The n-point correlation of quadratic forms

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by  Oliver Sargent

THE N-POINT CORRELATION OF QUADRATIC FORMS.
OLIVERSARGENT
Abstract. In this paper we investigate the distribution of the set of values of a quadratic form Q, at integral
points. In particular weareinterested inthe n-pointcorrelations of the this set. Theasymptotic behaviour of the
countingfunctionthatcountsthenumberofn-tuplesofintegralpoints(v1,...,vn),withboundednorm,suchthat
the n−1 differences Q(v1)−Q(v2),...Q(vn−1)−Q(vn), liein prescribed intervals is obtained. The results are
4 valid provided that the quadratic form has rank at least 5, is not a multiple of a rational form and n is at most
1 the rankof the quadratic form. For certainquadratic formssatisfyingDiophantine conditions weobtain aratefor
0 the limit. The proofs are based on those in the recent preprint ([GM13]) of F. Götze and G. Margulis, in which
2 theyprovean‘effective’versionoftheOppenheimConjecture. Inparticular,theproofsrelyonFourieranalysisand
estimates forcertainthetaseries.
n
a
J

7
1. Introduction.

]
T 1.1. Background. Let Q:Rd R be a quadratic form. It is interesting to understand the distribution of the set
N Q Zd inside R. If there exists→a multiple of Q such that its coefficients are all rational,then Q is called a rational
. form. InthecasewhenQisrational,Q Zd isadiscretesetinsideR. WhenQisnotarationalform,Qiscalledan
h (cid:0) (cid:1)
t irrational form. For irrational forms, the first milestone in understanding the distribution of the set Q Zd inside
a R was reached by G. Margulis in [Mar8(cid:0)9] w(cid:1)hen he provided a proof (shortly afterwards, refined by S.G. Dani and
m (cid:0) (cid:1)
G. Margulis in [DM89]) of the ‘Oppenheim Conjecture’. The modern statement of which is as follows: if d 3 and
[ Q is a nondegenerate, irrationaland indefinite form, then Q Zd is dense in R. ≥

1 Once it is knownthat Q Zd is dense in R, one canask for a moreprecise answerto the question ofhowQ Zd
(cid:0) (cid:1)
v is distributed in R. Let I R be any interval and E (I,T)= v Rd :Q(v) I, v T , then one can ask for
0 ⊂(cid:0) (cid:1) Q ∈ ∈ k k≤ (cid:0) (cid:1)
an asymptotic formula for the size of the set Zd E (I,T). The first results in this direction were obtained by
4 ∩ Q (cid:8) (cid:9)
3 Dani-Margulis in [DM93] who proved, if d 3 and Q is a nondegenerate, irrational and indefinite form and I is
≥
1 any interval, then
1. Zd E (I,T)
Q
lim ∩ 1.
0
4 T→∞(cid:12)Vol(EQ(I,T))(cid:12) ≥
(cid:12) (cid:12)
1 The situation regarding the upper bounds is more delicate and this was dealt with by the work of A. Eskin, G.
: Margulis and S. Mozes in [EMM98] who proved that if d 5 and Q is a nondegenerate, irrational and indefinite
v ≥
i form and I is any interval, then
X
Zd E (I,T)
r Q
(1.1) lim ∩ =1.
a
T→∞(cid:12)Vol(EQ(I,T))(cid:12)
(cid:12) (cid:12)
It should be noted that the actual results from [DM93] and [EMM98] are more general than stated above. The
situation regardingthe upper bounds for the case when d=3 or 4 is particularly interesting and is also considered
in [EMM98]. In the cases when Q has signature (2,1) or (2,2) no asymptotic formula of the form (1.1) is possible
for general quadratic forms, since in these cases there exist examples of quadratic forms for which (1.1) fails. In
[EMM05], quadratic forms of signature (2,2) satisfying a slightly modified version of (1.1) are characterised by
certain Diophantine conditions. The work of Eskin-Margulis-Mozes can be interpreted as providing conditions
which ensure the set Q Zd is equidistributed in R.
One can ask still finer questions about the distribution of the set Q Zd . Let e ,...,e be the standard basis
(cid:0) (cid:1) 1 nd
ofRnd,letp ...,p denote the projectionsonto e ,...,e ,..., e ,...,e respectively. Forv Rnd and
1 n h 1 di (n−1)(cid:0)d+1(cid:1) nd ∈
1 i n, we will write v =p (v). Let I ,...,I be intervals and
i i 1 n 1
≤ ≤ − (cid:10) (cid:11)
Pn(I ,...,I ,T)= v Rnd :Q(v ) Q(v ) I ,...,Q(v ) Q(v ) I , v T .
Q 1 n−1 ∈ 1 − 2 ∈ 1 n−1 − n ∈ n−1 k k≤
In order to understand the n-po(cid:8)int correlations of the set Q Zd , one asks for an asymptotic formula(cid:9)for the size
of the set Znd Pn(I ,...,I ,T). A more general problem about the distribution of values at integral points
∩ Q 1 n−1 (cid:0) (cid:1)
of systems of quadratic forms was studied by W. Müller in [Mül08]. In particular, it follows from Theorem 1 of
1
THE N-POINT CORRELATION OF QUADRATIC FORMS. 2
[Mül08] that if n 2, 4n d and Q is a nondegenerate and irrational form, then
≥ ≤
Znd Pn(I ,...,I ,T)
(1.2) lim ∩ Q 1 n−1 =1.
T→∞(cid:12)(cid:12)Vol PQn(I1,...,In−1,T) (cid:12)(cid:12)
(cid:16) (cid:17)
Whenn=2andd 3,itiseasytoseethat(1.2)followsfromthemainTheoremof[EMM98]. Forpositivedefinite
≥
forms the n-point correlation problem was also studied by Müller. In [Mül11], Müller obtains the following result:
if d 4 and Q is a nondegenerate, irrational and positive definite form, then (1.2) holds for every n. In [Mül11]
≥
Müllerformulatesthe probleminslightlydifferentlanguage,butitiseasilyseentobe equivalenttotheformstated
here up to a change of variables and modifications of the norms involved. The main result of this paper extends
the results of Müller to a larger range of n for indefinite forms.
