Random waves on \mathbb{T}^3: nodal area variance and lattice point correlations
aa r X i v : . [ m a t h . N T ] A ug RANDOM WAVES ON T : NODAL AREA VARIANCE AND LATTICEPOINT CORRELATIONS JACQUES BENATAR RICCARDO W. MAFFUCCI
Abstract.
We consider the ensemble of random Gaussian Laplace eigenfunctions on T = R / Z (‘3 d arithmetic random waves’), and study the distribution of their nodalsurface area. The expected area is proportional to the square root of the eigenvalue, or‘energy’, of the eigenfunction. We show that the nodal area variance obeys an asymptoticlaw. The resulting asymptotic formula is closely related to the angular distribution andcorrelations of lattice points lying on spheres. Keywords: nodal area, arithmetic random waves, Gaussian eigenfunctions, lattice pointcorrelations.
MSC(2010):
Contents
1. Introduction 21.1. Nodal volume of toral eigenfunctions 21.2. Arithmetic random waves 31.3. Prior work on this model 31.4. Statement of main results 51.5. Outline of the paper 7Acknowledgements 82. Lattice points on spheres and correlations 82.1. Preliminary results about the lattice point set 82.2. Equidistribution of lattice points on spheres 103. The proof of Theorems 1.6 and 1.7 123.1. Length four correlations 123.2. Length six correlations 143.3. Long correlations 154. Kac-Rice formulas 175. Proof of Proposition 1.4 195.1. The contribution of the singular set 195.2. Asymptotics for K on the non-singular set 215.3. Proof of Proposition 1.4 225.4. A note on arithmetic Berry cancellation 24 The research leading to these results has received funding from the European Research Council underthe European Union’s Seventh Framework Programme (FP7/2007-2013), ERC grant agreement n o
6. The leading term of the variance: proof of Lemma 5.10 246.1. Preparatory results 246.2. Proof of Lemma 5.10 28Appendix A. Berry’s method: proof of Lemma 5.8 29Appendix B. The incidence bound 33References 351.
Introduction
Nodal volume of toral eigenfunctions.
Let ( M , g ) be a d -dimensional compactRiemannian manifold and let ∆ denote the Laplace-Beltrami operator on M . We considerreal-valued functions F : M → R satisfying the differential equation(1.1) (∆ + E ) F = 0 , E > . The nodal set of F is the zero locus(1.2) { x ∈ M : F ( x ) = 0 } . It was proven by Cheng [10] that the nodal sets of Laplace eigenfunctions on M are smoothhypersurfaces, except for a subset of lower dimension. We let V := Vol( { x ∈ M : F ( x ) = 0 } )denote the nodal volume of F .A fundamental conjecture of Yau [31], [32] asserts that, for smooth M , the (( d − F with eigenvalue E obeys the sharpbounds(1.3) √ E ≪ M V ≪ M √ E. This conjecture was established for manifolds M with a real analytic metric (see Donnellyand Fefferman [12], and Br¨uning and Gromes [6], [7]), thus in particular it holds for thetorus T d = R d / Z d . The lower bound in Yau’s conjecture was proven for general smooth M by Logunov [22].For M = T d , the eigenspaces of the Laplacian are related to the theory of lattice pointson ( d − √ m S d − . Our main focus is the 3-dimensional torus T ; inthis setting we will call (1.2) a nodal surface and write(1.4) A := Vol( { x ∈ T : F ( x ) = 0 } )for the nodal area of F .The sequence of eigenvalues, or energy levels , of (1.1) is { E m = 4 π m } m ∈ V (3) , where V (3) := { m : 0 < m = a + b + c , a, b, c ∈ Z } . ANDOM WAVES ON T For m ∈ V (3) , let(1.5) E = E ( m,
3) := { µ ∈ Z : k µ k = m } be the set of all lattice points on the sphere of radius √ m . Their cardinality is the numberof ways that m may be written as a sum of three squares, and will be denoted by N = N m := |E| (one often writes r ( m )). Given an eigenvalue E = 4 π m , its eigenspace has dimension N ,and admits the L -orthonormal basis(1.6) { e ( µ · x ) } µ ∈E , where e ( z ) := e πiz . All the corresponding eigenfunctions may be written as linear combi-nations of these exponentials.1.2. Arithmetic random waves.
Let us consider the random Gaussian toral Laplaceeigenfunctions (‘arithmetic random waves’ [24], [28], [21])(1.7) F ( x ) = 1 √N X µ ∈E a µ e ( µ · x )with eigenvalue E = 4 π m , where a µ are complex standard Gaussian random variables(meaning that E ( a µ ) = 0 and E ( | a µ | ) = 1), independent except for the relations a − µ = a µ , which make F ( x ) real-valued. The arithmetic random wave (1.7) is thus a (centred)Gaussian random field, and the nodal area (1.4) is a random variable associated to F .The Gaussian field F is stationary , in the sense that its covariance function r depends on x only:(1.8) r ( x ) := E [ F ( y ) · F ( x + y )] = 1 N X µ ∈E e ( µ · x ) . Every (centred) Gaussian random field is determined by its covariance function via Kol-mogorov’s Theorem (see e.g. [11, § / √N in (1.7),which clearly has no impact on the nodal area, is chosen so that F is unit-variance (i.e. r (0) = 1).1.3. Prior work on this model.
Let us consider the general setting of arithmetic randomwaves in dimension d ≥ Definition 1.1.
For d ≥ , we denote (1.9) E ( m, d ) := { µ ∈ Z d : k µ k = m } the set of all lattice points on √ m S d − , and N ( d ) m their number. JACQUES BENATAR RICCARDO W. MAFFUCCI
If the index d is omitted it is understood that d = 3. On T d , the arithmetic randomwaves have the expression F : T d → R ,(1.10) F ( x ) = 1 q N ( d ) m X µ ∈E ( m,d ) a µ e ( µ · x ) , where m is the sum of d integer squares, and a µ are complex standard Gaussian randomvariables, independent save for the relations a − µ = a µ , which make F ( x ) real-valued.Rudnick and Wigman [28] investigated the d − T d . They computed the expected value to be, for d ≥ E [ V ] = I d √ m, where I d = r πd · Γ( d +12 )Γ( d )(see [28, Proposition 4.1]). Note that the order of magnitude of (1.11) agrees with Yau’sconjecture (1.3). They also gave the following bound for the variance: for d ≥ V ar ( V ) ≪ m √N as N → ∞ (see [28, Proposition 6.1]). As a consequence, the nodal volume concentrates around itsmean (see [21, Section 1.2]). Rudnick and Wigman ([28, Section 1]) conjectured that thestronger bound(1.13)
V ar ( V ) ≪ m N should hold.A deep result of Krishnapur, Kurlberg and Wigman [21] is the precise asymptotic behaviourof the variance on T (here the volume is the length of the nodal lines). The energy levelsfor d = 2 are the numbers expressible as a sum of two squares V (2) = { m : 0 < m = a + b , a, b ∈ Z } . For any subsequence of energies { m i } i ⊂ V (2) such that the multiplicities N m i → ∞ , itwas shown that (cf. [21, Theorem 1.1])(1.14) V ar ( V ) = c m i m i N m i (1 + o (1)) , where the positive real numbers c m i depend on the angular distribution of E ( m i ,
2) - theasymptotics for the variance are non-universal (see [21, Section 1.2]).Also remarkably, the order of magnitude of (1.14) is much smaller than expected (1.13),as the terms of order m N in the asymptotic expression for the nodal length variance can-cel perfectly; this effect was called arithmetic Berry cancellation in [21], after “Berry’scancellation phenomenon” [2]. ANDOM WAVES ON T Statement of main results.
The variance of the nodal area has the following preciseasymptotics.
Theorem 1.2. As m → ∞ , m , , , we have V ar ( A ) = m N · (cid:20) O (cid:18) N / − o (1) (cid:19)(cid:21) . The 3-dimensional torus exhibits arithmetic Berry cancellation like the 2-dimensionaltorus; the variance has the same order of magnitude as (1.14). See Section 5.4 for moredetails.We also remark that, unlike the 2-dimensional case, the leading order term does not fluc-tuate: this is because lattice points on spheres are equidistributed (see Section 2.2).We impose the natural condition m , , N → ∞ (see Section 2.1). Indeed, if m ≡ E ( m,
3) (1.5) is empty and E (4 m,
3) = { µ : µ ∈ E ( m, } (see e.g. [17, § m Definition 1.3.
For ℓ ≥ , the set of d -dimensional ℓ -th lattice point correlations , or ℓ - correlations for short, is C ( d ) m ( ℓ ) := n ( µ , . . . , µ ℓ ) ∈ E ( m, d ) ℓ : ℓ X i =1 µ i = 0 o . The set of non-degenerate ℓ -correlations is X ( d ) m ( ℓ ) := n ( µ , . . . , µ ℓ ) ∈ C ( d ) m ( ℓ ) : ∀H ( { , . . . , ℓ } , X i ∈H µ i = 0 o . We note that, for even ℓ , the set of ℓ -correlations is related to the ℓ -th moment of thecovariance function (1.8) as follows: R ( ℓ ) = |C ( d ) m ( ℓ ) |N ℓ , where we define(1.15) R ( ℓ ) = R ( d ) m ( ℓ ) := Z T d | r ℓ ( x ) | dx. To prove Theorem 1.2, we shall require the following arithmetic formula.
Proposition 1.4 (Arithmetic formula) . As m → ∞ , m , , , we have V ar ( A ) = m N · (cid:20) O (cid:18) N / − o (1) + |X (4) |N + |C (6) |N (cid:19)(cid:21) . We are naturally led to the following arithmetic problem:
JACQUES BENATAR RICCARDO W. MAFFUCCI
Question 1.5.
