwarz in (4.1),and we can assume that η lies in (0 , F g of order p > Z bN / aN / | F g | p d m = pN p/ Z ba η p − | E η | d η for 0 a b . (4.2)By definition of f and Parseval, we have ηN / | E η | h f , F g i = h f , ( g d σ Γ ) ∧ i = h b f , g i L (d σ Γ ) . By Cauchy-Schwarz and using the assumption k g k = 1, it follows that η N | E η | k b f k L (d σ Γ ) = h b f d σ Γ , b f i . By another application of Parseval, we conclude that η N | E η | h f ∗ F, f i . (4.3) This well-known inequality is the starting point of our argument.We now use the circle method to decompose the kernel F into two pieces, correspondingto the usual major and minor arcs. To bound F on minor arcs we will use the followingestimates of Weyl/Vinogradov type. Proposition 4.3.
Let k > be an integer and let τ, δ be real numbers with < τ < max(2 − k , / s k − ) and δ > kτ . Then if | F ( α ) | > N − τ and N is large enough withrespect to k, τ, δ , there exist integers q, a , . . . , a k such that q N δ , ( a , . . . , a k , q ) = 1 and | qα j − a j | N δ − k j for j k .Proof. When τ = 2 − k , this is [4, Theorem 5.1], with parameters M = 1, P = N − τ andchoosing the ε from that theorem small enough so that kτ + ε δ . When τ = 1 / s k − ,the proposition follows from the reasoning used in the proof of [47, Theorem 1.6] in [47,Section 8]. (cid:3) We adopt the convention that any implicit or explicit constant throughout the sectionmay depend on k , and we assume that N is large enough with respect to k when neededby the argument, without further indication. (Since k F a k ∞ N , we may certainlyassume that N is larger than any absolute constant in proving Proposition 4.1). Weset τ = if k = 3 and τ = max(2 − k , / s k − ) if k >
4, in accordance with the Weyl-type estimates we intend to use. We fix a small quantity ε ∈ (0 , τ ) and a constant δ = k ( τ − ε ). For k >
4, we can use the bound s k − > k ( k −
1) to deduce that δ < kτ max (cid:16) k k − , k s k − (cid:17) max (cid:16) k k − , k − (cid:17) , and the same bound holds for k = 3 trivially. We define the major and minor arcs in astandard fashion by M ( a , q ) = { α ∈ T k : k α j − a j /q k q − N δ − j (1 j k ) } , M = G q N δ G a ∈ [ q ] k :( a ,q )=1 M ( a , q ) , m = T k r M . (4.4)It is easy to check that we have indeed a disjoint union in (4.4) when δ < /
2. Weuse the fundamental domain U = ( N − δ , N − δ ] k containing the intervals a /q + Q j [ − q − N δ − j , q − N δ − j ] with 1 q N δ and a ∈ [ q ] k . DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 23
We first obtain a set of estimates for the exponential sum F on minor and major arcs.This involves the Gaussian sum and oscillatory integral defined respectively by S ( a , q ) = X u mod q e q ( a u + · · · + a k u k ) ( a ∈ Z kq ) ,I ( β , N ) = Z N e ( β x + · · · + β k x k )d x ( β ∈ R k ) . (4.5) Proposition 4.4.
For α ∈ U , we have | F ( α ) | = O ε ( N − τ +2 ε ) if α ∈ m ,q − S ( a , q ) I ( α − a /q, N ) + O ε ( N − τ +2 ε ) if α ∈ M ( a , q ) ⊂ M . Proof.
Consider a frequency α ∈ T k . If | F ( α ) | > N − ( τ − ε ) and N is large enough,then Proposition 4.3 with τ ← τ − ε and δ ← k ( τ − ε ) shows that α ∈ M . Therefore | F | . ε N − τ +2 ε on m .When α ∈ M ( a , q ) with 1 q N δ , a ∈ [ q ] k and ( a , q ) = 1, we have, for every j ∈ [ k ], | α j − a j /q | q − N δ − j (2 k ) − q − N − j , where we used the fact that δ < N is large in the last inequality. By a standardPoisson-based approximation formula [4, Lemma 4.4], we obtain the desired approxima-tion of F , noting that q − /k + ε . N − τ +2 ε for q N δ and ε small enough. (cid:3) In light of the previous proposition, we define a majorant function U p : U → C by U p = X q N δ X a ∈ [ q ] k :( a ,q )=1 | q − S ( a , q ) | p · M ( a ,q ) · τ − a /q | I ( · , N ) | p . (4.6)Our bounds on the exponential sum F can be phrased in the following form, where wewrote ε = 2 ε . Proposition 4.5.
We have a decomposition F = F + F with k F k ∞ . ε N − τ + ε and | F | p U p . Proof.
We naturally define F = X q N δ X ( a ,q )=1 q − S ( a , q ) τ − a /q I ( · , N ) · M ( a ,q ) and F = F − F . Since the arcs M ( a , q ) are disjoint for q N δ , ( a , q ) = 1, the requiredbounds follow from Proposition (4.4). (cid:3) Our argument is a modification of Bourgain’s [6], in which we directly use L bounds onthe major arc majorant U p to obtain L ∞ → L estimates for the operator of convolution with U p . In fact, we show that the L norm of U p is controlled by the following localmoments, where we define I ( β ) = I ( β , S p = X q > X a ∈ [ q ] k :( a ,q )=1 | q − S ( a , q ) | p , J p = Z R k | I ( ξ ) | p d ξ . (4.7) Lemma 4.6.
