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SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC

FORMULAE

VALENTIN BLOMER AND MAKSYM RADZIWI L L

Abstract. We establish an asymptotic formula, uniformly down to the Planckscale, for the number of small gaps between the first N eigenvalues of the Laplacianon almost all flat tori and also on almost all rectangular flat tori.

1. Introduction

What is the distribution of values at integer arguments of a randomly chosen pos-itive binary quadratic form? This number theoretic question has the following well-known dynamical interpretation. If Λ Ď R

2 is a lattice of rank 2, then the numbers4π2}ω}2, ω P Λ˚, are the eigenvalues of the Laplacian on R

2{Λ which are given bythe values of a positive binary quadratic form at integer arguments. The equationsof motions of a free particle moving through Λ are integrable, thus according to theconjectures of Berry-Tabor the energy levels of the corresponding quantized system(i.e., 4π}ω}2, ω P Λ˚) should behave like a sequence of points coming from a Poissonprocess, at least for generic Λ.The first to investigate this phenomenon systematically in the case of quadratic

forms was Sarnak [Sa] who showed that for almost all (in a Lebesgue sense) qua-dratic forms the pair correlation is Poissonian. To state this more precisely, let usfix some notation. Given α “ pα1, α2, α3q P R

3 with 4α1α3 ą α22 and α1 ą 0, let

qαpm,nq “ α1m2 ` α2mn` α3n

2 denote the corresponding positive binary quadraticform. Without loss of generality we may assume that qα is reduced, so that we canrestrict α P D with

D “ tpα1, α2, α3q P R3 | 0 ď α2 ď α1 ď α3u.

The precise measure that we choose on D is relatively unimportant, but it is mostnatural to choose the GL2pRq-invariant hyperbolic measure

dhypα “ dα

π3Dpαq3 “ dα1 dα2 dα3

p4α1α3 ´ α2q3{2.

Each quadratic form qα has the automorphism pm,nq ÞÑ p´m,´nq, so each positiveeigenvalue occurs with multiplicity at least 2. It is therefore natural to desymmetrize

2010 Mathematics Subject Classification. 35P20, 11J25, 11K36, 11E16.Key words and phrases. billiard, flat torus, small gaps, spectrum, Berry-Tabor conjecture, Poisson

statistics,The first author was partially supported by a SNF-DFG lead agency grant BL 915/2-2 and BL

915/5-1. The second author acknowledges support of a Sloan Fellowship and NSF grant DMS-1902063.

1

2 VALENTIN BLOMER AND MAKSYM RADZIWI L L

the spectrum and consider only the values qαpm,nq with m ą 0 or m “ 0 and n ě 0.We denote by 0 ă Λ1 ď Λ2 ď . . . the ordered set of values

qαpm,nqDpαq , Dpαq “ 1

π

b4α1α3 ´ α2

2,

where m ą 0 or m “ 0 and n ě 0. We usually suppress the dependence on α of thenumbers Λj. Asymptotically the average spacing between the Λi is one, thus we thinkof Λi as the properly rescaled multi-set of eigenvalues of the Laplacian on a suitablelattice. For an interval I Ď R and N ě 1 we write

(1.1) P pα, N, Iq “ 1

N#

pj, kq | Λj,Λk ď N, j ­“ k,Λj ´ Λk P I(

and denote by µpIq the length of I. Sarnak [Sa, Theorem 1] shows that almost all αsatisfy

P pα, N, Iq „ µpIqfor any fixed interval I as N Ñ 8. That this holds even for all diophantine α wasproved in [EMM, Theorem 1.7]. It is clear that it cannot hold for all α, for instanceit is clearly wrong for all integral forms qα and any nonempty I not containing zeroof size strictly less than Dpαq´1.Sarnak’s result is an effective asymptotic formula that comes with a(n unspecified)

power saving in the error term, so it is clear that one can shrink the interval I alittle bit with N . Our first result shows that the asymptotic formula remains truefor almost all α even for µpIq as small as N´1`ε. This is, up to the value of ε, thesmallest scale at which we expect that gaps exist at all, it corresponds to the Planckscale. In this sense Theorem 1 is best possible.

Theorem 1. Let η ą 0 be given. There exists a subset E Ď D of measure zero with

the following property: for all α P DzE we have

(1.2) P pα, N, r0,∆sq “ p1 ` op1qq∆, N Ñ 8,

uniformly in

(1.3) N´1`η ď ∆ ď N´η.

The restriction ∆ ď N´η could easily be removed. We included it for convenienceto streamline the argument as the main interest is certainly the case of small ∆.

Our next result concerns the thin subset of rectangular tori where α2 “ 0, i.e. thevalue distribution of diagonal quadratic forms. It was shown in [BBRR, Theorem1.2] that almost all rectangular tori have pairs of eigenvalues of size at most N withdifference at most N´1`ε. Here we upgrade the mere existence of small gaps to anasymptotic formula for its cardinality.Diagonal forms have four symmetries generated by pm,nq ÞÑ pm,´nq and pm,nq ÞÑ

p´m,nq, and we denote by 0 ă Λ1 ď Λ2 ď . . . the ordered sequence of values of

π

4?α1α3

qαpm,nq, m ą 0, n ě 0

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 3

and accordingly define P pα, N, Iq as in (1.1). Let R be the set of pα1, α3q P R2ą0

which is naturally equipped with the measure dα1 dα3

α1α3

.