How big are the sets C ( d ) m ( ℓ ) and X ( d ) m ( ℓ ) ? Firstly, it is easy to see that, for every d ,(1.16) |C ( d ) m (2) | = N ( d ) m . The case d = 2 of this problem was studied in detail by Bombieri and Bourgain [3]. Wehighlight two implications of their results since they are relevant to our own investigations.First, one has the unconditional bound |C (2) m (6) | = O ( N / ) as N → ∞ , which is proven via the Szemer´edi-Trotter Theorem (see [3, Section 2]). The second resultconcerns the even length correlations, where the number of tuples with pairwise vanishingcomponent vectors is of the order N ℓ/ : for a density 1 sequence of energy levels { m } ,it follows from [3, Theorem 17] (see also [4, Lemma 4]), that these tuples make up themajority of the set C (2) m ( ℓ ).Our next two theorems deal with the 3-dimensional setting. We provide an estimatefor the number of correlations and show that, for even ℓ ≥
8, the non-degenerate tuplesdominate those that cancel pairwise.
Theorem 1.6.
Letting m → ∞ , one has the estimate (1.17) |X (3) m (4) | ≪ N / o (1) . Theorem 1.7.
Letting m → ∞ , one has the estimate (1.18) |C (3) m (6) | ≪ N / o (1) . Corollary 1.8 (Long correlations) . For any even length ℓ ≥ one has the bounds N ℓ − − o (1) ≪ |X m ( ℓ ) | ≪ N ℓ − / o (1) . (1.19) as m → ∞ , m , , . The upper bound holds for all ℓ ≥ .Proof of Theorem 1.2. Insert the bounds of Theorems 1.6 and 1.7 into Proposition 1.4. (cid:3)
Remark.
We believe Theorems 1.6 and 1.7 to be of independent interest. In additionto their application in the proof of Theorem 1.2, they allow for the study of finer aspectsof A . In the companion paper [8], it is shown by way of Theorems 1.6 and 1.7, that inthe Wiener chaos expansion of A , only the fourth order chaos component is asymptoticallysignificant: its distribution is asymptotic to the distribution of A .The lower bound in Corollary 1.8 indicates that the value distribution of r ( x ), whenaveraged over the whole torus, is not Gaussian. For example the eighth moment of r ( x )blows up relative to the variance. ANDOM WAVES ON T Outline of the paper.
In the rest of this work, we prove Theorems 1.6 and 1.7,and Proposition 1.4. The proof of Proposition 1.4 begins in Section 4 and is concluded inSection 5, after the necessary preparatory results have been stated. The proof follows themethod employed in [21] for the 2-dimensional case. The arithmetic random wave F (asin (1.7)) is a Gaussian random field: the variance of the nodal area may be evaluated viathe Kac-Rice formulas , which are discussed in Section 4. To this purpose, it is necessaryto understand the (scaled) two-point correlation function K of F (defined in (4.1) and(4.5)). In Section 4, we express K in terms of the conditional Gaussian expectation ofthe 6 × ∇ F (0) , ∇ F ( x )) conditioned on F (0) = 0 , F ( x ) = 0; the resulting (scaled)covariance matrix, Ω, depends on the covariance function (1.8) and its (first and secondorder) derivatives.Next, in Section 5, we define a small set S ⊂ T (the singular set , cf. Definition 5.3),where it is possible to bound the contribution of K to the variance. We then establishasymptotics for K valid outside the set S : this computation involves the Taylor expansionof K as a 6-variate function of the matrix Ω around the identity matrix I ; in fact,we will show that, on T \ S , Ω is a small perturbation of I . The Taylor expansion iscarried out in Section A, using Berry’s method [2]. In Section 6 we perform the technicalcomputations needed to evaluate the leading constant of the nodal area variance; thenecessary background on spherical lattice points is covered in Section 2.Let us highlight similarities and differences with the 2-dimensional setting [21]. Boththe leading term and error term in Proposition 1.4 are of arithmetic nature, as in [21]: theleading term depends on the angular distribution of lattice points on spheres, while theerror term depends on the lattice point correlations of Definition 1.3. However, there aremarked differences between the 2- and 3-dimensional settings; first, as noted above, thenodal area variance obeys an asymptotic law, whereas the nodal length variance dependson arithmetic properties of the energy.Second, for the admissibility of the error term, we require a bound for |X (3) m (4) | whereas,in the 2-dimensional setting, |X (2) m (4) | = 0 for all m ∈ V (2) , which may be seen by noting that two circles intersect in at most two points (Zygmund’strick [33]). The bound for the length four correlations of Theorem 1.6 will be establishedin Section 3.1.One must also bound the total number of length six correlations C m (6). The proof ofTheorem 1.7 will be established in Section 3.2 via a theorem due to Fox, Pach, Sheffer, Sukand Zahl [15]. Their result allows one to bound the number of incidences between pointsand spheres in R , thereby playing the role of the Szemer´edi-Trotter Theorem employed indimension 2. Notation.
For functions f and g we will use Landau’s asymptotic notation f = O ( g )or equivalently f ≪ g to denote the inequality f ≤ Cg for some constant C . We may adda subscript e.g. f ≪ t g to emphasize the fact that C depends on the parameter t . Thestatement f ≍ g means g ≪ f ≪ g . JACQUES BENATAR RICCARDO W. MAFFUCCI
The letter µ will be reserved for elements of E m while τ will denote a member of E m + E m .Generic (deterministic) points x ∈ R will be underlined and we will write u · x for theEuclidean inner product of two vectors u and x . Finally, we will denote by S ( x, R ) ⊂ R the sphere of radius R centered at x . Acknowledgements.
R.M. was funded by a Graduate Teaching Scholarship, Departmentof Mathematics, King’s College London, and worked on this project as part of his PhDthesis, under the supervision of Igor Wigman. The authors wish to thank Igor Wigman forsuggesting this collaboration, for insightful remarks and corrections, and for his availability.The authors wish to thank Ze´ev Rudnick and Joshua Zahl for helpful communications andShagnik Das for pointing out the explicit bound in Theorem B.2.2.
Lattice points on spheres and correlations
Preliminary results about the lattice point set.
Recall the notation for thelattice point set E m and its cardinality N m . As mentioned in the introduction, E m is non-empty if and only if m is not of the form 4 l (8 k +7). We work with the assumption m , , µ (1) , µ (2) , µ (3) ) ∈ E m with µ (1) , µ (2) , µ (3) coprime. In this case, the quantities N m and m are related by the estimates(2.1) m / − o (1) ≪ N m ≪ m / o (1) (see e.g. [5, Section 1] or [29, Section 4]).We shall require a bound for the number of lattice points on circles centred in Z , κ ( m ) := max P |E m ∩ P | , where the maximum is taken over all planes P . It was shown by V. Jarn´ık [20] that κ ( m ) ≪ m o (1) . (2.2)Recall Definition 1.3 of the set of lattice point ℓ -correlations C = C ( d ) m ( ℓ ) and the subsetof non-degenerate correlations X = X ( d ) m ( ℓ ). In what follows, we may omit the index d when d = 3, as we will be mostly concerned with the 3-dimensional setting; we may alsosuppress the dependency on m . Definition 2.1.