For p > , we have Z U | U p | d m S p · J p · N p − K . Proof.
From the definition (4.6) of U p , we obtain effortlessly Z U | U p | d m S p · Z R k | I ( β , N ) | p d β . (4.8)By a linear change of variables in (4.5), we have I ( β , N ) = N Z e ( β N x + · · · + β k N k x k )d x = N · I ( β N, . . . , β k N k ) . By another linear change of variables, we find that Z R k | I ( β , N ) | p = N p Z R k | I ( β N, . . . , β k N k ) | p d β = N p − K Z R k | I ( ξ ) | p d ξ , and this can be inserted into (4.8) to finish the proof. (cid:3) Proposition 4.7.
Suppose that p > is such that S p < ∞ and J p < ∞ . Then | E η | . p N − K η − p if η > N − τ/ ε when N is large enough with respect to ε .Proof. Starting from the inequality (4.3), and using the decomposition of Proposition 4.5and H¨older’s inequality, we obtain η N | E η | h| F | ∗ f, f i + k F k ∞ k f k k| F | ∗ f k p k f k p ′ + O ε ( N − τ + ε | E η | ) . For η > N − τ/ ε , applying also Young’s inequality yields η N | E η | . k F k p k f k k f k p ′ k U p k /p | E η | − p , so that | E λ | . k U p k N − p η − p , and we obtain the desired bound upon invoking Lemma 4.6. (cid:3) DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 25
In the case of an even integer exponent p = 2 s , the two local moments in (4.7) arecalled respectively the singular series and the singular integral in Tarry’s problem, andthe problem of their convergence has been solved respectively by Hua [22] and Arkhipovet al. [3]. The following is [1, Theorems 1.3 and 2.4], and the method of proof used thereallows in fact for real exponents p . Proposition 4.8.
Let p > , k > and K = k ( k + 1) . The singular integral J p converges for p > K + 1 , and the singular series S p converges for p > K + 2 . In fact, the restriction estimates of Drury [14] for curves yield a distinct proof of theconvergence of the singular integral. We now have all the ingredients needed to derive atruncated restriction estimate.
Proof of Proposition 4.1.
Let θ = τ / ν >
0. Using the integration formula (4.2),and invoking Proposition 4.7 with p ← K + 2 + ν and Proposition 4.8, we obtain Z | F g | > N − θ + ε +1 / | F g | p d m ≍ p N p/ Z N − θ + ε η p − | E η | d η . p N p/ − K Z η p − K +2+ ν ) − d η. This last quantity is O p ( N p/ − K ) for p > K + 4 and ν small enough. (cid:3) We comment briefly on how the ε -removal lemma we have just proven extends to themultidimensional setting. Since we only need major arc information and any inequality ofWeyl type, we rely essentially on work of Arkhipov et al. [1] from the decade 1970–1980.We pick a finite subset E of N d r { } and consider the set S = { ( n j · · · n j d d ) ( j ,...,j d ) ∈ E : n , . . . , n d ∈ [ N ] } corresponding to the reduced system of polynomials P = ( x j , j ∈ E ) of degree k =max j ∈ E | j | and rank r = | E | . The exponential sums (1.2) become F ( P ) a ( α ) = X n ∈ [ N ] d a ( n ) e (cid:18) X j ∈ E α j n j (cid:19) , F ( P ) ( α ) = X n ∈ [ N ] d e (cid:18) X j ∈ E α j n j (cid:19) ( α ∈ T r ) , (4.9)when a : Z d → C is a certain weight function. We define the corresponding Gauss sumand oscillatory integral by S ( a , q ) = X u ∈ Z dq e q (cid:18) X j ∈ E a j u j (cid:19) ( a ∈ Z rq ) , I ( β ) = Z [0 , d e (cid:18) X j ∈ E β j x j (cid:19) d x ( β ∈ R r ) . By the multidimensional analogue of Hua’s bound [1, Theorem 2.6] and a standard vander Corput lemma [30, Corollary 2.3], we have | S ( a , q ) | . ε q d − /k + ε ( q > , ( a , q ) = 1) , (4.10) | I ( β ) | . (1 + | β | ) − /k ( β ∈ R r ) . (4.11)For p >
0, define the local moments S p = X q > X a ∈ [ q ] r :( a ,q )=1 | q − d S ( a , q ) | p , J p = Z R r | I ( β ) | p d β . By inserting the bounds (4.10) and (4.11) in these expressions, and using sphericalcoordinates to bound the second one, we find that S p < ∞ for p > k ( r + 1) and J p < ∞ for p > kr . Note also that estimates of Weyl type for the unweighted exponentialsum in (4.9) are available from early work of Arkhipov et al. [2, Theorem 3], but for ourpurposes it is more expedient to quote the work of Parsell [31, Lemma 5.3, Theorem 5.5].Using these ingredients as a replacement for Proposition 4.4, it is a straightforwarddeduction to obtain the following multidimensional analogue of Proposition 4.1. Proposition 4.9 (Truncated restriction estimate for monomial surfaces) . Let d > and let E be a finite non-empty subset of N d r { } . Consider the system of polynomials P = ( x j , j ∈ E ) of dimension d , rank r = | E | , degree k = max j ∈ E | j | and weight K = P j ∈ E | j | . There exists θ = θ ( d, r, k ) > such that, for p > k ( r + 1) , Z | F ( P ) a | > N d/ − θ k a k | F ( P ) a | p d m . p N p − K k a k p . With a few more linear algebraic considerations it is possible to obtain an absolutelyanalogous result for general translation-dilation invariant systems (where d, r, k, K retaintheir usual meaning), and we choose not to elaborate further on this point, which does notrequire any essentially new idea. Note that the above proposition misses the completesupercritical range p > K/d , but it suffices for our applications given the state ofknowledge [32] on multidimensional Vinogradov mean values.5.