1.2. Statement of results. Using the notation from the previous subsection we can now state the main results.
Theorem 1.1. Suppose that Q is not a multiple of a rational form and d 5 and 2 < n d. Then, for any
≥ ≤
intervals I ,...,I ,
1 n 1
−
Znd Pn(I ,...,I ,T)
lim ∩ Q 1 n−1 =1.
T (cid:12)Vol Pn(I ,...,I ,T) (cid:12)
→∞(cid:12) Q 1 n−1 (cid:12)
(cid:16) (cid:17)
Moreover, thereexistsapositive constantC , depending onlyon Qandn, suchthat foranyintervalsI ,...,I ,
Q,n 1 n 1
−
lim Znd∩PQn(I1,...,In−1,T) =C n−1 I .
T→∞(cid:12)(cid:12) Tnd−2(n−1) (cid:12)(cid:12) Q,n iY=1| i|
For quadratic forms Q satisfying the following Diophantine condition it is possible to prove an effective version
of Theorem 1.1. Let Q also denote the symmetric d d matrix that is associated to the quadratic form Q. Let
0<κ<1 and A>0, say that Q is of type (κ,A) if fo×r every M Mat (Z) and q Z 0 we have
d
∈ ∈ \{ }
inf Mq−1 tQ Aq−1−κ.
t [1,2] − ≥
∈
(cid:13) (cid:13)
The sizeof κ depends onhow wellQ canbe app(cid:13)roximatedby(cid:13)a rationalmatrix,ifκ is closeto 1,then Q is insome
sense close to a rational matrix.
Theorem 1.2. Suppose that Q is of Diophantine type (κ,A) and d 5 and 2<n d. Let δ(κ)= 2(d−4)(1−κ) .
≥ ≤ (1+nd)(d+1+κ)
Then, for any intervals I ,...,I there exists T >0 and a constant C such that for all T T ,
1 n 1 0 0
− ≥
Znd Pn(I ,...,I ,T)
∩ Q 1 n−1 1 Clogn−1(T)T−δ(κ).
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Vol PQn(I1,...,In−1,T) (cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)≤
(cid:12) (cid:16) (cid:17) (cid:12)
Remark 1.3. The constant C(cid:12)appearing in Theorem 1.2 depen(cid:12)ds on Q,n,A and the intervals I ,...,I .
(cid:12) (cid:12) 1 n 1
−
Remark 1.4. In Theorem 1.2 we use . to denote the Euclidean norm. The exponent δ(κ) depends on the choice
kk
of norm and is possibly non optimal. If the maximum norm was chosen, the bounds in subsection 5.2 could be
improved, and δ(κ) could be replaced with δ′(κ)= (2d(+11−+κκ)) at the cost of a factor of lognd(T) appearing.
Remark 1.5. ThesecondpartsofTheorems1.1and1.2followeasilyfromthefirstpartsandthefollowingassertion:
For any intervals I ,...,I there exists a positive constant C , depending only on Q and n, such that
1 n 1 Q,n
−
n 1
1 −
lim Vol Pn(I ,...,I ,T) =C I .
T→∞Tnd−2(n−1) Q 1 n−1 Q,n i=1| i|
(cid:0) (cid:1) Y
This statement is proved in Corollary 5.5.
1.3. The Berry-Tabor Conjecture. For positive definite forms there is a similar problem about the n-point
correlationsofthe normalisedvaluesofQatintegralpoints. This problemisdiscussedin[Mül08]andisinteresting
because it is related to the so called Berry-Tabor Conjecture (see [BT77]). A special case of this Conjecture
states that the spacings of eigenvalues of the Laplacian on ‘generic’ multidimensional tori should have a Poisson
distribution. This problemhas been studied in [Sar97] by P. Sarnak,in [Van99] and[Van00] by J.VanderKamand
in [Mar02] by J. Marklof.
THE N-POINT CORRELATION OF QUADRATIC FORMS. 3
1.4. Outline of paper and summary of the methods. One can try to prove Theorems 1.1 and 1.2 by using
the theory of unipotent flows, in analogy to what was done in [EMM98]. The problem one encounters, is that the
subgroup of linear transformations of Rnd stabilising the quadratic forms Q(v ) Q(v ),...,Q(v ) Q(v ) is
1 2 n 1 n
SO(Q)n and this seems too small to obtain the required statements. If one ha−d access to a precis−e q−uantitative
equidistribution statement, in the formofan explicitratefor the limit in (1.1), one couldhope to proveresults like
Theorems 1.1 and 1.2. Unfortunately, since the the results of [EMM98] relied on the equidistribution of unipotent
flows, no good error term was available. However, recently, F. Götze and G. Margulis proved such a statement in
the preprint [GM13] (see also [GM10] for an older version). Their methods do not rely on the equidistribution of
unipotentflows. InsteadtheyuseFourieranalysistoreducetheproblemtooneofobtainingasymptoticestimatesfor
certaintheta series. Inordertoestimate these theta series,they use someofthe techniquesdevelopedin[EMM98],
inparticularthe cruxoftheirproofreliesonanondivergencestatementaboutaverageofthe translatesoforbitsof
certain compact subgroups in the space of lattices. One cannot apply the results of [GM13] directly, since in order
to do this one would needthe errorto be uniform acrossallintervals. However,the proofs of Theorems1.1 and1.2
are based on the methods of [GM13].
The object of interest is
R 1 = 1 (v) 1 (v)dv,
(cid:16) PQn(I1,...,In−1,T)(cid:17) v∈XZnd PQn(I1,...,In−1,T) −ˆRnd PQn(I1,...,In−1,T)
where, here and throughout the rest of the paper, for any set S, 1 stands for the characteristic function of the
S
set S. Theorems 1.1 and 1.2 follow from suitable bounds for R 1 . To obtain these bounds, the
function 1 is replaced with a smoothened version atPQnt(hIe1,.c..o,Isnt−1o,fT)‘smoothing errors’ which can be
PQn(I1,...,In−1,T) (cid:12) (cid:0) (cid:1)(cid:12)
estimated in terms of volumes of certain regions of Rnd. This(cid:12)is carried out in sub(cid:12)sections 2.1 and 5.1. The next
stepis to use Fourieranalysisto transferthe problemintothe ‘frequencydomain’. After takingFouriertransforms,
the smoothened version of R 1 can be estimated by considering an integral over the ‘frequency
PQn(I1,...,In−1,T)
domain’, ω Rn 1, of the difference between a theta series, θ(ω) and its corresponding smooth version, ϑ(ω) (see
(2.9)). This∈step−is carried(cid:12)(cid:12)ou(cid:0)t in subsection 2(cid:1).(cid:12)(cid:12)2. In order to estimate the integral, the domain of integration is
split into two parts, namely a neighbourhood of the origin and its complement.