Denote by D = D ( d ) m ( ℓ ) the set of degenerate correlations so that C = D ˙ ∪X . Let ℓ be an even positive integer. We will call symmetric correlations D ′ those that cancelout in pairs. Further, denote by D ′′ the set of diagonal correlations of the form {± µ, . . . , ± µ } (with exactly ℓ/ plus signs). Note that D ′′ ⊆ D ′ ⊆ D . ANDOM WAVES ON T Let us analyse in detail the set C (4) = C (3) m (4), as several summations range over thisset in what follows. Let d = 3 and ℓ = 4 in Definitions 1.3 and 2.1. Then D (4) = D ′ (4) isthe set of quadruples ( µ , µ , µ , µ ) that cancel out in pairs, µ = − µ and µ = − µ , and permutations of the indices (i.e., each degenerate correlation is necessarily symmetricwhen ℓ = 4). The diagonal correlations D ′′ ⊂ D satisfy µ = µ = − µ = − µ for some permutation of the indices.With X (4) denoting as usual the set of non-degenerate correlations, a summation over C (4)may thus be treated by separating it as follows:(2.3) X C (4) = X µ = − µ µ = − µ + X µ = − µ µ = − µ + X µ = − µ µ = − µ + O X D ′′ + X X (4) . The proof of Theorem 1.6 will rely on a classical estimate regarding the size of the set I m ( r ) := (cid:8) ( µ , µ ) ∈ E m | µ · µ = r (cid:9) = (cid:8) ( µ , µ ) ∈ E m | k µ + µ k = 2( m + r ) (cid:9) . In fact, there is an exact formula for | I m ( r ) | (see [26, Section 7]) from which one can deducethe following bound. Theorem 2.2. [26]
For | r | < m one has that | I m ( r ) | ≪ gcd( r, m ) / m o (1) . Before proceeding to the next lemma we introduce some notation. Given a ∈ N write a m := gcd( a, m ), yielding the corresponding decomposition a = a m a ′ . For any interval J ⊂ (0 , m ) we may now introduce the collection J m ( J, a ) = (cid:8) τ ∈ E m + E m | k τ k ∈ J, k τ k ≡ a (cid:9) . Lemma 2.3. (i) For any
B ⊂ E m + E m satisfying the bound (cid:12)(cid:12)(cid:8) || τ || | τ ∈ B (cid:9)(cid:12)(cid:12) ≤ T one hasthat |B| ≪ N o (1) T / .(ii) Given any natural number a = a m a ′ and any interval J ⊂ (0 , m ) we have the estimate | J m ( J, a ) | ≤ N o (1) (cid:18) | J | ( a m ) / a ′ + N ( a ′ ) / (cid:19) . Proof. (i) Given any d | m the number of τ ∈ B for which gcd( ( || τ || − m ) / , m ) = d is atmost ′ X | l | Linnik conjectured (and provedunder GRH) that the projected lattice points E m / √ m ⊂ S become equidistributed as m → ∞ , m , , Lemma 2.4 ([25, Lemma 8]) . Let g ( z ) be a smooth function on S . For m → ∞ , m , , , we have N X µ ∈E g (cid:18) µ | µ | (cid:19) = Z z ∈S g ( z ) dz π + O g (cid:18) m / − o (1) (cid:19) . For each positive integer k , define the k -th moment of the normalised inner product oftwo lattice points(2.5) B k = B (3) k ( m ) := 1 m k N X µ ,µ ∈E ( µ · µ ) k . This arithmetic quantity arises naturally in the computation of the leading term of thevariance (see Section 6). By the equidistribution of lattice points on spheres, we will nowshow that each B k has a unique limit as m → ∞ . ANDOM WAVES ON T Lemma 2.5. We have: B k = for odd k ; for k = 2; k +1 + O (cid:16) m / − o (1) (cid:17) for even k ≥ , as m → ∞ , m , , . In particular, (2.6) B = 15 + O (cid:18) m / − o (1) (cid:19) . Proof. For odd k , the summands of (2.5) cancel out in pairs, by the symmetry of the set E . For k = 2, the result was shown in [28, Lemma 2.3]. It remains to prove the case ofeven k ≥ 4; we rewrite B k = 1 N X µ ,µ (cos( ϕ µ ,µ )) k , where ϕ µ ,µ is the angle between µ and µ . In a moment, we will show that, for all µ ,(2.7) 1 N X µ (cos( ϕ µ ,µ )) k = 1 k + 1 + O (cid:18) m / − o (1) (cid:19) , which implies 1 N X µ ,µ (cos( ϕ µ ,µ )) k = 1 N X µ (cid:18) k + 1 + O (cid:18) m / − o (1) (cid:19)(cid:19) hence the result of the present lemma. It remains to show (2.7); apply Lemma 2.4 with g ( · ) = cos k ( ϕ µ , · ):(2.8) 1 N X µ (cos( ϕ µ ,µ )) k = Z z ∈S (cos( ϕ µ ,z )) k dz π + O (cid:18) m / − o (1) (cid:19) . Write z = (sin( θ ) cos( ψ ) , sin( θ ) sin( ψ ) , cos( θ )), with spherical coordinates 0 ≤ θ ≤ π and0 ≤ ψ ≤ π . As the uniform probability measure dz π on S is rotation invariant, the integralin (2.8) is independent of µ , and we may rewrite(2.9) Z z ∈S (cos( ϕ µ ,z )) k dz π = Z z ∈S (cos( ϕ (0 , , ,z )) k dz π = 14 π Z π dψ Z π cos k θ sin θdθ = 1 k + 1 . Substituting (2.9) into (2.8) yields (2.7). (cid:3) The proof of Theorems 1.6 and 1.7 Length four correlations. In this subsection, we prove Theorem 1.6. For fixed µ , µ ∈ E m (with µ = − µ ), write τ = ( t , t , t ) := − ( µ + µ ). Clearly any pair ofpoints µ , µ ∈ E m satisfying µ + µ = τ (3.1)must both lie on the intersection S (0 , m / ) ∩ S ( τ, m / ). The resulting intersection isa circle of radius ρ = ( m − k τ k ) / , centered at τ and is confined to the plane (cid:8) x ∈ R (cid:12)(cid:12) τ · x = k τ k (cid:9) .As a consequence we may count the number of pairs ( µ , µ ) satisfying (3.1) by esti-mating the size of the set ˜ X ( τ ) consisting of those integer lattice points which lie in theplane P : τ · x = 0 x ∈ R (3.2)and have norm 2 ρ = (4 m − k τ k ) / . A bound for X m (4) is then given by |X m (4) | ≪ X τ ∈E m + E m X µ ,µ ∈E m µ + µ = τ (cid:16) | ˜ X ( τ ) | − (cid:17) + , (3.3)where the summation takes into account only those pairs ( µ , µ ) for which ˜ X ( τ ) containsat least two non-antipodal points. In the remainder of this subsection we will seek to boundthe size of the set T := { τ ∈ E m + E m | | ˜ X ( τ ) | > } . Proposition 3.1. With the above notation we have the estimate |T | ≪ N / o (1) . Let us first prove Theorem 1.6 assuming Proposition 3.1. Proof of Theorem 1.6. Following (2.2) one has the general upper bound | ˜ X ( τ ) | ≪ m o (1) whenever τ = 0. Inserting both this estimate and the bound of Proposition 3.1 into (3.3)we get (1.17). (cid:3) The proof of Proposition 3.1. In order to understand ˜ X ( τ ), we begin with a simpledescription of P ∩ Z . Recalling the notation τ = ( t , t , t ) let us first set gcd( t , t , t ) = s and write τ ′ = ( t ′ , t ′ , t ′ ) := s τ . Since τ ′ is primitive, the lattice P ∩ Z has determinant k τ ′ k (cf. the corollary [9, page 25]) and hence there exist vectors A, B ∈ Z with A × B = τ ′ .A generic lattice point in P may be expressed as kA + lB with k, l ∈ Z .Let us suppose τ ∈ T and write n := 4 m −k τ k . As τ ∈ T , there must be two non-antipodalvectors C = k A + l B and D = k A + l B for which k C k = k D k = n. Setting r := k l − k l we observe that C × D = r ( A × B ) = rs τ and record the inequality(3.4) k τ k ≤ s m r . ANDOM WAVES ON T Moreover, noting that k C × D k = n − ( C · D ) we obtain the identity s r (cid:0) n − ( C · D ) (cid:1) + n = 4 m. Multiplying both sides of the equation by 4 r s one gets the rearranged expression(2 s n + r ) − (2 s ( C · D )) = 16 mr s + r and hence (cid:0) s n + r − s ( C · D ) (cid:1) (cid:0) s n + r + 2 s ( C · D ) (cid:1) = 16 mr s + r . (3.5)Assuming the equation (3.5) has solutions, there must exist a positive d | mr s + r (givenby either factor on the LHS of (3.5)) so that4 s n + 2 r = d + 1 d (cid:0) mr s + r (cid:1) . (3.6)To count the number of vectors τ ∈ T we will consider equation (3.6) in each dyadicinterval r ∈ [ R, R ] , s ∈ [ S, S ]. Here R and S are dyadic powers in the ranges 1 ≤ R ≤ m and 1 ≤ S ≤ m / . Lemma 3.2. With R, S as above let T ( R, S ) denote the set of τ ∈ T which satisfy equation (3 . for some pair of integers ( r, s ) ∈ [ R, R ] × [ S, S ] . Then |T ( R, S ) | ≪ N o (1) min (cid:18) mS + N , Sm R + N , N R / S / (cid:19) . (3.7) Proof. Given τ ∈ T ( R, S ) with its associated quadruple ( n, r, s, d ) we recall that k τ k ≡ s . Setting s m := gcd( s, m ) we may write s = s m s ′ and put ν := gcd( s , m ). Clearly s m | ν and ν | ( s m ) so we are led to a decomposition of the form ν = s m σ , s m = σ σ which yields s = ν ( σ ( s ′ ) ). It follows from Lemma 2.3 part (ii) (with J = (0 , m ) and a = s ) and the inequality ( s m σ ) / σ ≥ ( s m σ σ ) / = s m that |T ( R, S ) | ≪ X s m | ms m ≤ S X s ′ ≍ S/s m X σ σ = s m N o (1) (cid:18) m ( s m σ ) / σ ( s ′ ) + N ( σ ) / s ′ (cid:19) (3.8) ≪ X s m | ms m ≤ S X s ′ ≍ S/s m N o (1) (cid:18) ms m ( s ′ ) + N s ′ (cid:19) ≪ N o (1) (cid:16) mS + N (cid:17) , yielding the first inequality in (3.7). In light of (3.4) we may reuse the estimates given in(3.8), this time applying Lemma 2.3 part (ii) with the interval J = (0 , s m /R ). Thebound |T ( R, S ) | ≪ N o (1) ( Sm/R + N ) follows readily. A brief inspection of (3.6) reveals that for each choice of ( r, s ) ∈ [ R, R ] × [ S, S ] and eachchoice of divisor d | mr s + r , the value of n is uniquely determined. In this manner weget O ( N o (1) RS ) possible values of n and hence the final estimate in (3.7) follows from anapplication of Lemma 2.3 part (i). (cid:3) To conclude the proof of Proposition 3.1, we note that |T | ≤ P R,S |T ( R, S ) | and applythe estimates of Lemma 3.2 to get |T | ≪ X R ≤ m,S ≤ m / dyadic N o (1) min (cid:18) N R / S / , Sm R , mS (cid:19) + N o (1) . For fixed S , the largest possible value of min( N R / S / , Sm /R ) occurs when R ≍ S / m / / N / . Recalling the relation between m and N (2.1),min( N R / S / , Sm /R ) ≪ N / o (1) S / . It follows that |T | ≪ X S ≤ m / N o (1) min (cid:16) N / S / , mS (cid:17) + N o (1) ≪ N / o (1) . Length six correlations. In this subsection, we prove Theorem 1.7. The key ingre-dient is the incidence bound [15, Theorem 6.4], which we state below in a simplified form.Given a collection of points P and a collection of varieties V , we define I ( P , V ) := |{ ( p, V ) ∈ P × V | p ∈ V }| to be the number of incidences between P and V .We will use the standard notation K s,t for complete bipartite graphs. Given graphs G and H , we say G is H -free if it does not contain an induced subgraph isomorphic to H . Theorem 3.3. [15] Let P ⊂ R be a set of k points and V a collection of n varieties ofbounded degree in R . Assuming the incidence graph of P × V is K s,t -free there exists, foreach ε > , a positive constant c = c ( ε ) so that I ( P , V ) ≤ stc (cid:18) k s s − + ε n s − s − + ( k + n ) (cid:19) . (3.9) Remark. The inequality (3.9) gives a polynomial dependence in t which will be crucialto the argument in this subsection. Although not explicitly stated in the above form onecan follow the proofs given in [15, Theorems 4.3 and 6.4] and keep track of all the constantsinvolved. We will carry out these straightforward modifications in Appendix B.To prove Theorem 1.7, we will apply Theorem 3.3 with the set of points P = E + E andvarieties S = (cid:8) S ( A, m / ) | A ∈ E + E + E (cid:9) . For fixed ε > m sufficiently large weset s = 2 and t = N ε and observe that, by (2.2), the incidence graph of P × S is K s,t -free.The remainder of the argument is carried out as in [3, Section 2] with Theorem 3.3 replacing ANDOM WAVES ON T the Szemer´edi-Trotter Theorem. For any dyadic power D ≥ S ( D ) the collectionof spheres S = S ( A, m / ) ∈ S for which | S ∩ P| ≍ D . Recalling (2.2) we gather that |C m (6) | ≪ ε N ε X D ≤N dyadic D |S ( D ) | . (3.10) Lemma 3.4. For D ≤ N we have the estimates ( i ) D |S ( D ) | ≪ N , ( ii ) D / |S ( D ) | ≪ N . Proof. (i) For each τ ∈ P = E + E denote by S τ ( D ) the collection of spheres in S ( D ) whichare incident to τ . Then we have the trivial bound D |S ( D ) | ≤ I ( P , S ( D )) ≤ X τ ∈P |S τ ( D ) | ≤ N . (ii) We first note the inequality |S ( D ) | ≤ |S ( D ) | / |P| / which follows easily from therearranged statement |S ( D ) | ≤ N . Applying Theorem 3.3 we get the bound D |S ( D ) | ≪ I ( P , S ( D )) ≪ ε |S ( D ) | / |P| / ε + |S ( D ) | + |P| ≤ |S ( D ) | / |P| / ε + |P| . (3.11)When the first term on the RHS of (3.11) dominates one finds that D |S ( D ) | ≪ |S ( D ) | / |N | / which gives D / |S ( D ) | ≪ N . When the second term on the RHS dominates we get D |S ( D ) | ≪ N so that D / |S ( D ) | ≤ D |S ( D ) |N / ≪ N / . (cid:3) Combining the estimates of the lemma with (3.10) we get |C m (6) | ≪ ε N ε X D ≤N D · min (cid:18) N , N D / (cid:19) ≪ ε N / ε , which completes the proof of Theorem 1.7.3.3. Long correlations. In this section we will prove Corollary 1.8 via an analytic argu-ment. We introduce the function f ( α ) := P µ ∈E m e ( µ · α ) and observe that, by orthogonality, |C m ( ℓ ) | = Z [0 , f ( α ) ℓ dα. (3.12)3.3.1. An upper bound for |C m ( ℓ ) | . Let ℓ ≥ | f ( α ) | ≤ N . Combining the estimate (1.18) with (3.12) it follows that |C m ( ℓ ) | ≤ N ℓ − Z | f ( α ) | dα ≪ ε N ℓ − / ε . Remark. To conclude the discussion of the upper bounds we record the straightforwardestimate |C m (5) | ≤ Z [0 , | f ( α ) | dα ≤ Z [0 , | f ( α ) | dα ! / Z [0 , | f ( α ) | dα ! / ≪ ε N / ǫ and note that |C m (2) | = N while |C m (3) | ≪ N o (1) (as a consequence of (2.2)).3.3.2. A lower bound for |X m ( ℓ ) | . Let ℓ ≥ D ( ℓ ) and D ′ ( ℓ )for the set of degenerate and symmetric tuples respectively. Observe that the degeneratetuples in C m ( ℓ ) number at most |D ( ℓ ) | = |D ′ ( ℓ ) | + X ≤ j ≤ ... ≤ j k ≤ ℓ − j + ... + j k = ℓ ≤ j k k Y i =1 |C m ( j i ) | ≪ N ℓ/ + N ( P i ≤ k j i ) − − / o (1) ≪ N ℓ − / o (1) with the largest contribution coming from the multi-index j = 2 , j = ℓ − 2. As a resultit will suffice to prove the asserted lower bound in (1.19) for |C m ( ℓ ) | .Consider the set A := (cid:8) α ∈ [0 , | | f ( α ) | ≥ N / (cid:9) . Since f (0) = N and f has partialderivatives of size at most m / N ≪ ε N ε , we gather that(3.13) | f ( α ) − N | = | f ( α ) − f (0) | ≤ ||∇ f || ∞ · || α || ≪ ε N ε || α || . It follows that | f ( α ) | ≥ N / || α || ≪ ε N − − ε and hence the Lebesgue measureof A is bounded from below by λ ( A ) ≫ ε N − − ε . Inserting this information into (3.12) wefind the desired estimate |C m ( ℓ ) | ≥ Z A f ( α ) ℓ dα ≥ λ ( A )( N / ℓ ≫ ε N ℓ − − ε . Remark. When d ≥ X ( d ) m (4) is much larger than D ( d ) m (4), as opposed to what happens indimensions 2 and 3. Indeed, for N = N ( d ) m we set f ( α ) := X µ ∈E ( d ) m e ( µ · α ) , A ( d ) := n α ∈ [0 , d (cid:12)(cid:12) | f ( α ) | ≥ N / o and recall the estimates ([19, Theorem 20.2]) m d/ − ≪ d N ( d ) m ≪ d m d/ − . Proceeding as in (3.13) one finds that | f ( α ) − N | ≪ m / N · || α || ≪ N ( d − / ( d − · || α || . As a result λ d ( A ( d ) ) ≫ N − d/ ( d − (where λ d denotes the d -dimensional Lebesgue measure)and hence |C ( d ) m (4) | ≥ Z A ( d ) f ( α ) dα ≫ N − d/ ( d − ≫ N / . ANDOM WAVES ON T Kac-Rice formulas For a smooth random field, the moments of the geometric measure of the nodal set aregiven by the Kac-Rice formulas (see [1], Theorems 6.8 and 6.9). The arithmetic randomwave F (1.7) is a Gaussian field; for each x ∈ T , let φ F ( x ) be the probability densityfunction of the (standard Gaussian) random variable F ( x ), and φ F ( x ) ,F ( y ) the joint densityof the random vector ( F ( x ) , F ( y )). We define the zero density function (also called firstintensity ) K : T → R and function (also called second intensity )˜ K : T × T → R of F as the conditional Gaussian expectations K = φ F ( y ) (0) · E [ k∇ F ( y ) k (cid:12)(cid:12) F ( y ) = 0]and(4.1) ˜ K ( x ) = φ F ( y ) ,F ( x + y ) (0 , · E [ k∇ F ( y ) k · k∇ F ( x + y ) k (cid:12)(cid:12) F ( y ) = F ( x + y ) = 0] , the latter defined for x = 0. The functions K and ˜ K do not depend on y , since F isstationary. The Kac-Rice formulas for the first and second moments of the nodal area are(4.2) E ( A ) = Z T K dx = K and(4.3) E ( A ) = Z T ˜ K ( x ) dx. As mentioned in the introduction, the expected nodal area was computed by Rudnick andWigman to be ((1.11) with d = 3)(4.4) E [ A ] = 4 √ √ m. It is more convenient to work with a scaled version of the second intensity,(4.5) K ( x ) := 3 E ˜ K ( x ) , where we recall that E = E m = 4 π m . Applying the Kac-Rice formulas, we obtain thefollowing precise expression for the variance of the nodal area. Proposition 4.1. (4.6) V ar ( A ) = E Z T (cid:18) K ( x ) − π (cid:19) dx. Proof. By (4.3) and (4.4), E [ A ] − ( E [ A ]) = Z T ˜ K ( x ) dx − m = Z T (cid:18) ˜ K ( x ) − m (cid:19) dx = E Z T (cid:18) K ( x ) − E m (cid:19) dx = E Z T (cid:18) K ( x ) − π (cid:19) dx. (cid:3) By the above arguments, to understand the nodal area variance of the arithmetic ran-dom wave F , we need to study the (scaled) two-point function K ; let us begin by intro-ducing the necessary notation. Recall the covariance function r of F is given by (1.8).Let(4.7) D ( x ) := ∇ r ( x ) = 2 πi N X µ ∈E e ( µ · x ) · µ, where for j = 1 , , D j ( x ) = ∂r∂x j ( x ) = 2 πi N X µ ∈E e ( µ · x ) µ ( j ) , µ = ( µ (1) , µ (2) , µ (3) ) . Further, denote(4.8) H ( x ) := − π N X µ ∈E e ( µ · x ) · µ t µ the Hessian 3 × r , where for j, k = 1 , , H jk ( x ) := ∂ r∂x j ∂x k ( x ) = − π N X µ ∈E e ( µ · x ) µ ( j ) µ ( k ) . The n × n identity matrix will be denoted I n . Proposition 4.2. The scaled two-point correlation function may be expressed as (4.9) K ( x ) = 12 π p − r ( x ) · E [ k w k · k w k ] , where w , w are three-dimensional random vectors with Gaussian distribution ( w , w ) ∼N (0 , Ω( x )) ; their covariance matrix is given by (4.10) Ω = I + (cid:18) X YY X (cid:19) , the × matrices X and Y being defined as (4.11) X ( x ) = − − r E · D t D and (4.12) Y ( x ) = − E · (cid:18) H + r − r · D t D (cid:19) . Proof. As F is stationary, (4.1) may be rewritten as(4.13) ˜ K ( x ) = φ F (0) ,F ( x ) (0 , · E [ k∇ F (0) k · k∇ F ( x ) k (cid:12)(cid:12) F (0) = F ( x ) = 0] . Since the covariance matrix of ( F (0) , F ( x )) is(4.14) A ( x ) = (cid:18) r ( x ) r ( x ) 1 (cid:19) , ANDOM WAVES ON T the joint Gaussian density equals(4.15) φ F (0) ,F ( x ) (0 , 0) = 12 π p − r ( x ) . By [28, Lemma 5.1], the covariance matrix of the eight-dimensional Gaussian vector( F (0) , F ( x ) , ∇ F (0) , ∇ F ( x ))is the block matrix Σ( x ) = (cid:18) A BB t C (cid:19) , with A as in (4.14), B ( x ) = (cid:18) × D × ( x ) − D × ( x ) 0 × (cid:19) and C = (cid:18) E I − H ( x ) − H ( x ) E I (cid:19) . By [24, Section 2.3] (also see [28, Proposition 2.4 (1)]), there are only finitely many x ∈ T such that r ( x ) = ± 1. Therefore, for almost all x ∈ T , the covariance matrix A ( x )is nonsingular. In view of [1, Proposition 1.2] (see also the hypotheses of [1, Theorem6.9]), the covariance matrix of ( ∇ F (0) , ∇ F ( x )) conditioned on F (0) = 0 , F ( x ) = 0 is˜Ω( x ) := C − B t A − B . We then have(4.16) E [ k∇ F (0) k · k∇ F ( x ) k (cid:12)(cid:12) F (0) = F ( x ) = 0] = E [ k v k · k v k ] , where v , v are three-dimensional random vectors with ( v , v ) ∼ N (0 , ˜Ω). Inserting (4.15)and (4.16) into (4.13) we obtain(4.17) ˜ K ( x ) = 12 π p − r ( x ) · E [ k v k · k v k ] , ( v , v ) ∼ N (0 , ˜Ω) . Lastly, to prove the expression (4.9) for the scaled two-point function, we rescale therandom vectors v i =: r E w i i = 1 , , and the matrix ˜Ω =: E X, Y as in (4.11) and (4.12). (cid:3) In the proof of the latter proposition, we saw that the distribution of ( w , w ) is non-degenerate (i.e., the matrix Ω( x ) is nonsingular) for almost all x . Also note that (4.10)expresses Ω( x ) as a perturbation of the identity matrix, in the sense that the entries of X ( x ) , Y ( x ) are small for ‘typical’ x ∈ T .5. Proof of Proposition 1.4 The contribution of the singular set. We will define a small subset of the torus,called the singular set S : outside of S , we will eventually establish precise asymptoticsfor the two-point correlation function K (recall (4.5) and (4.1)). The goal of the presentsubsection is to bound K on S , and also to control the measure of S . The definitions and results of the present section are borrowed from [24], [28] and [21]. Recall the notation E for the set of all lattice points on the sphere of radius √ m . Definition 5.1. We call the point x ∈ T positive singular (resp. negative singular )if there exists a subset E x ⊆ E with density |E x ||E| > such that cos(2 π ( µ · x )) > (resp. cos(2 π ( µ · x )) < − ) for all µ ∈ E x . For instance, the origin (0 , , 0) is a positive singular point. Take q ≍ √ m and partitionthe torus into q cubes, each centred at a/q , a ∈ Z , of side length 1 /q . Note that thecubes have disjoint interiors. Definition 5.2. We call the cube Q ⊂ T positive singular (resp. negative singular ) if itcontains a positive (resp. negative) singular point. Definition 5.3. The singular set S is the union of all positive and negative singularcubes. The main result of the present subsection is the bound for the integral of K on S , forwhich we shall need two lemmas. The covariance function r of the arithmetic random wave F satisfies | r ( x ) | ≤ 1. The following lemma shows that, on S , r is bounded away from 0. Lemma 5.4 ([24, Lemmas 6.4 and 6.5]) . (1) For all positive (resp. negative) singular cubes Q , there exists a subset E Q ⊆ E withdensity |E x ||E| > such that for all y ∈ Q and for all µ ∈ E Q , we have cos(2 π ( µ · y )) > (resp. cos(2 π ( µ · y )) < − / ).(2) For all y ∈ S : | r ( y ) | > . Recall the definitions (4.11) and (4.12) for the matrices X ( x ) and Y ( x ). Lemma 5.5 (cf. [21, Lemma 3.2]) . We have uniformly (entry-wise) (5.1) X ( x ) = O (1) , Y ( x ) = O (1) . One immediate consequence is (5.2) K ( x ) ≪ p − r ( x ) . Recall the notation R ( ℓ ) (1.15) for the ℓ -th moment of the covariance function r . Proposition 5.6 (cf. [24, Section 6.3] and [21, Lemma 4.4]) . (1) The contribution of the singular set to (4.6) has the following bound: Z S | K ( x ) | dx ≪ meas ( S ) . ANDOM WAVES ON T (2) For all integers ℓ ≥ : meas ( S ) ≪ R ( ℓ ) . We end this subsection with a property of the covariance function outside the singular set. Lemma 5.7 ([24, Lemma 6.5]) . For all x / ∈ S , | r ( x ) | is bounded away from : | r ( x ) | ≤ − . Thanks to the lemma, on the non-singular set T \ S we have the following approxima-tions:(5.3) 1 √ − r = 1 + 12 r + 38 r + O ( r )and(5.4) 11 − r = 1 + r + O ( r ) . Asymptotics for K on the non-singular set.Lemma 5.8. Let ( w , w ) ∼ N (0 , Ω) , Ω = I + (cid:18) X YY X (cid:19) , with rank ( X ) = 1 . Then: E [ k w k · k w k ] = 8 π · (cid:20) tr ( X )3 + tr ( Y )18 − tr ( XY )45 − tr ( X )45+ tr ( Y )900 + tr ( Y ) − tr ( X ) tr ( Y )90 (cid:21) + O ( tr ( X ) + tr ( Y )) . The proof of Lemma 5.8 is quite lengthy and takes up the whole of Appendix A. Assumingit, we arrive at the asymptotics for K on T \ S . Proposition 5.9. For x ∈ T such that r ( x ) is bounded away from ± , we have thefollowing asymptotics for the (scaled) two point correlation function: K ( x ) = 4 π + L ( x ) + ǫ ( x ) where (5.5) L ( x ) := 4 π (cid:20) r + tr ( X )3 + tr ( Y )18 + 38 r − tr ( XY )45 − tr ( X )45+ tr ( Y )900 + tr ( Y ) − tr ( X ) tr ( Y )90 + 16 r tr ( X ) + 136 r tr ( Y ) (cid:21) and ǫ ( x ) := O [ r + tr ( X ) + tr ( Y )] . Proof of Proposition 5.9 assuming Lemma 5.8. By Proposition 4.2, we have (4.9); for thefirst factor of (4.9), as r ( x ) is bounded away from ± 1, we may use the expansion (5.3). Onthe second factor of (4.9), apply Lemma 5.8 with X, Y as in (4.11) and (4.12). (cid:3) Later we will need to integrate L term-wise. Notation. To simplify the formulation of our next result, We will write f ∼ ψ g if | f − g | = O (cid:18) |X (4) |N + |C (6) |N (cid:19) and we will write f ∼ ϕ g if | f − g | = O (cid:18) m / − o (1) · N + |X (4) |N + |C (6) |N (cid:19) . Lemma 5.10. We have the following estimates: (1) Z T trX ( x ) dx ∼ ψ − N − N . (2) Z T trY ( x ) dx ∼ ψ N − N . (3) Z T tr ( XY )( x ) dx ∼ ψ − N . (4) Z T tr ( X )( x ) dx ∼ ψ N . (5) Z T tr ( Y )( x ) dx ∼ ϕ · N . (6) Z T ( trY ( x )) dx ∼ ϕ · N . (7) Z T ( trX · trY )( x ) dx ∼ ψ − N . (8) Z T ( r trX )( x ) dx ∼ ψ − N . (9) Z T ( r tr ( Y ))( x ) dx ∼ ψ N . (10) Z T tr ( X )( x ) dx = O (cid:18) |C (6) |N (cid:19) . (11) Z T tr ( Y )( x ) dx = O (cid:18) |C (6) |N (cid:19) . The proof of Lemma 5.10 is given in Section 6.5.3. Proof of Proposition 1.4. Assuming the above preparatory results, we arrive