Additive equations of large degree
In this section we derive Theorems 1.1, 1.4 and 1.5 on systems of equations of largedegree. We start by establishing a few simple facts about translation-dilation invariantsystems of polynomials.
Lemma 5.1.
Suppose that P is a translation-dilation invariant system of r polynomialsof dimension d and degree k . Then x P ( x ) is injective and r > k . DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 27
Proof.
We first show that k r . Recall from [32, Section 2] that P = ( P , . . . , P r ) is atranslation-dilation invariant system when the polynomials P , . . . , P r are homogeneousof degree k i >
1, and when there exist integer polynomials c jℓ ( ξ ) in d variables for1 j r , 0 ℓ < j such that P j ( x + ξ ) − P j ( x ) = c j ( ξ ) + j − X ℓ =1 c jℓ ( ξ ) P ℓ ( x ) ( x , ξ ∈ Z d ) . Performing a Taylor expansion of the left-hand side at x , and choosing ξ = e i for anindex i ∈ [ d ] such that x i appears in a monomial of highest degree of P j , we may ensurethat the left-hand side is a polynomial of degree k j − x , while the right-hand side isa linear combination of polynomials of degrees 0 , k , . . . , k j − . Consequently, we obtainthe recursive bounds k k j max ℓ
2, so that upon iterating wederive k j j for 1 j r , and in particular k = max k j r as desired.Next, note that the system of equations P ( x ) − P ( y ) = 0 in variables x , y ∈ Z d istranslation-invariant. Consider two fixed integers x , y ∈ Z d such that P ( x ) = P ( y ).Then we have P ( x + ξ ) = P ( y + ξ ) for every ξ ∈ Z d , and therefore for every ξ ∈ R d byconsidering polynomials in the variable ξ . By Taylor expansion at x and y , we find that ∂ α P j ( x ) = ∂ α P j ( y ) for every α ∈ N d and every j ∈ [ r ]. Since we assumed that at leastone polynomial P j involves the variable x i for each i ∈ [ d ], it follows that x = y . (cid:3) Using an interpolation argument of Parsell et al. [32, Section 11], we also find that thenumber of subset-sum solutions is always negligible when a bound of the correct orderof magnitude is available for the relevant unweighted exponential sum.
Lemma 5.2.
Let s > and λ , . . . , λ s ∈ Z r { } be such that λ + · · · + λ s = 0 . Supposethat P is a translation-dilation invariant system of r polynomials of dimension d , degree k and weight K . Suppose that, for an integer s > K/d , k F ( P ) k ss . ε N ds − K + ε . Then the number of subset-sum solutions x ∈ [ N ] d to (1.1) is bounded up to a constantfactor by N ds − K − c , where c = c ( s, r, d, k ) > .Proof. By injectivity of P (Lemma 5.1) and orthogonality we have immediately k F ( P ) k = N d . Consider now a partition [ s ] = E F · · · F E ℓ with ℓ > P i ∈ E j λ i = 0 for all j ∈ [ ℓ ]. Since the λ i are nonzero, we have m j = | E j | ∈ [2 , s ) for every j ∈ [ ℓ ]. We write N ( E i ) ( N ) for the number of solutions n i ∈ [ N ] d to the equations P i ∈ E j λ i P ( n i ) = 0, j ∈ [ ℓ ]. By orthogonality, H¨older’s inequality and 1-periodicity, we have N ( E i ) ( N ) = Q ℓj =1 R T r Q i ∈ E j F ( λ i α )d α Q ℓj =1 Q i ∈ E j (cid:2) R T r | F ( λ i α ) | m j d α (cid:3) m j = Q ℓj =1 k F k m j m j . Interpolating between L s and L , and observing that P ℓj =1 m j = s , we deduce that N ( E i ) ( N ) Q ℓj =1 (cid:0) k F ( P ) k ss (cid:1) m j − s − (cid:0) k F ( P ) k (cid:1) s − m j s − . ε ( N ds − K + ε ) s − ℓs − ( N d ) ℓs − ss − = ( N ds − K + ε ) − ℓ − s − ( N d ) s ( ℓ − s − . With further rearranging, we obtain N ( E i ) ( N ) . ε N ds − K + ε ( N K − ds − ε ) ℓ − s − . Since ℓ >
2, this last term is at most O ( N ds − K − c ) for a certain c = c ( s, r, d, k ) > s > K/d , which is precisely our assumption. (cid:3)
With these preliminaries in place, and from the results of Sections 3 and 4, we canrecover the theorems of the introduction on systems of large degree.