Theintegralovertheregionboundedawayfromtheoriginisdealtwithbyconsideringtheintegralofθ(ω)andthe
integral of ϑ(ω) separately. The integral of θ(ω) contributes the main term in the bound for R 1
PQn(I1,...,In−1,T)
and it contains the arithmetic information about Q. This term is dealt with in subsection 3.3. The integral of
(cid:12) (cid:0) (cid:1)(cid:12)
ϑ(ω)onlycontributesalowerordertermto the boundfor R 1 andisdealtw(cid:12)ithinsubsection3.1(cid:12).
PQn(I1,...,In−1,T)
These two integrals can be estimated using techniques and results from [GM13]. The reason for this, is that θ(ω)
(cid:12) (cid:0) (cid:1)(cid:12)
and ϑ(ω) can be written as a product of n sums/integrals(cid:12)of the form studied(cid:12)in [GM13] (see (2.10) and (2.11)).
The integral,overthe neighbourhoodofthe origin,isdealtwithinsubsection3.2. This termcontributesalower
orderterm,butitgrowswithn,fasterthanthemainterm. Forn>dthistermdominatesthemainterm,explaining
why the assumption n d is needed. The reason for this, is that here we consider the difference, θ(ω) ϑ(ω).
Poissonsummationisus≤edtoconvertthisintoasumoverZnd 0 ,theproblemthatarisesisthatform Z−nd 0
\{ } ∈ \{ }
we can still have m = 0 for some 1 i d. Therefore, although it is still possible to take advantage of the fact
i
≤ ≤
that the sum obtained by Poisson summation can be written as a product of n sums, an additional argument is
needed to deal with the fact that 0 could be included in each of the sums in the product.
Finally in Section 4 all of the bounds are collected and Theorems 1.1 and 1.2 are proved. The bounds obtained
in Section 3 depend on the L1 norm of a certain function which depends on a smoothing parameter. In order to
prove Theorem 1.2 we need a precise estimates for this norm in terms of the smoothing parameter. This is carried
out in subsection 5.2.
2. Set up.
For the rest of the paper let n and d be natural numbers with n d. In the case when n = 2, there is
≤
only one quadratic form and the conclusions of Theorems 1.1 and 1.2 follow from the results of [EMM98] and
[GM13]. Hence, throughout the rest of the paper we suppose that n 3. For 1 i n 1, fix intervals I and
i
Q : Rd R a nondegenerate quadratic form, suppose that d 5 an≥d keep the≤nota≤tion−from the introduction.
→ ≥
Let Q = Q2, hence Q corresponds to a positive definite quadratic form. Let sp(Q) denote the spectrum of Q,
+ +
λ =min λ and λ =max λ. Since the problem is unaffected by rescaling Q, we may suppose
min λ sp(Q) max λ sp(Q)
that λ ∈n 1,|th|is supposition will∈be use|d|in the proof of Lemma 3.13. Define B(T)= v Rnd : v T
min
≥ − ∈ k k≤
and B (T) = v Rnd : v T , where we use . to denote the Euclidean norm and . to denote the
∞ ∈ k k∞ ≤ kk (cid:8)kk∞ (cid:9)
(cid:8) (cid:9)
THE N-POINT CORRELATION OF QUADRATIC FORMS. 4
maximum norm. Let
Pn(I ,...,I )= v Rnd :Q(v ) Q(v ) I ,...,Q(v ) Q(v ) I .
Q 1 n−1 ∈ 1 − 2 ∈ 1 n−1 − n ∈ n−1
Note that Pn(I ,...,I ,T) = P(cid:8)n(I ,...,I ) B(T). As is standard, we use the notation(cid:9) fˆto denote the
Q 1 n−1 Q 1 n−1 ∩
Fourier transform of a function f. We will also make heavy use the Vinogradov asymptotic notation f(s) g(s),
≪
which means that there exists some constant C > 0 such that f(s) Cg(s) for all values of s indicated. The
≤
constantC willbeindependentofthoseparametersbutwillusuallydependond,n,QandtheintervalsI ,...,I .
1 n 1
−
2.1. Smoothing. For any i N, let ki = ki(v)dv be a probability measure on Ri with the properties that it is
symmetric around 0, ki v ∈Ri : v 1 =1 and
∈ k k≤
(2.1) (cid:0)(cid:8) (cid:9)(cid:1) ki(v) exp c v
≤ − k k
for some positive constant c and all v ∈ Ri. F(cid:12)(cid:12)bor an(cid:12)(cid:12)y τ > 0(cid:16), letpkτi de(cid:17)note the rescaled measure such that kτi (A) =
ki τ 1A for any measurable set A. Note that (2.1) implies that
−
(2.(cid:0)2) (cid:1) ki (v) exp c τ v .
τ ≤ − k k
For an interval I =[a,b] and ǫ R, define I(cid:12)(cid:12)ǫc=[a(cid:12)(cid:12) ǫ,b+(cid:16)ǫ]. Fpor any(cid:17)τ >0, T >0 and v Rnd, let
∈ − ∈
w (v)=1 knd(v) and w (v)=w T 1v .
±τ B(1±τ)∗ τ ±τ,T ±τ −
For any ǫ>0 and ω Rn 1, let (cid:0) (cid:1)
−
∈
S±ǫ(ω)=1I1±ǫ×···×In±−ǫ1 ∗kǫΠ(ω),
where kΠ(ω)=k1( ω,e )...k1( ω,e ). For v Rnd, let
ǫ ǫ h 1i ǫ h n−1i ∈
SQ (v)=S (Q(v ) Q(v ),...,Q(v ) Q(v )).
