The n-point correlation of quadratic forms.

The n-point correlation of quadratic forms.

Oliver Sargent Department Of Mathematics, University Walk, Bristol, BS8 1TW, UK.
Abstract.

In this paper we investigate the distribution of the set of values of a quadratic form , at integral points. In particular we are interested in the -point correlations of the this set. The asymptotic behaviour of the counting function that counts the number of -tuples of integral points , with bounded norm, such that the differences , lie in prescribed intervals is obtained. The results are valid provided that the quadratic form has rank at least 5, is not a multiple of a rational form and is at most the rank of the quadratic form. For certain quadratic forms satisfying Diophantine conditions we obtain a rate for the limit. The proofs are based on those in the recent preprint ([GM13]) of F. Götze and G. Margulis, in which they prove an ‘effective’ version of the Oppenheim Conjecture. In particular, the proofs rely on Fourier analysis and estimates for certain theta series.

1. Introduction.

1.1. Background

Let be a quadratic form. It is interesting to understand the distribution of the set inside . If there exists a multiple of such that its coefficients are all rational, then is called a rational form. In the case when is rational, is a discrete set inside . When is not a rational form, is called an irrational form. For irrational forms, the first milestone in understanding the distribution of the set inside was reached by G. Margulis in [Mar89] when he provided a proof (shortly afterwards, refined by S.G. Dani and G. Margulis in [DM89]) of the ‘Oppenheim Conjecture’. The modern statement of which is as follows: if and is a nondegenerate, irrational and indefinite form, then is dense in .

Once it is known that is dense in , one can ask for a more precise answer to the question of how is distributed in . Let be any interval and , then one can ask for an asymptotic formula for the size of the set . The first results in this direction were obtained by Dani-Margulis in [DM93] who proved, if and is a nondegenerate, irrational and indefinite form and is any interval, then

 limT→∞∣∣Zd∩EQ(I,T)∣∣Vol(EQ(I,T))≥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 and is a nondegenerate, irrational and indefinite form and is any interval, then

 (1.1) limT→∞∣∣Zd∩EQ(I,T)∣∣Vol(EQ(I,T))=1.

It should be noted that the actual results from [DM93] and [EMM98] are more general than stated above. The situation regarding the upper bounds for the case when or is particularly interesting and is also considered in [EMM98]. In the cases when has signature or 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 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 is equidistributed in .

One can ask still finer questions about the distribution of the set . Let be the standard basis of , let denote the projections onto respectively. For and , we will write . Let be intervals and

 PnQ(I1,…,In−1,T)={v∈Rnd:Q(v1)−Q(v2)∈I1,…,Q(vn−1)−Q(vn)∈In−1,∥v∥≤T}.

In order to understand the -point correlations of the set , one asks for an asymptotic formula for the size of the set . A more general problem about the distribution of values at integral points of systems of quadratic forms was studied by W. Müller in [Mül08]. In particular, it follows from Theorem 1 of [Mül08] that if , and is a nondegenerate and irrational form, then

 (1.2) limT→∞∣∣Znd∩PnQ(I1,…,In−1,T)∣∣Vol(PnQ(I1,…,In−1,T))=1.

When and , it is easy to see that (1.2) follows from the main Theorem of [EMM98]. For positive definite forms the -point correlation problem was also studied by Müller. In [Mül11], Müller obtains the following result: if and is a nondegenerate, irrational and positive definite form, then (1.2) holds for every . In [Mül11] Müller formulates the problem in slightly different language, but it is easily seen to be equivalent to the form stated 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 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 is not a multiple of a rational form and and . Then, for any intervals ,

 limT→∞∣∣Znd∩PnQ(I1,…,In−1,T)∣∣Vol(PnQ(I1,…,In−1,T))=1.

Moreover, there exists a positive constant , depending only on and , such that for any intervals ,

 limT→∞∣∣Znd∩PnQ(I1,…,In−1,T)∣∣Tnd−2(n−1)=CQ,nn−1∏i=1|Ii|.