Proof of Theorem 1.4.
We want to apply Theorem 1.3 with P = ( x, . . . , x k ) and Z defined as the set of projected or subset-sum solutions to (1.12). We write F = F ( x,...,x k ) and F a = F ( x,...,x k ) a , and we let N ( N ) denote the number of solutions n , . . . , n s ∈ [ N ]to (1.12). Via the circle method [47, Section 9], and assuming the existence of nonsingularreal and p -adic solutions to (1.12), one can obtain an asympotic formula of the form N ( N ) ∼ S · J · N s − K for k > s > s k , for certain constants S > J > n = · · · = n s ,and there are at most N = N s − K − ( s − K − such solutions, where s − K − > s > s k > K . By Lemma 5.2 and the estimate k F k s s . N s − K + ε for s > s k > K , the number of subset-sum solutions is also O ( N s − K − c ) for a certain c = c ( s, k ) >
0. Therefore the assumption (1.9) is satisfied for s > s k .Finally, the restriction estimate (1.10) is valid for any s ′′ > s k , via the bound k F a k s s k F k s s k a k s ∞ = J s,k ( N ) k a k s ∞ ( s ∈ N ) . The estimate (1.11), on the other hand, holds for some θ > s ′ > K + 4,by Proposition 4.1. Therefore, the assumptions of Theorem 1.3 are satisfied for s > max(2 K +4 , s k ), and indeed for s > K +4 upon using the result s k = 2 K from [12]. (cid:3) DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 29
Proof of Theorems 1.1 and 1.5.
We start by proving the more general Theorem 1.5,again by verifying the assumptions of Theorem 1.3. For s > r ( k + 1) and s > K + d ,the work of Parsell et al. [32, Section 11] shows that the assumptions (1.9) hold with aconstant ω = ω ( s, r, d, k ) when Z is defined as the set of projected solutions or subset-sum solutions to (1.1) (one may instead use Lemma 5.2 and [32, Theorem 2.1] to boundthe number of subset-sum solutions).Assumption (1.10) holds for s ′′ > r ( k + 1) by [32, Theorem 2.1] and using once morethe inequality k F ( P ) a k s s k F ( P ) k s s k a k s ∞ = J s ( N, P ) k a k ∞ ( s ∈ N ) . The truncated restriction estimate (1.11) holds for s ′ > k ( r + 1) by the natural gener-alization of Proposition 4.9 to arbitrary reduced translation-dilation invariant systems P , which we chose not to state. Since r > k by Lemma 5.1, we have 2 r ( k + 1) > k ( r + 1), and therefore this does not impose any additional constraint. After choos-ing max(2 r ( k + 1) , K + d ) < s ′′ < s ′ < s , Theorem 1.3 applies and gives the desiredconclusion. In the special case P = ( x j , | j | k ), it is explained in [32, Section 11]that 2 r ( k + 1) > K + d , so that the assumption s > K + d becomes redundant, andTheorem 1.1 follows. In that case the required estimate (1.11) was explicitely stated asProposition 4.9, taking E = { j ∈ N d : 1 | j | k } . (cid:3) We conclude this section with a small remark, which is that the usual argument [41,Section 7] by which one obtains a lower bound of the correct order of magnitude for J s,k ( N ) also shows that a system of equations of the form (1.1) with symmetric coeffi-cients has the expected density of solutions in any subset of [ N ] d . This phenomenon wasfirst observed by Rusza in the linear case [35, Theorem 3.2]. Proposition 5.3.
Let t > and µ , . . . , µ t ∈ Z r { } . Suppose that P is a system of r polynomials having dimension d , degree k and weight K . Suppose that A is a subset of [ N ] d of density δ and let N ( A, P , µ ) denote the number of solutions n i , m i ∈ A to thesystem of equations µ P ( n ) + · · · + µ t P ( n t ) = µ P ( m ) + · · · + µ t P ( m t )(5.1) in s = 2 t variables. Then N ( A, P , µ ) & P , µ δ s N ds − K . (5.2) In particular, there exist constants C ( P , µ ) > and c ( s, r, d, k ) > such that if δ > C ( P , µ ) N − c ( s,r,d,k ) , then A contains a solution to (5.1) , which is neither a projected nora subset-sum solution, provided also that • P = ( x, . . . , x k ) and s > s k + 2 , or • P = ( x j , | j | k ) and s > r ( k + 1) + 2 , or • P is an arbitrary system of polynomials and s > max(2 r ( k + 1) , K + d ) + 2 .Proof. We write P = ( P , . . . , P r ) and k i = deg P i . For a set E ⊂ R r and γ ∈ R , wewrite γ · E = { γx, x ∈ E } , and we also use traditional sumset notation in the proof. Wedefine P ( A ) = { P ( n ) , n ∈ A } and a number-of-representations function R ( u ) = { n , . . . , n t ∈ A : µ P ( n ) + · · · + µ t P ( n t ) = u } ( u ∈ Z r ) . Summing over all u ∈ Z r , we obtain | A | t = X u ∈ µ · P ( A )+ ··· + µ t · P ( A ) R ( u ) . By Cauchy-Schwarz, it follows that | A | t | µ · P ( A ) + · · · + µ t · P ( A ) | · X u ∈ Z r R ( u ) . Observing that µ · P ( A ) + · · · + µ t · P ( A ) ⊂ (cid:2) − O ( N k ) , O ( N k ) (cid:3) × · · · × (cid:2) − O ( N k r ) , O ( N k r ) (cid:3) , where the implicit constants depend on P and µ , we have therefore δ t N dt . P , µ N K · N ( A, P , µ ) . We recover (5.2) after some rearranging.In the various cases stated at the end of the proposition, we have seen previously inthis section that the number of projected or subset-sum solutions is O P , µ ( N ds − K − c ( s,r,d,k ) )for some constant c ( s, r, d, k ) >
0, and therefore we obtain solutions which are not ofthis kind for δ > C ( P , µ ) N − c ′ ( s,r,d,k ) , for some C ( P , µ ) > c ′ ( s, r, d, k ) > (cid:3) The parabola system
Fix d > s > λ , . . . , λ s ∈ Z r { } , not necessarily summing upto zero. We let N ( N, λ ) denote the number of solutions x i ∈ [ N ] d to the system ofequations λ x + · · · + λ s x s = 0 ,λ | x | + · · · + λ s | x s | = 0 , (6.1)where | · | denote the Euclidean norm on R d . This corresponds to the reduced translation-dilation invariant system of polynomials P = ( x , . . . , x d , x + · · · + x d ) of dimension d ,rank d +1, degree 2 and weight d +2. We first observe that N ( N, λ ) can be easily boundedfrom below by inserting the linear equation into the quadratic one, and invoking classicalresults on diagonal quadratic forms of rank at least five. DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 31
Proposition 6.1.