±ǫ ±ǫ 1 − 2 n−1 − n
For a measurable function f on Rnd define
(2.3) R(f)= f(v) f(v)dv.
v∈XZnd −ˆRnd
Note thatR(f) is only welldefined ifboth the quantities onthe righthand side of(2.3)are finite. Let ν andν
T τ,T
be measures on Rnd and Rn 1 respectively, defined by
−
ˆRndfdνT =ˆRndf T−1v 1PQn(I1,...,In−1)(v)dv
(cid:0) (cid:1)
and
fdν = f(Q(v ) Q(v ),...,Q(v ) Q(v ))w (v)dv.
ˆRn−1 τ,T ˆRnd 1 − 2 n−1 − n ±τ,T
In the next two Lemmas we approximate R 1 1 by a smoothened version.
PQn(I1,...,In−1) B(T)
(cid:16) (cid:17)
Lemma 2.1. For all τ >0 and T >0,
R 1 1 max R 1 w + 1 1 dν .
Proof. Defi(cid:12)(cid:12)(cid:12)ne(cid:16)aPmQne(Ia1s,u..r.,eInµ−T1)oBn(TR)n(cid:17)d(cid:12)(cid:12)(cid:12),≤by±τ (cid:12)(cid:12)(cid:12) (cid:16) PQn(I1,...,In−1) ±τ,T(cid:17)(cid:12)(cid:12)(cid:12) ˆRnd(cid:0) B(1+2τ)− B(1−2τ)(cid:1) T
fdµ = f T 1v 1 (v).
ˆRnd T v∈XZnd (cid:0) − (cid:1) PQn(I1,...,In−1)
Define functions on Rnd by f =1 and f =1 . Note that
B(1) τ B(1 τ)
± ±
(2.4) R 1 1 = fd(µ ν ) .
From the definition of kτnd it follo(cid:12)(cid:12)(cid:12)ws(cid:16)thPaQnt(I1,...,In−1) B(T)(cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)ˆRnd T − T (cid:12)(cid:12)(cid:12)(cid:12)
f f knd f f knd f .
−2τ ≤ −τ ∗ τ ≤ ≤ +τ ∗ τ ≤ +2τ
THE N-POINT CORRELATION OF QUADRATIC FORMS. 5
Since all the functions in the previous inequality are bounded and have compact support and the measureµ ν
T T
−
is locally finite, by integrating with respect to µ ν we obtain
T T
−
fd(µ ν ) f kndd(µ ν )+ f knd f dν
ˆRnd T − T ≤ˆRnd +τ ∗ τ T − T ˆRnd +τ ∗ τ − T
(cid:0) (cid:1)
f kndd(µ ν )+ (f f )dν .
≤ˆRnd +τ ∗ τ T − T ˆRnd +2τ − −2τ T
Similarly
fd(µ ν ) f kndd(µ ν) (f f )dν .
ˆRnd T − T ≥ˆRnd −τ ∗ τ − −ˆRnd +2τ − −2τ T
Inview of(2.4)andthe definitionofw the conclusionofthe Lemmafollowsfromthe previoustwoinequalities.
τ,T
± (cid:3)
Lemma 2.2. For all ǫ>0, τ >0 and T >0,
(cid:12)R(cid:16)1PQn(I1,...,In−1)w±τ,T(cid:17)(cid:12)≤m±aǫx(cid:12)R(cid:16)S±Qǫw±τ,T(cid:17)(cid:12)+ˆRn−1(cid:16)1I12ǫ×···×In2ǫ−1 −1I1−2ǫ×···×In−−2ǫ1(cid:17)dντ,T.
Proof. De(cid:12)fine a measure µ on R(cid:12)n 1, by(cid:12) (cid:12)
(cid:12) τ,T (cid:12) − (cid:12) (cid:12)
fdµ = f(Q(v ) Q(v ),...,Q(v ) Q(v ))w (v).
ˆRn−1 τ,T v∈XZnd 1 − 2 n−1 − n ±τ,T
Define functions on Rnd by f =1I1×···×In−1 and f±ǫ =1I1±ǫ×···×In±−ǫ1. Note that
(2.5) R 1 w = fd(µ ν ) .
From the definition of kǫΠ it fol(cid:12)(cid:12)(cid:12)low(cid:16)s tPhQna(tI1,...,In−1) ±τ,T(cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)ˆRn−1 τ,T − τ,T (cid:12)(cid:12)(cid:12)(cid:12)
f (ω) f kΠ(ω) f(ω) f kΠ(ω) f .
−2ǫ ≤ −ǫ∗ ǫ ≤ ≤ +ǫ∗ ǫ ≤ +2ǫ
Sinceallthefunctionsinthepreviousinequalityareboundedandhavecompactsupportandthemeasureµ ν
τ,T τ,T
−
is locally finite, by integrating with respect to µ ν as in the proof of Lemma 2.1 we obtain
τ,T τ,T
−
f kΠd(µ ν ) (f f )dν
ˆRn−1 −ǫ∗ ǫ τ,T − τ,T −ˆRn−1 +2ǫ− −2ǫ τ,T
fd(µ ν )
≤ˆRn−1 τ,T − τ,T
f kΠd(µ ν )+ (f f )dν .
≤ˆRn−1 +ǫ∗ ǫ τ,T − τ,T ˆRn−1 +2ǫ− −2ǫ τ,T
In view of (2.5) and the definition of SQ the conclusion of the Lemma follows from the previous inequality. (cid:3)
ǫ
±
In subsection 5.1 we will obtain bounds for the smoothing errors
ˆRnd 1B(1+2τ)−1B(1−2τ) dνT and ˆRn−1 1I12ǫ×···×In2ǫ−1 −1I1−2ǫ×···×In−−2ǫ1 dντ,T
that arise from Lemm(cid:0)as 2.1 and 2.2. (cid:1) (cid:16) (cid:17)
2.2. Fourier transforms. Toobtainboundsfor R SQ w thestrategyof[GM13]willbeused,inparticular
ǫ τ,T
± ±
we proceed via the Fourier transform. Numerou(cid:12)s t(cid:16)ext books(cid:17)o(cid:12)n Fourier analysis are available, for instance see
[Gra08]. Let S Rd denote the space of Schwartz(cid:12)(cid:12)functions on R(cid:12)(cid:12)d (see Section 2.2 of [Gra08]) and let C0∞ Rd be
smooth functions with compact support on Rd. We note that Rd Rd L1 Rd and that Rd is in-
(cid:0) (cid:1) C0∞ ⊂ S ⊂ S (cid:0) (cid:1)
variantundertheFouriertransform. Forv Rnd,letQ (v)= n Q (v )andζ (v)=w (v)exp(Q (v)).