atthe asymptotics for the nodal area variance. Proof of Proposition 1.4. In the expression for the variance of Proposition 4.1, we separatethe domain of integration over the singular set S ⊂ T of Definition 5.3 and its complement:(5.6) V ar ( A ) = E Z T \ S (cid:18) K ( x ) − π (cid:19) dx + E Z S (cid:18) K ( x ) − π (cid:19) dx. By Lemma 5.7, the asymptotics for K of Proposition 5.9 hold outside the singular set:(5.7) Z T \ S (cid:18) K ( x ) − π (cid:19) dx = Z T \ S L ( x ) dx + O Z T \ S | ǫ ( x ) | dx. ANDOM WAVES ON T Note that the constant term 4 /π of the nodal area variance cancels out with the expecta-tion squared. Next, recall Proposition 5.6:(5.8) Z S | K ( x ) | dx ≪ meas ( S ) ≪ R (6) = |C (6) |N . Inserting (5.7) and (5.8) into (5.6) gives(5.9) V ar ( A ) = E Z T \ S L ( x ) dx + E O Z T \ S | ǫ ( x ) | dx ! + O (cid:18) |C (6) |N (cid:19)! . The former error term is redundant by Lemma 5.10, parts 10 and 11. Using | r ( x ) | ≤ L , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T \ S L ( x ) dx − Z T L ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ Z S | L ( x ) | dx ≪ meas ( S )which together with (5.9) and (5.8) implies(5.10) V ar ( A ) = E Z T L ( x ) dx + O (cid:18) E · |C (6) |N (cid:19) . We integrate (5.10) term-wise (recall the expression (5.5) for L ), and, as the integral isover the whole torus, we may apply the considerations Z T r ( x ) dx = 1 N , Z T r ( x ) dx = 3 N + O (cid:18) N + |X (4) |N (cid:19) (see Lemma 6.1), and the estimates of Lemma 5.10, to deduce: V ar ( A ) = E π Z T (cid:20) r + tr ( X )3 + tr ( Y )18 + 38 r − tr ( XY )45 − tr ( X )45 + tr ( Y )900+ tr ( Y ) − tr ( X ) tr ( Y )90 + 16 r tr ( X ) + 136 r tr ( Y ) (cid:21) dx + O (cid:18) E · |C (6) |N (cid:19) = E π (cid:20) N (cid:18) − · ( − 3) + 118 · (cid:19) + 1 N (cid:18) · ( − 3) + 118 · ( − 6) + 38 · − 145 ( − − · 15 + 1900 · · − 190 ( − 27) + 16 ( − 3) + 136 · (cid:19) + O (cid:18) m / − o (1) · N + |X (4) |N + |C (6) |N (cid:19)(cid:21) , where we note the error term m/ N is negligible. The terms of order m/ N cancel perfectly:as noted in the introduction, the 3-dimensional torus exhibits arithmetic Berry cancellation (see the next section for more details). The terms of order m/ N sum up to E · π · N · · m N , hence, recalling (2.1), the claim of the present proposition. (cid:3) A note on arithmetic Berry cancellation. Let us analyse in more detail thevanishing of the term of order m/ N of the nodal area variance (cf. [21, Section 4.2]). Theleading term of K ( x ) − /π is (recall (5.5), (4.11) and (4.12))4 π (cid:20) r + tr ( X )3 + tr ( Y )18 (cid:21) ∼ π (cid:20) r + 13 (cid:18) E DD t (cid:19) + 118 (cid:18) E tr ( H ) (cid:19)(cid:21) = 2 π v ( x ) , having defined v ( x ) := r ( x ) − E ( DD t )( x ) + 1 E tr ( H ( x )) . The latter expression has the same shape as the two-dimensional case [21, (39)]: theremainder of this discussion is essentially identical to [21, Section 4.2]. One rewrites v ( x ) = 4 N X µ ,µ ∈E e ([ µ + µ ] · x ) · cos (cid:16) ϕ µ ,µ (cid:17) , where ϕ µ ,µ is the angle between the two lattice points µ , µ . On integrating over thetorus (4.6), all summands such that µ + µ = 0 vanish (see also (6.1) to follow). As ϕ µ , − µ = π , the arithmetic cancellation phenomenon is tantamount to cos ( ϕ/ 2) vanishingat π , similarly to the two-dimensional problem.6. The leading term of the variance: proof of Lemma 5.10 Preparatory results. Recall the expression of the covariance function (1.8) and itsderivatives (4.7) and (4.8); also recall the notation of Definition 1.3 for the set of latticepoint correlations. Lemma 6.1. We have the following estimates, where ∼ ρ means up to an error O (cid:18) N + |X (4) |N (cid:19) and ∼ σ means up to an error O (cid:18) m / − o (1) · N + |X (4) |N (cid:19) : (1) Z T r ( x ) dx = 1 N ; Z T r ( x ) dx ∼ ρ N . (2) E Z T ( DD t )( x ) dx = 1 N ;1 E Z T ( DD t ) ( x ) dx ∼ ρ · N . ANDOM WAVES ON T (3) E Z T ( r DD t )( x ) dx ∼ ρ N . (4) E Z T tr ( H ( x )) dx = 1 N ;1 E Z T ( r tr ( H ))( x ) dx ∼ ρ · N . (5) E Z T tr ( H ( x )) dx ∼ σ · N ;1 E Z T tr ( H ( x )) dx ∼ σ · N . (6) E Z T ( DD t tr ( H ))( x ) dx ∼ ρ N . (7) E Z T ( rDHD t )( x ) dx ∼ ρ − · N . (8) E Z T ( DH D t )( x ) dx ∼ ρ · N . (9) E Z T ( DD t ) ( x ) dx ≪ |C (6) |N . (10) E Z T ( r DD t )( x ) dx ≪ |C (6) |N . (11) E Z T tr ( H ( x )) dx ≪ |C (6) |N . (12) E Z T ( rDD t DHD t )( x ) dx ≪ |C (6) |N . Proof. The various estimates are obtained with the following common strategy. Firstly,one rewrites the integrand using the expressions (1.8), (4.7) and (4.8) for the covariancefunction and its (first and second order) derivatives. Next, the integral over the torus intaken, invoking the orthogonality relations of the exponentials:(6.1) Z T e ( µ · x ) dx = ( µ = 00 µ = 0 . We are thus left with a summation over the set of ℓ -correlations C ( ℓ ), where ℓ = 2 , k = 6 we need onlyan upper bound. The most delicate computations are for 4-correlations, when we splitthe summation exploiting the structure of C (4) (see (2.3)). This leads to computing k -th moments (for k = 1 , , , or 4) of the normalised inner product of two lattice points,applying Lemma 2.5.We now present the details of the proof for some of the estimates of the present lemma;the remaining computations apply the same ideas (outlined above), and we will omit themhere. We begin with part 1, first statement, which is an immediate consequence of (1.16): Z T r ( x ) dx = R (2) = |C (2) |N = 1 N . The second statement of part 1 follows from the structure of C (4) (2.3): Z T r ( x ) dx = R (4) = |C (4) |N = 3 N + O (cid:18) N + |X (4) |N (cid:19) . Let us show part 2 of the present lemma, starting with the first statement. By (4.7), wemay rewrite the integrand as(6.2) DD t = tr ( D t D ) = − π N · X µ ,µ e ([ µ + µ ] · x )( µ · µ ) . We take the integral over T , bearing in mind (6.1), and compute the resulting summationover the set of 2-correlations, using (1.16): Z T ( DD t )( x ) dx = − π N · X C (2) ( µ · µ ) = − π N · X µ ( − µ · µ ) = E N , as claimed. For the second statement of part 2, we begin by squaring (6.2):( DD t ) = (4 π ) N · X E m e ([ µ + µ + µ + µ ] · x ) · ( µ · µ ) · ( µ · µ ) . By (6.1),(6.3) Z T ( DD t ) dx = (4 π ) N · X C (4) ( µ · µ ) · ( µ · µ ) . To treat the resulting summation over 4-correlations, we split it with (2.3). The contribu-tion over diagonal and non-degenerate quadruples is bounded via Cauchy-Schwartz: X D ′′ ˙ ∪X (4) ( µ · µ ) · ( µ · µ ) ≤ X D ′′ ˙ ∪X (4) ( √ m ) ≪ m · ( N + |X (4) | ) . There are three more contributions to the summation in (6.3), that arise from symmetric(and non-diagonal) 4-correlations; we directly compute the first of these contributions: X µ = − µ µ = − µ ( µ · µ ) · ( µ · µ ) = X µ ,µ ( − µ · µ ) · ( − µ · µ ) = m N . For the remaining two summations, we invoke Lemma 2.5 with k = 2: X µ = − µ µ = − µ ( µ · µ ) · ( µ · µ ) = X µ = − µ µ = − µ ( µ · µ ) · ( µ · µ ) = X µ X µ ( µ · µ ) = m N . The various contributions yield(6.4) X C (4) ( µ · µ ) · ( µ · µ ) = 53 · m N + O ( m N ) + O ( m · |X (4) | ) . ANDOM WAVES ON T Inserting (6.4) into (6.3) we arrive at the second statement of part 2 of the present lemma.The proof of part 3 is very similar to that of part 2, second statement, except Lemma 2.5is applied with k = 1.To prove part 4, first statement, recall (4.8) and (6.1) to directly compute Z T tr ( H ( x )) dx = (4 π ) N · X C (2) tr ( µ t µ µ t µ ) = (4 π ) N · X µ ( µ · µ ) = E N . For part 4, second statement, (1.8), (4.8) and (6.1) imply Z T ( r tr ( H ))( x ) dx = (4 π ) N · X C (4) tr ( µ t µ µ t µ ) = (4 π ) N · X C (4) ( µ · µ ) ;one now splits the sum and proceeds as in the proof of part 2.Let us prove part 5 of the present lemma, first statement. By (4.8) and (6.1), we have Z T tr ( H ( x )) dx = (4 π ) N X C (4) tr ( µ t µ µ t µ µ t µ µ t µ )= (4 π ) N " X µ ,µ tr ( µ t µ µ t µ µ t µ µ t µ ) + X µ ,µ tr ( µ t µ µ t µ µ t µ µ t