Suppose that λ + · · · + λ s = 0 and at least two of the λ i are positiveand at least two are negative, and s > max(4 , d ) . Then N ( N, λ ) & N ds − ( d +2) . Proof.
We rewrite (6.1) as x s = − λ s (cid:16) s − X j =1 λ j x j (cid:17) , s − X j =1 λ s λ j | x j | + (cid:12)(cid:12)(cid:12)(cid:12) s − X j =1 λ j x j (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (6.2)We only consider solutions ( x i ) with x s as above and x i = λ s y i for 1 i < s , with y i ∈ [ − cN, cN ] d for a small enough constant c = c ( λ ) >
0. By translation-invarianceof (6.1), such solutions may be shifted to fit in the box [ N ] d . Unfolding the squarednorm in the right-hand side of (6.2), we obtain a quadratic equation s − X j =1 λ s λ j | y j | + X j,k ∈ [ s − λ j λ k y j · y k = 0 ⇔ d X i =1 (cid:20) s − X j =1 λ s λ j y ij + X j,k ∈ [ s − λ j λ k y ij y ik (cid:21) = 0 ⇔ e y T B e y = 0 , (6.3)where e y = [ [ y j ] j ∈ [ s − . . . [ y dj ] j ∈ [ s − ] T and B = A . . . A ∈ Z d ( s − × d ( s − , A = [ λ j ( λ k + δ jk λ s )] j,k ∈ [ s − ∈ Z ( s − × ( s − . Under our assumptions on the λ i , it is established in the proof of [20, Proposition 7.3]that the quadratic form z z T A z is indefinite of rank s −
2, and therefore e y e y T B e y is an indefinite quadratic form in d ( s −
1) variables of rank d ( s − > s > d . By diagonalizing B and invoking classical results on diagonal quadratic forms [13,Chapter 8], we find & N d ( s − − = N ds − ( d +2) solutions y ∈ [ − cN, cN ] d ( s − to (6.3), andthere are at least as many solutions x ∈ [ N ] ds to the original system (6.1). (cid:3) Remark 6.2.
Via the same method, one can show that when P si =1 λ i = 0 , the numberof solutions to (6.1) in [ − N, N ] d ∩ Z d is at least cN ds − ( d +2) , as long as s > d andthere exists a nonzero real solution to (6.1) . We do not insist on this point since we haveopted to work with quadrants [ N ] d throughout the article. Let us quote a crucial restriction estimate that will be used in this section.
Theorem 6.3 (Bourgain [8], Bourgain-Demeter [11]) . Suppose that d > and P =( x , . . . , x d , x + · · · + x d ) . Then the estimates (1.3) and (1.5) hold respectively for p =2( d + 2) /d and p > d + 2) /d . We also define an unweighted exponential sum F ( α, θ ) = F ( P ) ( α, θ ) = X n ∈ [ N ] d e ( α | n | + θ · n ) (( α, θ ) ∈ T d +1 )(6.4)associated to the ( d + 1)-dimensional parabola. The estimate k F k pp . p N dp − ( d +2)+ ε for p > p d = 2 + 4 d , (6.5)which follows from Theorem 6.3, will be used in a few places. It can be proven in asimpler way by the method of Hu and Li [21, Theorem 1.3].First, we turn our attention to the problem of bounding the number of trivial solutions,and we need a complement to Proposition 5.2. For distinct indices i, j ∈ [ s ], we let N i,j ( N, λ ) denote the number of solutions x , . . . , x s ∈ [ N ] d to (6.1) with x i = x j . Proposition 6.4.
For s > max(4 , d ) , there exists c = c ( d, s ) > such that, for everypair of distinct indices i, j ∈ [ d ] , N i,j ( N, λ ) . N ds − ( d +2) − c . Proof.