Note that if f Rd and g = 1 , whe∈re A is a com++pact subsei(cid:0)t=1of(cid:1)R+d tih(cid:0)en(cid:1)g ±fτ (cid:0) (cid:1)Rd±τ. This fa(cid:0)ct+i(cid:1)+mplies
∈C0∞ A P ∗ ∈ C0∞
that S Rn 1 and ζ Rnd , therefore it is possible to use the Fourier inversion formula. Hence
ǫ ∈S − (cid:0) (cid:1)±τ ∈S (cid:0) (cid:1)
(2.6) (cid:0) S±Qǫ((cid:1)v)=(2π)1−nˆ(cid:0)Rn−1(cid:1)S±ǫ(ω)exp(ih(Q(v1)−Q(v2),...,Q(vn−1)−Q(vn)),ωi)dω
and b
(2.7) ζ±τ (v)=(2π)−ndˆRndζ±τ(y)exp(ihv,yi)dy.
b
THE N-POINT CORRELATION OF QUADRATIC FORMS. 6
Therefore, by using the definition (2.3) and (2.6) we obtain
(2.8) R S±Qǫw±τ,T =(2π)1−nˆRn−1R(eQ,ωw±τ,T)S±ǫ(ω)dω,
(cid:16) (cid:17)
where e (v) =exp(i (Q(v ) Q(v ),...,Q(v ) Q(v )),ω ). Alsobusing the definitions of w and ζ
Q,ω 1 2 n 1 n τ,T τ
h − − − i ± ±
we have
R(e w )=R(e (v)ζ (v/T)exp( Q (v/T))).
Q,ω τ,T Q,ω τ ++
± ± −
Combining this with (2.7) gives
R(eQ,ωw±τ,T)=(2π)−ndˆRndR(eQ,ω(v)exp(2πihv/T,yi−Q++(v/T)))ζ±τ(y)dy
=(2π)−ndˆRndR(eQ,ωe˜Q,T,y)ζ±τ (y)dy b
where e˜ (v)=exp(i v/T,y Q (v/T)). We now wribte
Q,T,y ++
h i−
R(e e˜ )=θ (ω) ϑ (ω),
Q,ω Q,T,y T,y T,y
−
where θ (ω) and ϑ (ω) are defined as follows: For x R, let Q (x,v )=i(xQ(v )+ v /T,y ) T 2Q (v )
T,y T,y T,y i i i − + i
and for ω Rn 1 define ∈ h i −
−
∈
n
Q (ω,v)= Q (ω ω ,v ),
T,y T,y i− i−1 i
i=1
X
where ω =ω =0.
0 n
Remark 2.3. For the rest of the paper the convention that ω = ω = 0, will be used in order to simplify the
0 n
notation.
For ω Rn 1 and y Rnd, let
−
∈ ∈
(2.9) θ (ω)= exp Q (ω,v) and ϑ (ω)= exp Q (ω,v) dv,
T,y T,y T,y ˆ T,y
v∈XZnd (cid:0) (cid:1) Rnd (cid:0) (cid:1)
From (2.9) and the definition of Q , it follows that
T,y
n
(2.10) θ (ω)= exp(Q (ω ω ,v ))
T,y T,y i i 1 i
− −
iY=1vXi∈Zd
and
n
(2.11) ϑ (ω)= exp(Q (ω ω ,v ))dv .
T,y ˆRd T,y i− i−1 i i
i=1
Y
Next we define a certain bounded region of Rn 1. Let
−
(T)= ω Rn 1 : ω ω T 1 for 1 i n .
− i i 1 −
B ∈ | − − |≤ ≤ ≤
Decomposing the integral over Rn 1 in(cid:8)(2.8) into regions we see that (cid:9)
−
(2.12) R SQ w E ( τ, ǫ,T)+E ( τ, ǫ,T)+E ( τ, ǫ,T)
±ǫ ±τ,T ≪ 0 ± ± 1 ± ± 2 ± ±
(cid:12) (cid:16) (cid:17)(cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
E ( τ, ǫ,T)= S (ω) ϑ (ω)ζ (y)dydω
0 ± ± (cid:12)ˆRn−1 (T) ±ǫ ˆRnd T,y ±τ (cid:12)
(cid:12) \B (cid:12)
(cid:12)(cid:12) b b (cid:12)(cid:12)
E ( τ, ǫ,T)=(cid:12) R(e w )S (ω)dω (cid:12)
1 ± ± (cid:12)ˆ (T) Q,t ±τ,T ±ǫ (cid:12)
(cid:12) B (cid:12)
(cid:12)(cid:12) b (cid:12)(cid:12)
E ( τ, ǫ,T)=(cid:12) S (ω) θ (ω)(cid:12)ζ (y)dydω .