For quadratic forms satisfying the following Diophantine condition it is possible to prove an effective version of Theorem 1.1. Let also denote the symmetric matrix that is associated to the quadratic form . Let and , say that is of type if for every and we have

 inft∈[1,2]∥∥Mq−1−tQ∥∥≥Aq−1−κ.

The size of depends on how well can be approximated by a rational matrix, if is close to , then is in some sense close to a rational matrix.

Theorem 1.2.

Suppose that is of Diophantine type and and . Let Then, for any intervals there exists and a constant such that for all ,

 ∣∣ ∣ ∣∣∣∣Znd∩PnQ(I1,…,In−1,T)∣∣Vol(PnQ(I1,…,In−1,T))−1∣∣ ∣ ∣∣≤Clogn−1(T)T−δ(κ).
Remark 1.3.

The constant appearing in Theorem 1.2 depends on and the intervals .

Remark 1.4.

In Theorem 1.2 we use to denote the Euclidean norm. The exponent depends on the choice of norm and is possibly non optimal. If the maximum norm was chosen, the bounds in subsection 5.2 could be improved, and could be replaced with at the cost of a factor of appearing.

Remark 1.5.

The second parts of Theorems 1.1 and 1.2 follow easily from the first parts and the following assertion: For any intervals there exists a positive constant , depending only on and , such that

 limT→∞1Tnd−2(n−1)Vol(PnQ(I1,…,In−1,T))=CQ,nn−1∏i=1|Ii|.

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 -point correlations of the normalised values of at integral points. This problem is discussed in [Mül08] and is interesting 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 problem has been studied in [Sar97] by P. Sarnak, in [Van99] and [Van00] by J. VanderKam and in [Mar02] by J. Marklof.

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 stabilising the quadratic forms is and this seems too small to obtain the required statements. If one had access to a precise quantitative equidistribution statement, in the form of an explicit rate for the limit in (1.1), one could hope to prove results 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 unipotent flows. Instead they use Fourier analysis to reduce the problem to one of obtaining asymptotic estimates for certain theta series. In order to estimate these theta series, they use some of the techniques developed in [EMM98], in particular the crux of their proof relies on a non divergence statement about average of the translates of orbits of certain compact subgroups in the space of lattices. One cannot apply the results of [GM13] directly, since in order to do this one would need the error to be uniform across all intervals. However, the proofs of Theorems 1.1 and 1.2 are based on the methods of [GM13].

The object of interest is

 R(1PnQ(I1,…,In−1,T))=∑v∈Znd1PnQ(I1,…,In−1,T)(v)−∫Rnd1PnQ(I1,…,In−1,T)(v)dv,

where, here and throughout the rest of the paper, for any set , stands for the characteristic function of the set . Theorems 1.1 and 1.2 follow from suitable bounds for . To obtain these bounds, the function is replaced with a smoothened version at the cost of ‘smoothing errors’ which can be estimated in terms of volumes of certain regions of . This is carried out in subsections 2.1 and 5.1. The next step is to use Fourier analysis to transfer the problem into the ‘frequency domain’. After taking Fourier transforms, the smoothened version of can be estimated by considering an integral over the ‘frequency domain’, , of the difference between a theta series, and its corresponding smooth version, (see (2.9)). This step is carried out in subsection 2.2. In order to estimate the integral, the domain of integration is split into two parts, namely a neighbourhood of the origin and its complement.

The integral over the region bounded away from the origin is dealt with by considering the integral of and the integral of separately. The integral of contributes the main term in the bound for and it contains the arithmetic information about . This term is dealt with in subsection 3.3. The integral of only contributes a lower order term to the bound for and is dealt with in subsection 3.1. These two integrals can be estimated using techniques and results from [GM13]. The reason for this, is that and can be written as a product of sums/integrals of the form studied in [GM13] (see (2.10) and (2.11)).