We first show that, for a certain c ( t, s, d ) > k F k tt . N s − ( d +2) − c ( t,s,d ) for 2 t < s. (6.6)Indeed, by interpolation between L and L s , and via (6.5), we obtain k F k tt ( k F k ss ) − s − ts − ( k F k ) s − ts − . ( N ds − ( d +2)+ ε ) − s − ts − ( N d ) s − ts − . N ds − ( d +2)+ ε ( N − ( s − d − ε ) s − ts − , which is . N ds − ( d +2) − c ( t,s,d ) since s > d .Next, note that for distinct indices i, j ∈ [ s ], we have N i,j ( N, λ ) N ( N, µ ) with µ ∈ ( Z r { } ) t and t = s − t = s − λ i + λ j = 0 or not.Observe also that N i,j ( N, µ ) = Z T d +1 F ( µ α ) · · · F ( µ t α )d α k F k tt . We have s − > s − > s >
4, and by (6.6) it follows that N i,j ( N, µ ) . N ds − ( d +2) − c ( d,s ) for a certain c ( d, s ) > (cid:3) DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 33
At this stage we have developed enough machinery to solve the system of equa-tions (6.1) in a thin subset of [ N ] d . Proof of Theorem 1.6.
We wish to apply again Theorem 1.3. The bounds (1.9) areprovided by Propositions 6.1 and 6.4 as well as Lemma 5.2 (which is applicable thanksto (6.5)), provided that s > max(4 , d ), a condition equivalent to the one stated in thetheorem. The full L → L p estimate of Theorem 6.3 implies of course (1.10) and (1.11)for some real numbers s ′ , s ′′ with p d = 2 + d < s ′′ < s ′ < s . (cid:3) Remark 6.5.
For P = ( x , . . . , x d , x + · · · + x d ) , Bourgain [8, Propositions 3.6, 3.110and 3.114] proved that k F ( P ) a k pp . N dp/ − ( d +2) k a k p when d = 1 and p > , or d > and p > , or d > and p > d . This can be usedto obtain the conclusion of Theorem 1.6 respectively for d = 1 and s > , or d > and s > , or d > and s > . In the second part of this section, we apply a traditional blend of the circle methodto derive an asymptotic formula for N ( N, λ ). The bound (6.5) allows us to control thecontribution of minor arcs, and therefore most of our attention is devoted to the majorarc piece. We define the Weyl sum G ( α, θ ) = X n ∈ [ N ] e ( αn + θn ) (( α, θ ) ∈ T ) , so that by (6.4) and splitting of variables, we have F ( α, θ ) = d Y j =1 G ( α, θ j ) . (6.7)We also define a Gaussian sum and an oscillatory integral respectively by S ( a, b ; q ) = X u mod q e q ( au + bu ) ( q > , a, b ∈ Z q ) ,I ( β, ξ ; N ) = Z N e ( βx + ξx )d x ( β, ξ ∈ R ) , and we write I ( β, ξ ) = I ( β, ξ ; 1). By a change of variables, we have I ( β, ξ ; N ) = N · I ( N β, N ξ ) ( β, ξ ∈ R ) . (6.8)For a parameter Q >
1, we define individual major arcs of level Q by M Q ( a, b ; q )= { ( α, θ ) ∈ T d +1 : k α − a/q k QN − , k θ j − b j /q k QN − (1 j d ) } , for any q > a, b ) ∈ [ q ] d +1 . We define the major and minor arcs of level Q by M Q = G q > G ( a, b ) ∈ [ q ] d +1 ( a, b ,q )=1 M Q ( a, b , q ) , m Q = T d +1 r M Q , (6.9)where one can check the union is indeed disjoint when Q N / . When the needarises, we will work with the fundamental domain U = ( N − / , N − / ] d +1 of T d +1 .The reason for this choice is of course that, for Q N / ,( a, b ) /q + [ QN − , QN − ] × [ QN − , QN − ] d ⊂ U for 1 q Q, ( a, b ) ∈ [ q ] d +1 . We start by deriving major and minor arc bounds for the exponential sum (6.7).
Proposition 6.6.
Suppose that N / Q N / . For every q Q , ( a, b ) ∈ [ q ] d +1 , and ( α, θ ) ∈ M Q ( a, b , q ) ∩ U , we have F ( α, θ ) = d Y j =1 q − S ( a, b j ; q ) I ( α − a/q, θ j − b j /q ; N ) + O ( Q − / N d ) . For ( α, θ ) ∈ m Q , we have | F ( α, θ ) | . Q − / N d . Proof.