2 ± ± (cid:12)ˆRn−1 (T) ±ǫ ˆRnd T,y ±τ (cid:12)
(cid:12) \B (cid:12)
(cid:12)(cid:12) b b (cid:12)(cid:12)
(cid:12) (cid:12)
THE N-POINT CORRELATION OF QUADRATIC FORMS. 7
3. Bounding the integrals
In this section we obtain bounds for the integrals E , E and E , in terms of ζ . Precise bounds for ζ
0 1 2 τ 1 τ 1
are givenin subsection 5.2. We will only consider the case when ǫ>0 and τ >0 since the other three cases can be
(cid:13) (cid:13) (cid:13) (cid:13)
dealt with in an identical manner. The following Theorem will be proved in Propo(cid:13)sbit(cid:13)ions 3.4, 3.7 and 3.13. (cid:13)b (cid:13)
Theorem 3.1. For all 0<ǫ<1 and τ >0, there exists T >0 such that for all T >T ,
0 0
E (τ,ǫ,T) ζ T(n 1)d 2(n 2)
0 ≪ τ 1 − − −
E1(τ,ǫ,T)≪(cid:13)(cid:13)bζ(cid:13)(cid:13)τ 1+τ(1−nd)/2 T(n−1)d−n+1
E2(τ,ǫ,T)≪(cid:16)ζ(cid:13)(cid:13)τb1(cid:13)(cid:13)Aǫ(T)Tnd−2((cid:17)n−1),
where for any fixed ǫ > 0 we have limT Aǫ(cid:13)(T)(cid:13)= 0 provided that Q is irrational. (See (3.28) for a precise
definition of A (T).) →∞ (cid:13)b (cid:13)
ǫ
The bounds for E and E contribute only to lower order terms. Note that the bound for E is of smaller order
0 1 0
of magnitude than Tnd 2(n 1) all n N. The bound for E is of smaller order of magnitude than Tnd 2(n 1) only
− − 1 − −
∈
for n d. Using the fact that ϑ (t) can be split as in (2.11) the required bound for E is relatively simple to
y 0
≤
obtain. The bound for E is slightly more involved since the formula (3.9) is used. This means that, although one
1
can still take advantage of the splitting given in (2.10), the sums in the product may include 0 and this causes
extra difficulties. To overcomethese difficulties we employ Lemma 3.6, which enables us to bound the minimum of
certainquantities frombelowby aweightedaverage. The boundforE contributesto the maintermandthis term
2
depends on the arithmetic properties of Q. Using (2.10), the bound for E follows reasonably directly from results
2
in [GM13].
3.1. Bound for E . The following bound will be used in subsections 3.1 and 3.2 to obtain bounds for E and E .
0 0 1
It is relatively straightforward to prove via a direct computation involving Gaussian integrals (see Formula (3.28)
in [GM13]). The notation Q−+1 will stand for the positive definite quadratic form that corresponds to the matrix
Q−+1.
Lemma 3.2. For 1 i n 1, all y Rnd, x R and T >0,
≤ ≤ − ∈ ∈
1
(cid:12)ˆRdexp(QT,y(x,vi))dvi(cid:12)≪gT (x)Td/2exp(cid:18)−4gT (x)Q−+1(yi/T)(cid:19),
(cid:12) (cid:12)
where gT (x)=T/ 1+(cid:12)(cid:12) (xT2)2. (cid:12)(cid:12)
q
We will need estimates for the Fourier transform of the smoothened characteristic function.
Lemma 3.3. For all ǫ>0 and ω Rn 1,
−
∈
n 1
− 1
S (ω) min 1, exp c ǫω .
ǫ i
≪ ω − | |
(cid:12) (cid:12) Yi=1 (cid:26) (cid:12)(cid:12) i(cid:12)(cid:12)(cid:27) (cid:16) p (cid:17)
(cid:12)b (cid:12) (cid:12) (cid:12)
Proof. Using the definition of Sǫ we get Sǫ(ω) ≤ 1I1ǫ×(cid:12)···×I(cid:12)nǫ−1(ω) kǫ1(ω1)...kǫ1(ωn−1) . Then a simple compu-
tation and (2.1) gives
(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)
(cid:12)b (cid:12) (cid:12)b (cid:12)(cid:12)c c (cid:12)
n 1
− 1
S (ω) sin ω π I exp c ǫω .
ǫ i i i
≪ ω − | |
(cid:12) (cid:12) iY=1(cid:12)(cid:12) i (cid:0) (cid:12) (cid:12)(cid:1)(cid:12)(cid:12) (cid:16) p (cid:17)
Since sin(kx) min k, 1/x fo(cid:12)rball x(cid:12) R, the(cid:12)(cid:12)claim of th(cid:12)e L(cid:12)e(cid:12)(cid:12)mma follows. (cid:3)
x ≤ { | |} ∈
(cid:12) (cid:12)
The(cid:12)bound(cid:12)forE0 isobtainedbyusingLemmas3.2and3.3,togetherwithsomeelementaryestimatesofintegrals
of pow(cid:12)ers of g(cid:12) (x). For ω Rn 1, define G (ω)= n g (ω ω ).
T ∈ − T i=1 T i− i−1
Proposition 3.4. For all ǫ>0, τ >0 and T >0, Q
E0(τ,ǫ,T)≪ ζτ 1T(n−1)d−2(n−2).
Proof. Using (2.11) and Lemma 3.2, (cid:13) (cid:13)
(cid:13)b (cid:13)
ϑ (ω) Tnd/2G (ω)d/2.
T,y T
| |≪
Note that for any x R and T R, gT (x) Tx−1. It follows that for all ω Rn−1 and y Rnd and 1 i,j n
∈ ∈ ≤| | ∈ ∈ ≤ ≤
with i=j,
6
THE N-POINT CORRELATION OF QUADRATIC FORMS. 8
d/2
G (ω)
(3.1) ϑT,y(ω) T(n−2)d/2 T (ωi ωi 1 ωj ωj 1 )−d/2.
| |≪ (cid:18)gT (ωi−ωi−1)gT (ωj −ωj−1)(cid:19) | − − || − − |
Choose 1 l n with l =i or j. Since, we assume that n 3, this is always possible. Note that g (x) T and
T
≤ ≤ 6 ≥ ≤
hence
d/2
G (ω)
(3.2) T Td/2G (ω)d/2,
T
g (ω ω )g (ω ω ) ≤
(cid:18) T i− i−1 T j − j−1 (cid:19)
where e
G (ω)
T
G (ω)= .
T
g (ω ω )g (ω ω )g (ω ω )
T i i 1 T j j 1 T l l 1
− − − − − −
Let Tn =(n 1)T and e
−
= , where = ω Rn 1 : ω ω >T 1, ω ω >T 1 .