The integral, over the neighbourhood of the origin, is dealt with in subsection 3.2. This term contributes a lower order term, but it grows with , faster than the main term. For this term dominates the main term, explaining why the assumption is needed. The reason for this, is that here we consider the difference, . Poisson summation is used to convert this into a sum over , the problem that arises is that for we can still have for some . Therefore, although it is still possible to take advantage of the fact that the sum obtained by Poisson summation can be written as a product of sums, an additional argument is needed to deal with the fact that 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 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 and be natural numbers with . In the case when , 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 . For , fix intervals and a nondegenerate quadratic form, suppose that and keep the notation from the introduction. Let , hence corresponds to a positive definite quadratic form. Let denote the spectrum of , and . Since the problem is unaffected by rescaling , we may suppose that , this supposition will be used in the proof of Lemma 3.13. Define and , where we use to denote the Euclidean norm and to denote the maximum norm. Let

 PnQ(I1,…,In−1)={v∈Rnd:Q(v1)−Q(v2)∈I1,…,Q(vn−1)−Q(vn)∈In−1}.

Note that . As is standard, we use the notation to denote the Fourier transform of a function . We will also make heavy use the Vinogradov asymptotic notation , which means that there exists some constant such that for all values of indicated. The constant will be independent of those parameters but will usually depend on and the intervals .

2.1. Smoothing.

For any , let be a probability measure on with the properties that it is symmetric around , and

 (2.1) ∣∣ˆki(v)∣∣≤exp(−c√∥v∥)

for some positive constant and all . For any , let denote the rescaled measure such that for any measurable set . Note that (2.1) implies that

 (2.2) ∣∣ˆkiτ(v)∣∣≤exp(−c√τ∥v∥).

For an interval and , define . For any , and , let

 w±τ(v)=1B(1±τ)∗kndτ(v) and w±τ,T(v)=w±τ(T−1v).

For any and , let

 S±ϵ(ω)=1I±ϵ1×⋯×I±ϵn−1∗kΠϵ(ω),

where . For , let

 SQ±ϵ(v)=S±ϵ(Q(v1)−Q(v2),…,Q(vn−1)−Q(vn)).

For a measurable function on define

 (2.3) R(f)=∑v∈Zndf(v)−∫Rndf(v)dv.

Note that is only well defined if both the quantities on the right hand side of (2.3) are finite. Let and be measures on and respectively, defined by

 ∫RndfdνT=∫Rndf(T−1v)1PnQ(I1,…,In−1)(v)dv

and

 ∫Rn−1fdντ,T=∫Rndf(Q(v1)−Q(v2),…,Q(vn−1)−Q(vn))w±τ,T(v)dv.

In the next two Lemmas we approximate by a smoothened version.

Lemma 2.1.

For all and ,

 ∣∣R(1PnQ(I1,…,In−1)1B(T))∣∣≤max±τ∣∣R(1PnQ(I1,…,In−1)w±τ,T)∣∣+∫Rnd(1B(1+2τ)−1B(1−2τ))dνT.
Proof.

Define a measure on , by

Define functions on by and . Note that

 (2.4) ∣∣R(1PnQ(I1,…,In−1)1B(T))∣∣=∣∣∣∫Rndfd(μT−νT)∣∣∣.

From the definition of it follows that

 f−2τ≤f−τ∗kndτ≤f≤f+τ∗kndτ≤f+2τ.

Since all the functions in the previous inequality are bounded and have compact support and the measure is locally finite, by integrating with respect to we obtain

 ∫Rndfd(μT−νT) ≤∫Rndf+τ∗kndτd(μT−νT)+∫Rnd(f+τ∗kndτ−f)dνT ≤∫Rndf+τ∗kndτd(μT−νT)+∫Rnd(f+2τ−f−2τ)dνT.

Similarly

 ∫Rndfd(μT−νT)≥∫Rndf−τ∗kndτd(μ−ν)−∫Rnd(f+2τ−f−2τ)dνT.

In view of (2.4) and the definition of the conclusion of the Lemma follows from the previous two inequalities. ∎

Lemma 2.2.

For all , and ,

 ∣∣R(1PnQ(I1,…,In−1)w±τ,T)∣∣≤max±ϵ∣∣R(SQ±ϵw±τ,T)∣∣+∫Rn−1(1I2ϵ1×⋯×I2ϵn−1−1I−2ϵ1×⋯×I−2ϵn−1)dντ,T.
Proof.

Define a measure on , by

 ∫Rn−1fdμτ,T=∑v∈Zndf(Q(v1)−Q(v2),…,Q(vn−1)−Q(vn))w±τ,T(v).

Define functions on by and . Note that

 (2.5) ∣∣R(1PnQ(I1,…,In−1)w±τ,T)∣∣=∣∣∣∫Rn−1fd(μτ,T−ντ,T)∣∣∣.

From the definition of it follows that

 f−2ϵ(ω)≤f−ϵ∗kΠϵ(ω)≤f(ω)≤f+ϵ∗kΠϵ(ω)≤f+2ϵ.

Since all the functions in the previous inequality are bounded and have compact support and the measure is locally finite, by integrating with respect to as in the proof of Lemma 2.1 we obtain

In view of (2.5) and the definition of the conclusion of the Lemma follows from the previous inequality. ∎

In subsection 5.1 we will obtain bounds for the smoothing errors

 ∫Rnd(1B(1+2τ)−1B(1−2τ))dνTand∫Rn−1(1I2ϵ1×⋯×I2ϵn−1−1I−2ϵ1×⋯×I−2ϵn−1)dντ,T

that arise from Lemmas 2.1 and 2.2.

2.2. Fourier transforms

To obtain bounds for the strategy of [GM13] will be used, in particular we proceed via the Fourier transform. Numerous text books on Fourier analysis are available, for instance see [Gra08]. Let denote the space of Schwartz functions on (see Section 2.2 of [Gra08]) and let be smooth functions with compact support on . We note that and that is invariant under the Fourier transform. For , let and . Note that if and , where is a compact subset of then . This fact implies that and , therefore it is possible to use the Fourier inversion formula. Hence

 (2.6) SQ±ϵ(v)=(2π)1−n∫Rn−1ˆS±ϵ(ω)exp(i⟨(Q(v1)−Q(v2),…,Q(vn−1)−Q(vn)),ω⟩)dω

and

 (2.7)

Therefore, by using the definition (2.3) and (2.6) we obtain

 (2.8) R(SQ±ϵw±τ,T)=(2π)1−n∫Rn−1R(eQ,ωw±τ,T)ˆS±ϵ(ω)dω,

where . Also using the definitions of and we have

 R(eQ,ωw±τ,T)=R(eQ,ω(v)ζ±τ(v/T)exp(−Q++(v/T))).

Combining this with (2.7) gives

 R(eQ,ωw±τ,T) =(2π)−nd∫RndR(eQ,ω(v)exp(2πi⟨v/T,y⟩−Q++(v/T)))ˆζ±τ(y)dy =(2π)−nd∫RndR(eQ,ω~eQ,T,y)ˆζ±τ(y)dy

where . We now write

 R(eQ,ω~eQ,T,y)=θT,y(ω)−ϑT,y(ω),

where and are defined as follows: For , let and for define

 ¯¯¯¯QT,y(ω,v)=n∑i=1QT,y(ωi−ωi−1,vi),

where .

Remark 2.3.

For the rest of the paper the convention that , will be used in order to simplify the notation.

For and , let

 (2.9)

From (2.9) and the definition of , it follows that

 (2.10) θT,y(ω)=n∏i=1∑vi∈Zdexp(QT,y(ωi−ωi−1,vi))

and

 (2.11) ϑT,y(ω)=n∏i=1∫Rdexp(QT,y(ωi−ωi−1,vi))dvi.

Next we define a certain bounded region of . Let

 B(T)={ω∈Rn−1:|ωi−ωi−1|≤T−1 for 1≤i≤n}.