By Dirichlet’s principle, we may find 1 a q N with ( a, q ) = 1 suchthat | α − a/q | − q − N − q − . If q > Q , it follows by Weyl’s inequality [41,Lemma 2.4] that | G ( α, θ j ) | . ε Q − / N ε . Q − / N for all j ∈ [ d ], and therefore | F ( α, θ ) | . Q − d/ N d by (6.7).Next, fix a parameter η ∈ (0 ,
1] whose value shall be determined shortly. If q Q andthere exists j ∈ [ d ] such that | G ( α, θ j ) | ηN , then clearly | F ( α, θ ) | ηN d by (6.7).In the case where q Q and | G ( α, θ j ) | > ηN for all j ∈ [ d ], we show that ( α, θ ) ∈ M Q for a certain value of η . By a final coefficient lemma [4, Lemma 4.6], and assuming that Q / ηN − ε for some ε >
0, we may find an integer 1 t j for every j ∈ [ d ] suchthat, writing q j = t j q , we have q j . ε η − N ε , k q j α k . ε η − N − ε , k q j θ j k . η − N − ε . We let q = [ q , . . . , q k ], and since we have k q γ k ( q /q j ) k q j γ k for every γ ∈ T and j ,we deduce that q . ε η − N ε , k q α k . ε η − N − ε , k q θ j k . η − N − ε . Finally, choose η = Q − / − ε for an ε ∈ (0 , N large and ε small we have( α, θ ) ∈ M Q .Working now with ( α, θ ) ∈ U ∩ M Q ( a, b , q ), with q Q and ( a, b ) ∈ [ q ] d +1 , wehave | α − a/q | QN − and | θ j − b j /q | QN − for all j . By the usual approximation DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 35 formula [41, Theorem 7.2], it follows that G ( α, θ j ) = q − S ( a, b j ; q ) I ( α − a/q, θ j − b j /q ; N ) + O ( Q )for all j ∈ [ d ], and we have Q Q − / N . Taking the product over j ∈ [ d ], we obtainthe required approximation of F on M Q ( a, b , q ), again by (6.7). (cid:3) We treat in advance certain local moments that will arise in our analysis.
Proposition 6.7.
For p > and i ∈ [ s ] , let S i,p = X q > X ( a, b ) ∈ [ q ] d +1 :( a, b ,q )=1 d Y j =1 (cid:12)(cid:12) q − S (cid:0) λ i ( a, b j ); q (cid:1)(cid:12)(cid:12) p , (6.10) J i,p = Z R d +1 d Y j =1 (cid:12)(cid:12) I (cid:0) λ i ( β, ξ j ) (cid:1)(cid:12)(cid:12) p d β d ξ . (6.11) Then S i,p < ∞ for p > d and J i,p < ∞ for p > d .Proof. By Lemma A.1 and writing h = ( a, q ) and λ = λ · · · λ s in (6.10), we obtain S i,p . λ i X q > X a,b ,...,b d q :( a,b ,...,b d ,q )=1 h | λ ( b ,...,b d ) h dp/ q − dp/ . λ i X q > q d +1 − dp/ since h | λ ( a, b , . . . , b d , q ) implies | h | | λ | , and the last sum is absolutely convergentprecisely for p > d + 2) /d .By the usual van der Corput estimate, and integrating first in the variables ξ j in (6.11),we also have J i,p . Z R d Y j =1 (cid:20) Z R (1 + | β | + | ξ j | ) − p/ d ξ j (cid:21) d β. Note that R ∞ (1 + a + x ) − p/ dx ≍ p (1 + a ) − p/ for a > p >
2, and therefore underthis assumption we have J i,p . Z R (1 + | β | ) d (1 − p/ d β. This last integral is absolutely convergent for p > d . (cid:3) We define the singular series and singular integral truncated at the level T > S ( T ) = X q T X ( a, b ,q )=1 s Y i =1 d Y j =1 q − S (cid:0) λ i ( a, b j ); q (cid:1) , (6.12) J ( T ) = Z [ − T,T ] d +1 s Y i =1 d Y j =1 q − I (cid:0) λ i ( β, ξ j ) (cid:1) d β d ξ , (6.13)and when those converge absolutely we write S = S (+ ∞ ) and J = J (+ ∞ ). By H¨older’sinequality applied to products over i ∈ [ s ], and by Proposition 6.7, it follows that wehave absolute convergence in (6.12) and (6.13) for s > d . We now have all themoment bounds needed to carry out our main estimation. Proposition 6.8.
For s > d , we have S , J ∈ [0 , ∞ ) and there exists ν > such that N ( N, λ ) = S · J · N ds − ( d +2) + O ( N ds − ( d +2) − ν ) . Proof.