B1 Bi,j Bi,j ∈ − | i− i−1| n− | j − j−1| n−
1 i,j n
≤[≤ (cid:8) (cid:9)
i=j
6
Note that Rn 1 (T) . (Or, equivalently, Rn 1 (T).) To see this, suppose that ω Rn 1 ,
− 1 − 1 − 1
\B ⊂ B \B ⊂ B ∈ \B
then ω ω > T 1 for at most one 1 i n. If ω ω T 1 for all 1 i n, then clearly ω (T).
| i− i−1| n− ≤ ≤ | i− i−1| ≤ n− ≤ ≤ ∈ B
Suppose 1 l n is such that ω ω >T 1. Now note that ω ω = (ω ω ) T 1 by
≤ ≤ | l− l−1| n− | l− l−1| 1≤i≤n,i6=l i− i−1 ≤ −
the triangle inequality and hence ω (T). (cid:12) (cid:12)
∈B (cid:12)P (cid:12)
Using the definition of E , (cid:12) (cid:12)
0
E (τ,ǫ,T)= S (ω) ϑ (ω)ζ (y)dydω
0 ˆ ǫ ˆ T,y τ
(cid:12) Rn−1 (T) Rnd (cid:12)
(cid:12) \B (cid:12)
(cid:12)(cid:12) b b (cid:12)(cid:12)
≤(cid:12)ˆRn−1 (T)(cid:12)Sǫ(ω)ˆRndϑT,y(ω)ζτ (y)dy(cid:12)dω(cid:12)
\B (cid:12) (cid:12)
(cid:12)b b (cid:12)
(3.3) ≤1≤Xi,j≤nˆBi,j(cid:12)(cid:12)(cid:12)Sǫ(ω)ˆRndϑT,y(ω)ζτ(y)d(cid:12)y(cid:12)(cid:12)dω.
i=j (cid:12)b b (cid:12)
6 (cid:12) (cid:12)
From Lemma 3.3 we get that for all ǫ>0 we have the uniform bound, S (ω) 1. Hence using (3.1) and (3.2),
ǫ
≪
(cid:12) (cid:12)
(3.4) ˆBi,j(cid:12)(cid:12)Sǫ(ω)ˆRndϑT,y(ω)ζτ (y)dy(cid:12)(cid:12)dω ≪(cid:13)ζτ(cid:13)1T(n−1)d/2ˆBi,jGT(cid:12)b(ω)d/2(cid:12)(|ωi−ωi−1||ωj −ωj−1|)−d/2dω.
(cid:12)b b (cid:12) (cid:13)b (cid:13) e
(cid:12) (cid:12) ξi if i<l
By doing the change of variables ϕ:ω ω we get
i i 1
− − →(ξi 1 if i>l
−
(3.5) ˆBi,jGT (ω)d/2(|ωi−ωi−1||ωj −ωj−1|)−d/2dω ≤ˆϕ(Bi,j)1≤kY≤n−1gT (ξk)d/2(|ξi||ξj|)−d/2dξ,
e k=i,j
6
where ϕ( )= ξ Rn 1 : ξ T 1, ξ T 1 . Note that
Bi,j ∈ − | i|≥ n− | j|≥ n−
(cid:8) ∞ x(cid:9) d/2dx Td/2 1.
− −
ˆTn−1 ≪
By making change of variables T2x=sinhy, we get
1
g (x)d/2dx=Td/2 2 dy Td/2 2.
ˆR T − ˆR coshd/4−1y ≪ −
The last two observations, (3.4) and (3.5) imply that for all 1 i,j n with i=j,
≤ ≤ 6
ˆBi,j(cid:12)(cid:12)Sǫ(ω)ˆRndϑT,y(ω)ζτ(y)dy(cid:12)(cid:12)dω ≤(cid:13)ζτ(cid:13)1T(n−1)d−2(n−2).
The conclusion of the Lemma fo(cid:12)lblows from (3.3). b (cid:12) (cid:13)b (cid:13) (cid:3)
(cid:12) (cid:12)
THE N-POINT CORRELATION OF QUADRATIC FORMS. 9
3.2. Bound for E . We will need two preliminary Lemmas. The first is probably standard, but for completeness,
1
a proof is provided.
Lemma 3.5. For any c>0, there exists a positive constant B such that, for any y Rnd
∈
exp c y m 2 <B.
− k − k
m∈ZXnd\{0} (cid:16) (cid:17)
Proof. For v [ 1/2,1/2]nd , v 2 nd/4. Hence
∈ − k k ≤
exp c u+v 2 dv exp c u 2 ndc/4 exp( 2c u,v )dv.
ˆ − k k ≥ − k k − ˆ − h i
[ 1/2,1/2]nd [ 1/2,1/2]nd
− (cid:16) (cid:17) (cid:16) (cid:17) −
It is easy to check that for any u Rnd, exp( 2c u,v )dv 1 and hence we get the inequality
∈ [ 1/2,1/2]nd − h i ≥
´−
exp c u 2 exp c u+v 2 dv.
− k k ≪ˆ − k k
[ 1/2,1/2]nd
(cid:16) (cid:17) − (cid:16) (cid:17)
Hence
exp c y m 2 exp c y m+v 2 dv
m∈ZXnd\{0} (cid:16)− k − k (cid:17)≪m∈ZXnd\{0}ˆ[−1/2,1/2]nd (cid:16)− k − k (cid:17)
= exp c z 2 dz
m∈ZXnd\{0}ˆ[−1/2,1/2]nd+y−m (cid:16)− k k (cid:17)
exp c z 2 dz < .
≪ˆRnd − k k ∞
(cid:16) (cid:17)
(cid:3)
The secondpreliminary Lemma is the crucialstep in obtaining the estimate for E . For ω Rn 1, let G (ω)=
1 ∈ − ∗T
min g (ω ω )2.
1 i n T i i 1
≤≤ − −
Lemma 3.6. Let ω (T) and s ,...,s = ω ω . Suppose s = max s . Then, there exist
positive constants c ,∈..B.,c , and b{,1...,b ns}uch {thait−, fori−a1l}l1T≤i≤n1, 1 1≤i≤n| i|
1 n 2 n
≥
n
c
G (ω) i g (s )2,
∗T ≥ Tbi T i
i=1
X
where b =0 and n b 2(n 2).