Decomposing the integral over in (2.8) into regions we see that

 (2.12) ∣∣R(SQ±ϵw±τ,T)∣∣≪E0(±τ,±ϵ,T)+E1(±τ,±ϵ,T)+E2(±τ,±ϵ,T)

where

 E0 E1 (±τ,±ϵ,T)=∣∣∣∫B(T)R(eQ,tw±τ,T)ˆS±ϵ(ω)dω∣∣∣ E2

3. Bounding the integrals

In this section we obtain bounds for the integrals , and , in terms of . Precise bounds for are given in subsection 5.2. We will only consider the case when and since the other three cases can be dealt with in an identical manner. The following Theorem will be proved in Propositions 3.4, 3.7 and 3.13.

Theorem 3.1.

For all and , there exists such that for all ,

 E0(τ,ϵ,T) E1(τ,ϵ,T) E2(τ,ϵ,T) ≪∥∥ˆζτ∥∥1Aϵ(T)Tnd−2(n−1),

where for any fixed we have provided that is irrational. (See (3.28) for a precise definition of .)

The bounds for and contribute only to lower order terms. Note that the bound for is of smaller order of magnitude than all . The bound for is of smaller order of magnitude than only for . Using the fact that can be split as in (2.11) the required bound for is relatively simple to obtain. The bound for is slightly more involved since the formula (3.9) is used. This means that, although one can still take advantage of the splitting given in (2.10), the sums in the product may include and this causes extra difficulties. To overcome these difficulties we employ Lemma 3.6, which enables us to bound the minimum of certain quantities from below by a weighted average. The bound for contributes to the main term and this term depends on the arithmetic properties of . Using (2.10), the bound for follows reasonably directly from results in [GM13].

3.1. Bound for E0.

The following bound will be used in subsections 3.1 and 3.2 to obtain bounds for and . It is relatively straightforward to prove via a direct computation involving Gaussian integrals (see Formula (3.28) in [GM13]). The notation will stand for the positive definite quadratic form that corresponds to the matrix .

Lemma 3.2.

For , all , and ,

 ∣∣∣∫Rdexp(QT,y(x,vi))dvi∣∣∣≪gT(x)Td/2exp(−14gT(x)Q−1+(yi/T)),

where .

We will need estimates for the Fourier transform of the smoothened characteristic function.

Lemma 3.3.

For all and ,

 ∣∣ˆSϵ(ω)∣∣≪n−1∏i=1min{1,∣∣∣1ωi∣∣∣}exp(−c√|ϵωi|).
Proof.

Using the definition of we get . Then a simple computation and (2.1) gives

 ∣∣ˆSϵ(ω)∣∣ ≪n−1∏i=1∣∣∣1ωisin(ωiπ∣∣Ii∣∣)∣∣∣exp(−c√|ϵωi|).

Since for all , the claim of the Lemma follows. ∎

The bound for is obtained by using Lemmas 3.2 and 3.3, together with some elementary estimates of integrals of powers of . For , define .

Proposition 3.4.

For all , and ,

 E0(τ,ϵ,T)≪∥∥ˆζτ∥∥1T(n−1)d−2(n−2).
Proof.

Using (2.11) and Lemma 3.2,

 ∣∣ϑT,y(ω)∣∣≪Tnd/2GT(ω)d/2.

Note that for any and , . It follows that for all and and with ,

 (3.1)

Choose with or . Since, we assume that , this is always possible. Note that and hence

 (3.2)

where

 ˜GT(ω)=GT(ω)gT(ωi−ωi−1)gT(ωj−ωj−1)gT(ωl−ωl−1).

Let and

 B1=⋃1≤i,j≤ni≠jBi,j, where Bi,j={ω∈Rn−1:|ωi−ωi−1|>T−1n,∣∣ωj−ωj−1∣∣>T−1n}.

Note that . (Or, equivalently, .) To see this, suppose that , then for at most one . If for all , then clearly . Suppose is such that