Throughout the proof, we use the letter ν to denote a small positive constantwhose value may change from line to line, but which remains bounded away from zero interms of d and s . The letter ε denotes a positive constant which may be taken arbitrarilysmall, and whose value may also change from line to line. We fix Q = N / , althoughthe precise value is unimportant. For a measurable subset E of T d +1 , we define themultilinear operator T E ( K , . . . , K s ) = Z E K · · · K s d m acting on functions K i : T d +1 → C . For p d = 2 + d and any i ∈ [ s ], we will use the bound | T E ( K , . . . , K s ) | (cid:20) k K i k s − p d L ∞ ( E ) k K i k p d p d Y j ∈ [ s ] r { i } k K j k ss (cid:21) s (6.14)which follows from H¨older’s and Young’s inequalities. We define F i = F ( λ i · ), so that N ( N, λ ) = T T d +1 ( F , . . . , F s ) . (6.15)Note that for any P > λ ∈ Z r { } , ( α, θ ) ∈ M P implies λ ( α, θ ) ∈ M | λ | P ,and therefore λ i ( α, θ ) ∈ m Q implies ( α, θ ) ∈ m Q/ | λ i | for any i ∈ [ s ]. By Proposition 6.6,we have therefore | F i | . Q − / N d for all i ∈ [ s ] on m Q . From (6.14) and (6.5), it follows DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 37 that | T m Q ( F , . . . , F s ) | . (cid:2) ( N d − / ) s − p d N dp d − ( d +2)+ ε ( N ds − ( d +2)+ ε ) s − (cid:3) /s . N ε − (1 / − p d /s ) N ds − ( d +2) . N ds − ( d +2) − ν . (6.16)We now evaluate T M Q ( F , . . . , F s ), by replacing the exponential sums F i with theirusual major arc approximation. For i ∈ [ s ], we define the function V i : U → C by V i ( α, θ ) = d Y j =1 q − S (cid:0) λ i ( a, b j ); q (cid:1) I ( α − a/q, θ j − b j /q ; N ) for ( α, θ ) ∈ M Q ( a, b ; q ) , (6.17)for every q > a, b ) ∈ [ q ] d +1 such that ( a, b , q ) = 1, and we define V i = 0 on m Q .Via Proposition 6.7 and (6.8), it is a simple matter to check that k V i k pp . N dp − ( d +2) for p > d . Observe that if ( α, θ ) ∈ M Q ( a, b , q ) then λ i ( α, θ ) ∈ M | λ i | Q ( λ i a, λ i b , q ) for any i ∈ [ s ].Therefore, by Proposition 6.6, we have | F i − V i | . N d − / on M Q . Expanding F i = V i + ( F i − V i ) by multilinearity, and using a minor variant of (6.14), it follows that | T M Q ( F , . . . , F s ) − T M Q ( V , . . . , V s ) | . max i ∈ [ s ] (cid:20) k F i − V i k s − p d − ε ∞ k F i − V i k p d + εp d + ε Y j ∈ [ s ] r { i } max( k F j k ss , k V j k ss ) (cid:21) /s . N ε − (1 / − p d /s ) N ds − ( d +2) . N ds − ( d +2) − ν . (6.18)for ε small enough. Recall (6.17) and (6.8), so that by integrating over the fundamentaldomain U and summing over all the major arcs in (6.9), we obtain T M Q ( V , . . . , V s )= X q Q X ( a, b ,q )=1 s Y i =1 d Y j =1 q − S (cid:0) λ i ( a, b j ); q (cid:1)Z [ − QN − ,QN − ] Z [ − QN − ,QN − ] d s Y i =1 d Y j =1 q − N I (cid:0) λ i ( N β, N ξ j ) (cid:1) d β d ξ = S ( Q ) · J ( Q ) · N ds − ( d +2) , (6.19)where we have operated a change of variables β ← N β , ξ ← N ξ in the last step.From the discussion following the introduction of the singular series (6.12) and (6.13), it follows that for p > d , we have S , J < ∞ and S ( Q ) = S + O ( N − ν ) , J ( Q ) = J + O ( N − ν ) . Inserting this into (6.19), and recalling (6.15), (6.16) and (6.18), we obtain finally N ( N, λ ) = T m Q ( F , . . . , F s ) + ( T M Q ( F , . . . , F s ) − T M Q ( V , . . . , V s )) + T M Q ( V , . . . , V s )= S · J · N ds − ( d +2) + O ( N ds − ( d +2) − ν ) . (cid:3) Proof of Theorem 1.7.
Starting from Proposition 6.8, it suffices to carry out a classicalanalysis [24, Chapter 20] of the singular series S and the singular integral J , after whichone would find that S > J > − N, N ] d ∩ Z d instead, we would have obtained an asymptoticformula for the number of solutions to (6.1) in that larger box, and by Remark 6.2 wecould deduce that the corresponding singular factor is positive whenever a nonzero realsolution to (6.1) is known. (cid:3) Appendix A. A uniform bound on Gauss sums
Here we include the proof of a well-known estimate that we could not locate preciselyin the literature.
Lemma A.1.
For q > and a, b ∈ Z q , let S ( a, b ; q ) = P u mod q e q ( au + bu ) . Uniformlyin q, a, b , we have | S ( a, b ; q ) | . ( a,q ) | b ( a, q ) / q / . Proof.
We let h = ( a, q ), a ′ = a/h , q ′ = q/h . We have S ( a, b ; q ) = X x mod q e q ′ ( a ′ x ) e q ( bx )= X u mod q ′ e q ′ ( a ′ u ) X x mod q : x ≡ u mod q ′ e q ( bx ) . (A.1)Writing x = u + q ′ y with y ∈ Z h , we find that X x mod q : x ≡ u mod q ′ e q ( bx ) = e q ( bu ) X y mod h e h ( by ) = e q ( bu ) · h h | b . DDITIVE EQUATIONS IN DENSE VARIABLES VIA TRUNCATED RESTRICTION ESTIMATES 39
Inserting this back into (A.1), we find that S ( a, b ; q ) = 0 if h ∤ b , and else we write b = hb ′ and obtain S ( a, b ; q ) = h X u mod q ′ e q ′ ( a ′ u + b ′ u ) . Since ( a ′ , q ′ ) = 1 and q ′ = q/h , the usual squaring-differencing argument then gives | S ( a, b ; q ) | . h ( q/h ) / = ( hq ) / . (cid:3) References
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Department of mathematics, University of British Columbia, Room 121, 1984 Mathe-matics Road, Vancouver BC V6T 1Z2, Canada