1 i=2 i ≤ −
Proof. For ω (PT), let s ,...,s = ω ω and s =max s . Thus, G (ω)=g (s )2. Let
∈B { 1 n} { i− i−1}1≤i≤n 1 1≤i≤n| i| ∗T T 0
log s /logT if s 1/T2,1/T
i i
α = − | | | |∈
i (2 if |si|∈(cid:2)0,1/T2 . (cid:3)
Note that 1 α 2 for all 1 i n and α = min α . Fo(cid:2)r all s (cid:3)such that s 1/T2,1/T we have
i 1 1 i n i i i
si =1/Tαi ≤. Thu≤s ≤ ≤ ≤≤ | | ∈
| | (cid:2) (cid:3)
gT (s0) 2 1+T4−2αi 1
(3.6) = .
(cid:18)gT (si)(cid:19) 1+T4−2α0 ≥ 2T2(αi−α1)
For all s such that s 0,1/T2 we have g (s )2 T2/2,T2 . Thus
i i T i
| |∈ ∈
(cid:2) (cid:3) g (s ) 2 (cid:2) 1 (cid:3) 1
T 1
(3.7) .
(cid:18)gT (si)(cid:19) ≥ 1+T4−2α1 ≥ 2T2(αi−α1)
Let δ =α α . Note that for all 1 i n 1, 0 δ 1. Moreover, n s =0 and hence
i i− 1 ≤ ≤ − ≤ i ≤ i=1 i
s s , P
1 i
| |≤ | |
si∈{s1,.X..,sn}\{s1}
which implies that
n 1 n 1
1 − 1 − 1
= .
Tα1 ≤ Tαi Tδi+α1
i=2 i=2
X X
THE N-POINT CORRELATION OF QUADRATIC FORMS. 10
Thus
n 1
− 1
(3.8) 1 .
≤ Tδi
i=2
X
Let in=−21δi = ∆ and note that for all 2 ≤ i ≤ n−1, ∆−δi ≤ n−2. Hence by multiplying by the denominators
in (3.8) we get
P
n 1
−
T∆ T∆−δi (n 2)Tn−2.
≤ ≤ −
i=2
X
Since this holds for all T ≥1 we get in=−21δi ≤n−2. Finally, note that from (3.6) and (3.7), we obtain
P
1 1
G (ω)=g (s )2 g (s )2+ g (s )2 ,
∗T T 0 ≥ n T 1 2T2(αi−α1) T i 
si∈{s1,.X..,sn}\{s1}
 
since si∈{s1,...,sn}\{s1}2(αi−α1)= in=−212δi ≤2(n−2) the claim of the Lemma follows. (cid:3)
UsiPng the preceding two results wePcan now obtain a bound for E . This is done by using Poisson summation
1
to convert the difference θ (ω) ϑ (ω), into a sum over m Znd 0 . An estimate for each term in the sum
T,y T,y
can be obtained provided that y−/T m is bounded from belo∈w. Th\e{in}tegral over y Rnd is then decomposed
into boxes centred at each poinkt of −Znd.kThus, the estimates for the summands can b∈e used and there will be
additional term coming from the point at the centre of the box under consideration. Lemma 3.6 is used to bound
a function of the form G (ω)d/2exp( G (ω)). Finally, Lemma 5.8 is used to estimate the term that arises for
T − ∗T
small y/T m .
k − k
Proposition 3.7. For all ǫ>0 , τ >0 and T τ 1,
−
≥
E (τ,ǫ,T) ζ +τ(1 nd)/2 T(n 1)d n+1.
1 ≪ τ 1 − − −
(cid:16)(cid:13) (cid:13) (cid:17)
Proof. Recall (see Section 2) (cid:13)b (cid:13)
E (τ,ǫ,T)= R(e w )S (ω)dω
1 ˆ Q,t τ,T ǫ
(cid:12) (T) (cid:12)
(cid:12) B (cid:12)
and (cid:12)(cid:12) b (cid:12)(cid:12)
(cid:12) (cid:12)
R(e w )= R(e e˜ )ζ (y)dy = (θ (ω) ϑ (ω))ζ (y)dy.
Q,ω τ,T ˆRnd Q,ω Q,T,y τ ˆRnd T,y − T,y τ
Note that e e˜ Rnd and thus, there exisbts a constant C, so that e e˜ b(v) + e \e˜ (v)
Q,ω Q,T,y Q,ω Q,T,y Q,ω Q,T,y
∈ S ≤
C(1+kxk)−(n+1). Hence,(cid:0)usin(cid:1)g Poissonsummation (Theorem 3.1.17 in [Gra0(cid:12)(cid:12)8]) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
(3.9) θ (ω) ϑ (ω)= ϑ (ω).
T,y T,y T,y Tm
− −
m∈ZXnd\{0}
By using (3.9),
(3.10) E (τ,ǫ,T)= S (ω) ϑ (ω)ζ (y)dydω .
1 (cid:12)(cid:12)(cid:12)ˆB(T) ǫ ˆRndm∈ZXnd\{0} T,y−Tm τ (cid:12)(cid:12)(cid:12)
(cid:12) b b (cid:12)
Let Σ = ϑ (ω(cid:12)). By Lemma 3.2 we have (cid:12)
T,ω,y m∈Znd\{0} T,y−Tm (cid:12) (cid:12)
P Σ ϑ (ω) Tnd/2G (ω)d/2 exp Q (y,m) ,
| T,ω,y|≪ | y−Tm |≪ T − ∗T,ω
m∈ZXnd\{0} m∈ZXnd\{0} (cid:0) (cid:1)
where
1
Q∗T,ω(y,m)= 4 gT (ωi−ωi−1)2Q−+1(yi/T −mi).
1 i n
≤X≤
Note Q−+1 is a positive definite quadratic form and because ω ∈ B(T) we have (using that T > 1) 1/2 <
g (ω ω )2 for1 i n. Recall,λ isthemaximum(intermsofabsolutevalue)eigenvalueofQ. Therefore,
T i i 1 max
− − ≤ ≤