µ )+ X µ ,µ tr ( µ t µ µ t µ µ t µ µ t µ ) + E · O (cid:18) N + |X (4) |N (cid:19) = (4 π ) N " X µ ,µ m ( µ · µ ) + X µ ,µ ( µ · µ ) + X µ ,µ m ( µ · µ ) + E · O (cid:18) N + |X (4) |N (cid:19) . One computes the three summations on the RHS of the latter expression via Lemma 2.5,with k = 2 , Z T tr ( H ( x )) dx = E N (cid:20) 13 + 15 + 13 + O (cid:18) m / − o (1) (cid:19) + O (cid:18) |X (4) |N (cid:19)(cid:21) , where we note the error term E / N is negligible by (2.1). The second statement of part5, and parts 6, 7 and 8 of the present lemma are all derived in a similar fashion, and wewill omit these proofs here.Let us prove part 12 of the present lemma, parts 9, 10 and 11 being similar. By (1.8),(4.7), (4.8) and (6.1), Z T ( rDD t DHD t )( x ) dx = − (4 π ) N X C (6) ( µ · µ ) · ( µ · µ ) · ( µ · µ ) ≪ E N · |C (6) | (for summations over 6-correlations, an upper bound via the Cauchy-Schwartz inequalityis sufficient for our purposes). (cid:3) Proof of Lemma 5.10. Proof of Lemma 5.10. To prove part 1, recall Lemma 5.5 (uniform boundedness of X ) andwrite Z T trX ( x ) dx = Z T \ S trX ( x ) dx + O ( meas S ) . Recall the expression of X (4.11); one uses the approximation (5.4) on T \ S , and Propo-sition 5.6 to bound the contribution of the singular set: Z T trX ( x ) dx = − E (cid:18)Z T DD t dx + Z T r DD t dx (cid:19) + O (cid:18) E Z T r DD t dx (cid:19) + O (cid:18) |C (6) |N (cid:19) . To compute the three integrals on the RHS of the latter expression, apply Lemma 6.1,parts 2, 3 and 10. Here and elsewhere the error term 1 / N (arising from several of theestimates of Lemma 6.1) is negligible compared to |C (6) | / N . Part 2 of the present lemmais derived in a similar way.Let us show part 3 of the present lemma, parts 4, 7, 8 and 9 being similar. By Lemma5.5, (5.4) and Proposition 5.6, Z T tr ( XY )( x ) dx = − E (cid:20)Z T tr ( DH D t ) dx + O Z T rDD t DHD t dx (cid:21) + O (cid:18) |C (6) |N (cid:19) . Now Lemma 6.1, parts 8 and 12, yields Z T tr ( XY )( x ) dx = 1 N (cid:20) − O (cid:18) |X (4) |N + |C (6) |N (cid:19)(cid:21) , which concludes the proof of part 3 of the present lemma.We now prove part 5, part 6 being similar. By Lemma 5.5 and Proposition 5.6, we have Z T tr ( Y )( x ) dx = 81 E Z T tr ( H ) dx + O (cid:18) |C (6) |N (cid:19) . Now Lemma 6.1, part 5 yields Z T tr ( Y )( x ) dx = 3515 · N + O (cid:18) m / − o (1) · N + |X (4) |N + |C (6) |N (cid:19) , hence the claim of part 5 of the present lemma. Lastly, we show part 10, part 11 beingsimilar. By Lemma 5.5 and Proposition 5.6, we have Z T tr ( X )( x ) dx = − E Z T ( DD t ) ( x ) dx + O (cid:18) |C (6) |N (cid:19) ≪ |C (6) |N , where in the last step we applied Lemma 6.1, part 9. (cid:3) ANDOM WAVES ON T Appendix A. Berry’s method: proof of Lemma 5.8 In this section, we establish Lemma 5.8: following [2] and [21], we regard E [ k w kk w k ](recall the notation in the statement of the lemma) as a function of the entries of thematrices X (4.11) and Y (4.12), and perform a Taylor expansion about X = Y = 0. Weemploy Berry’s method as opposed to computing the Taylor polynomial by brute force,which would result in a longer computation. Lemma A.1. Let: w , w ∈ R , ( w , w ) ∼ N (0 , Ω) with Ω = I + (cid:18) X YY X (cid:19) . Then (A.1) E [ k w kk w k ] = 12 π Z Z R ( f (0 , − f ( t, − f (0 , s ) + f ( t, s )) dtds ( ts ) with (A.2) f ( t, s ) = 1 p det ( I + J ( t, s )) , where (A.3) I + J = (cid:18) (1 + t ) I + tX √ tsY √ tsY (1 + s ) I + sX (cid:19) is a perturbation of the identity matrix I .Proof. We begin with [2, (24)]: k w i k = 1 √ π Z R + (cid:16) − e − t k w i k / (cid:17) dtt , i = 1 , . The LHS of (A.1) becomes E [ k w kk w k ] = 12 π Z Z R E h (1 − e − t k w i k / )(1 − e − s k w i k / ) i dtds ( ts ) = 12 π Z Z R (cid:16) E [1] + E h − e − t k w k / i + E h − e − s k w k / i + E h e − ( t k w k + s k w k ) / i(cid:17) dtds ( ts ) . Setting f ( t, s ) = f X,Y ( t, s ) := E [exp( − ( t k w k + s k w k ) / , it remains to show that f ( t, s ) may be rewritten as in (A.2). By definition of expectation, f ( t, s ) = Z R × R p (2 π ) · √ det Ω ·· exp( − ( t k w k + s k w k ) / · exp (cid:18) − (cid:0) w w (cid:1) Ω − (cid:18) w w (cid:19)(cid:19) dw dw = 1 p (2 π ) det Ω Z R × R exp (cid:18) − (cid:0) w w (cid:1) (cid:20)(cid:18) tI sI (cid:19) + Ω − (cid:21) (cid:18) w w (cid:19)(cid:19) dw dw = 1 √ det Ω · s det (cid:20)(cid:18) tI sI (cid:19) + Ω − (cid:21) − = 1 p det ( I + J ( t, s )) , with I + J ( t, s ) as in (A.3). (cid:3) We will need the following expansions for a square matrix P , as P → I + P ) − = I − P + O ( P )and(A.5) (det( I + P )) − = 1 − trP + 14 tr ( P ) + 18 ( trP ) + O (cid:18) max i,j ( P ) ij (cid:19) . Proof of Lemma 5.8. By Lemma A.1, we get the expression (A.1), and require the Taylorexpansion of f X,Y ( t, s ) = det( I + J ) − around X = Y = 0. By (A.3) and the formula forthe determinant of a block matrix:det( I + J ) = det((1 + t ) I + tX ) · det[(1 + s ) I + sX − √ tsY ((1 + t ) I + tX ) − √ tsY ] , hence(A.6) f X,Y ( t, s ) = det( I + J ) − = det((1 + t ) I + tX ) − · det[(1 + s ) I + sX − tsY ((1 + t ) I + tX ) − Y ] − . Bearing in mind that I and X are 3 × t ) I + tX ) = (1 + t ) · det (cid:18) I + t t X (cid:19) and thus rewrite the first factor on the RHS of (A.6) as(A.7) 1(1 + t ) det (cid:18) I + t t X (cid:19) − . Sincedet[(1 + s ) I + sX − tsY ((1 + t ) I + tX ) − Y ] } = det ( (1 + s ) " I + s s X − ts (1 + t )(1 + s ) Y (cid:18) I + t t X (cid:19) − Y , ANDOM WAVES ON T the second factor on the RHS of (A.6) equals1(1 + s ) det " I + s s X − ts (1 + t )(1 + s ) Y (cid:18) I + t t X (cid:19) − Y − ;applying (A.4) with P = t t X , we further rewrite the second factor on the RHS of (A.6)as:(A.8) 1(1 + s ) · det (cid:20) I + s s X − ts (1 + t )(1 + s ) Y + t s (1 + t ) (1 + s ) Y XY + O ( Y X Y ) (cid:21) − . Next, we apply (A.5) to both (A.7) and (A.8), with P = t t X and P = s s X − ts (1 + t )(1 + s ) Y + t s (1 + t ) (1 + s ) Y XY + O ( Y X Y )respectively. The above computations on the two factors of (A.6) yield(A.9) f X,Y ( t, s ) = 1(1 + t ) (1 + s ) · (cid:20) − (cid:18) t t + s s (cid:19) tr ( X )+ 12 · ts (1 + t )(1 + s ) tr ( Y ) − · ts (1 + t )(1 + s ) (cid:18) t t + s s (cid:19) tr ( XY )+ (cid:18) t (1 + t ) + 38 s (1 + s ) + 14 ts (1 + t )(1 + s ) (cid:19) tr ( X )+ 14 · t s (1 + t ) (1 + s ) tr ( Y ) + 18 t s (1 + t ) (1 + s ) tr ( Y ) − · ts (1 + t )(1 + s ) (cid:18) t t + s s (cid:19) tr ( X ) tr ( Y ) (cid:21) + O ( tr ( X ) + tr ( Y )) , where we have used the assumption rank ( X ) = 1 so that tr ( X ) = tr ( X ). The integrandin (A.1) is h X,Y ( t, s ) := f (0 , − f ( t, − f (0 , s ) + f ( t, s );to compute the Taylor polynomial for h around X = Y = 0, first note that, except for theterms in 1 , tr ( X ) , tr ( X ), the various terms in the expansion of h are the same as those inthe expansion of f : this is because each term in (A.9), save for those in 1 , tr ( X ) , tr ( X ),vanishes when t = 0 or s = 0. Next, we directly compute the terms in 1 , tr ( X ) , tr ( X ) ofthe expansion of h to be respectively (cid:18) − t ) / (cid:19) · (cid:18) − s ) / (cid:19) , (cid:20) t (1 + t ) / (cid:18) − s ) / (cid:19) + s (1 + s ) / (cid:18) − t ) / (cid:19)(cid:21) , and − t (1 + t ) / (cid:18) − s ) / (cid:19) − s (1 + s ) / (cid:18) − t ) / (cid:19) + 14 ts (1 + t ) / (1 + s ) / . To perform the integration(A.10) E [ k w kk w k ] = 12 π Z Z R h ( t, s ) dtds ( ts ) term-wise, we need to improve the error term O ( tr ( X ) + tr ( Y )) in the expansion of h sothat it depends on t and s , as Z Z R dtds ( ts ) is divergent at the origin. To do this, we note that, for all X and Y , h vanishes when t = 0or s = 0; hence, for t, s ≥ 0, we may write h X,Y ( t, s ) = O X,Y ( ts ) . We may then improve the error term in the expansion of h to O (cid:0) min( t, · min( s, · ( tr ( X ) + tr ( Y )) (cid:1) . Therefore, h X,Y ( t, s ) = (cid:18) − t ) / (cid:19) · (cid:18) − s ) / (cid:19) (A.11) + 12 (cid:20) t (1 + t ) / (cid:18) − s ) / (cid:19) + s (1 + s ) / (cid:18) − t ) / (cid:19)(cid:21) tr ( X )+ 12 t (1 + t ) / s (1 + s ) / tr ( Y ) − (cid:18) t (1 + t ) / s (1 + s ) / + t (1 + t ) / s (1 + s ) / (cid:19) tr ( XY ) + (cid:20) − t (1 + t ) / (cid:18) − s ) / (cid:19) − s (1 + s ) / (cid:18) − t ) / (cid:19) + 14 ts (1 + t ) / (1 + s ) / (cid:21) tr ( X )+ 14 t s (1 + t ) / (1 + s ) / tr ( Y ) + 18 t s (1 + t ) / (1 + s ) / tr ( Y ) − (cid:18) ts (1 + t ) / (1 + s ) / + t s (1 + t ) / (1 + s ) / (cid:19) tr ( X ) tr ( Y )+ O (cid:0) min( t, · min( s, · ( tr ( X ) + tr ( Y )) (cid:1) . ANDOM WAVES ON T Lastly, we insert (A.11) into (A.1), and compute the integrals Z ∞ (cid:18) − t ) / (cid:19) dtt / = 4 , Z ∞ dt (1 + t ) / √ t = 43 , Z ∞ √ t dt (1 + t ) / = 415 , Z ∞ min( t, dtt / = 4 , to obtain the statement of the present lemma. (cid:3) Appendix B. The incidence bound In this section we briefly explain how one can modify the proof of [15, Theorem 1.2] toobtain Theorem 3.3. As in [15] the result will follow from a more general statement. Firstwe recall the notions of degree and dimension of a real algebraic variety since they are keyfeatures of the proof of Theorem 3.3. The degree and dimension of a real variety. Let V ⊂ R d be a real algebraic variety.Letting I ( V ) denote the ideal of polynomials vanishing on V , we define dim( V ) to be theKrull dimension of the quotient ring R d /I ( V ).Let V ∗ denote the complexification of V (i.e. the Zariski closure of V , viewed as a subsetof C d ). As discussed in [15, Section 4.1], the notion of degree is well-defined for complexvarieties, so we may take deg( V ) to be the degree of the complex variety V ∗ . One has thefollowing relationship between the complexity and degree of a complex variety W (see [30,Lemmas 4.2 and 4.3]): • Any irreducible variety W ⊂ C d of degree at most D can be expressed as the zeroset (cid:8) z ∈ C d : g i ( z ) = 0 ∀ i ≤ r (cid:9) where each polynomial g i ∈ C [ z , ..., z d ] has degreeat most D and r = O d,D (1). • Suppose W = (cid:8) z ∈ C d : g i ( z ) = 0 ∀ i ≤ r (cid:9) is cut out by polynomials of degree atmost D . Then W can be decomposed into O r,d,D (1) irreducible varieties each havingdegree O r,d,D (1).Given a polynomial f we will denote by Z ( f ) its zero set. Theorem B.1 ([15, Theorem 6.4], quantitative in s, t ) . Let P ⊂ R d be a set of k points and V a collection of n algebraic varieties of bounded degree in R d . Suppose the incidence graphof P × V is K s,t -free and that P is contained in some irreducible variety X of dimension e and degree D . Lastly, suppose that no variety V ∈ V contains X . Then for any ε > there are positive constants c ( e ) = c ( ε, d, D, e ) and c ( e ) = c ( ε, d, D, e ) so that (B.1) I ( P , V ) ≤ st (cid:18) c ( e ) k s ( e − es − + ε · n e ( s − es − + c ( e )( k + n ) (cid:19) . Observe that Theorem 3.3 follows immediately, taking X = R and d = 3. The proofof Theorem B.1 is carried out exactly as in [15, Theorems 4.3 and 6.4] except that onerequires a quantitative version of the classical K˝ov´ari-S´os-Tur´an Theorem. Here we are assuming there is a constant C such that each V ∈ V can be written as an algebraic set (cid:8) x ∈ R d : p i ( x ) = 0 ∀ i ≤ r (cid:9) where r ≤ C and each p i ∈ R [ x , ..., x d ] has degree at most C . Theorem B.2 ([18]) . Let G = ( V , V , E ) be a bipartite graph with | V | = k , | V | = n andsuppose that G does not contain a copy of K s,t . Then | E | ≤ ( t − /s kn − /s + ( s − n . The final ingredient is a result from [30]. Theorem B.3. [30, Theorem A.2] Let V ⊂ R d be an irreducible variety of dimension h and degree D and let f ∈ R [ x , ..., x d ] be a polynomial of degree M ≥ . Then V \ Z ( f ) has at most O D ( M h ) connected components. Sketch of the proof of Theorem B.1. The inequality (B.1) is established by meansof a two-step induction argument on the quantities e and k + n . Base cases. When e = 0, the irreducible variety X consists of a single point andhence (B.1) is easily satisfied. On the other hand, when k + n is small (regardless of thedimension e ) we can choose c ( e ) , c ( e ) to be sufficiently large, thereby satisfying (B.1).Let r be a large number to be determined later. By the polynomial partitioning method(in the modified form [15, Theorem 4.2]) there exists a polynomial f ∈ R [ x , ..., x d ] \ I ( X )of degree at most O d,D ( r /e ) so that each connected component of R d \ Z ( f ) contains atmost k/r points of P .Defining X f := X ∩ Z ( f ), we may now split the set of incidences I ⊂ P × V into threeparts (recall that X is not contained in any V ∈ V ):- I is given by those ( p, V ) ∈ I for which p ∈ Z ( f ) and V properly intersects eachirreducible component of X f that contains p .- I is given by those ( p, V ) ∈ I for which p lives in some irreducible component of X f that contains V .- I consists of all remaining incidences, i.e. those ( p, V ) ∈ I with p / ∈ X f .Throughout the remainder of the argument we may assume that n ≤ k s . Indeed, when k s < n we get that n − /s < k − and hence by Theorem B.2 I ( P , V ) ≤ t /s kn − /s + sn ≤ ( t + s ) n, yielding the desired estimate (B.1). We record the inequality(B.2) n = n e − es − n e ( s − es − ≤ k s ( e − es − n e ( s − es − . Let k f = |P ∩ X f | . Since X is irreducible of dimension e and f / ∈ I ( X ) we have thatdim( X f ) =: e ′ ≤ e − 1. By [27, Theorem 2] (note that the results in [27] are described interms of both complexity and degree) and the discussion preceding Theorem B.1 we candecompose X f into l irreducible varieties, each of degree at most l and dimension at most e ′ . Here l depends on the quantities d, D, e, r . A bound for I . Applying the induction hypothesis it follows that(B.3) |I | ≤ stl (cid:18) c ( e − k s ( e − e − s − + ε n ( e − s − e − s − + c ( e − k f + n ) (cid:19) . Invoking the estimate [15, (12)] one has(B.4) k s ( e − e − s − + ε n ( e − s − e − s − ≤ k s ( e − es − + ε n e ( s − es − . ANDOM WAVES ON T Choosing c ( e ) ≥ lc ( e − , c ( e ) ≥ lc ( e − 1) and inserting (B.4) into (B.3) we get(B.5) |I | ≤ st (cid:18) c ( e ) k s ( e − es − + ε n e ( s − es − + c ( e )( k f + n ) (cid:19) . A bound for I . Since the incidence graph of P × V is K s,t -free, each irreduciblecomponent of X f either contains at most s − P or is contained in at most t − V . As a result |I | ≤ lsn + tk f . Since we have already assumed c ( e ) ≥ l , it follows that(B.6) |I | ≤ st c ( e )( k f + n ) . A bound for I . Let k ′ f := k − k f . By Theorem B.3 the set R d \ Z ( f ) can bepartitioned into connected components (or cells ) Ω , ..., Ω J with J = O D,d ( r ). Given any( p, V ) ∈ I we first note that the variety V must properly intersect one of the cells Ω j .Among the O (1) polynomials defining V there must be at least one, say g , for which Z ( g ) does not fully contain X . Since X is irreducible we have that dim( Z ( g ) ∩ X ) ≤ e − Z ( g ) ∩ X intersects at most O D,d ( r ( e − /e ) cells. As a consequence V ∩ X intersects at most O D,d ( r ( e − /e ) cells. Introduce for each j = 1 , ..., J V ( j ) X = { V ∩ X | V ∈ V and V intersects Ω j } , P ( j ) = P ∩ Ω j . Noting that |I | = J X j =1 I (cid:16) P ( j ) , V ( j ) X (cid:17) , the argument proceeds in precisely the same manner as [15, Theorem 4.3] and one gets J X j =1 I (cid:16) P ( j ) , V ( j ) X (cid:17) ≤ st (cid:18) c ( e ) κ r − ε k s ( e − es − + ε n e ( s − es − + c ( e )( k ′ f + κ r ( e − /e n ) (cid:19) , where κ , κ depend only on the parameters D, d, e . Taking κ c ( e ) r ( e − /e ≤ c ( e ) / r to be sufficiently large with respect to ε and κ , it follows from (B.2) that(B.7) |I | = J X j =1 I (cid:16) P ( j ) , V ( j ) X (cid:17) ≤ st (cid:18) c ( e )3 k s ( e − es − + ε n e ( s − es − + c ( e ) k ′ f (cid:19) . It remains to collect the three estimates (B.5), (B.6) and (B.7). 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