Theorem 2. Let η ą 0 be given. There exists a subset F Ď R of measure zero with

the following property: for all α P RzF we have

P pα, N, r0,∆sq “ p1 ` op1qq∆, N Ñ 8,

uniformly in

(1.4) N´1`η ď ∆ ! 1.

The proofs of Theorems 1 and 2 use a variety of techniques. Both of them start withFourier analysis and transform the problem at hand into a diophantine question. Thearithmetic part of the proof of Theorem 1 is mainly based on lattice point argumentsand the geometry of numbers. Theorem 2 uses more advanced machinery. The desiredasymptotic formula follows without much difficulty from the Generalized RiemannHypothesis, or even from a Lindelof-type bound for the 8th moment of the Riemannzeta function on the half-line. In [BBRR] the use of GRH was avoided by introducingan additional bilinear structure (and hence a second set of variables) along with thebest known bounds for the Riemann zeta function close to the one-line. This comes atthe price of losing density in the asymptotic formula and returns only a lower boundfor P pα, N, r0,∆sq. Therefore we need a new idea. The plan is to restrict the secondset of variables to primes and employ ideas from [MR] together with an analysis ofnumbers without small and large prime factors.The problem of determining the smallest ∆ for which

lim infNÑ8

P pα, N, r0,∆sq ą 0 almost surely in α

has attracted some attention, in the context of Theorem 2. Recently Aistleitner, El-Baz and Munsch [AEM] showed that for almost α there exist gaps that are at mostplogNq2c{N with c “ 1 ´ 1`log log 2

log 2« 0.086 the Erdos-Tenenbaum-Ford constant. In

this direction we note that with more effort Theorem 2 can be shown to still hold for∆ “ plogNqA{N and A ą 0 some large fixed constant.

2. Proof of Theorem 1

Since D can be covered by countably many compact sets, it suffices to considera compact subset D0 Ď D and show that almost all α P D0 satisfy (1.2). Next weobserve that P pα, N, Iq “ P pλα, N, Iq for every λ ą 0, so we can de-homogenize bysetting α2 “ 1, as the set of α P D0 with α2 “ 0 has measure 0. From now on wewrite

qαpm,nq “ α1m2 ` mn ` α3n

2

where α “ pα1, 1, α3q is contained in a compact domain D˚ inside

(2.1) tpα1, 1, α3q P R2 | 1 ď α1 ď α3u.

4 VALENTIN BLOMER AND MAKSYM RADZIWI L L

Correspondingly we write

d˚hypα “ dα1 dα3

p4α1α3 ´ 1q3{2“ dα

π3Dpαqwhere now Dpαq “ Dppα1, 1, α3qq “ π´1p4α1α3 ´ 1q1{2. Let V,W be fixed smooth,real-valued functions with compact support in p´8,8q respectively. For M,T ě 1we define

GαpM,T q :“ 1

4

ÿ

x1,x2,y1,y2PZpx1,y1q­“˘px2,y2q

W´T ¨ qαpx1, y1q ´ qαpx2, y2q

Dpαq¯V´qαpx1, y1qM2Dpαq

¯.

As the argument of V is non-negative, we define

(2.2) V ˚pxq “ δxě0V pxq.As a precursor to Theorem 1 we show the following proposition.

Proposition 3. Fix η ą 0 and suppose that

(2.3) Mη ď T ď M2´η.

Then for all ε ą 0 we haveż

D˚

ˇˇGαpM,T q ´ xV ˚p0qxW p0qM

2

T

ˇˇ2

d˚hypα !ε

M4´ 1

44`ε

T 2` M2`ε

T.

Here and in the following we denote by pf the Fourier transform of f . We postponethe proof of Proposition 3 to Section 3 and complete the proof of Theorem 1. For0 ă δ ă 1 and M,T as in (2.3) let S`

δ pM,T´1q be the set of α such that

#

pj, kq | Λj,Λk ď M2, j ­“ k, 0 ď Λj ´ Λk ď T´1(

ě p1 ` δqM2

T.

We specialize V,W to be smooth, non-negative functions such that V,W ě 1 on r0, 1sand 1 ď pV ˚p0q,xW p0q ď 1 ` δ{3. Then S`

δ pM,T q is contained in the set of α suchthat

GαpM,T q ě p1 ` δqM2

T,

so thatµhyppS`

δ pM,T´1qq ! δ´2M´η{2

if (2.3) holds and η ď 1{23. In the same way we can bound the measure of the setS´δ pM,T´1q of α such that

#

pj, kq | Λj,Λk ď M2, j ­“ k, 0 ď Λj ´ Λk ď T´1(

ď p1 ´ δqM2

T.

Hence if SδpN,∆q is the set of α such thatˇˇ#

pj, kq | Λj,Λk ď N, j ­“ k, 0 ď Λj ´ Λk ď ∆

(´ N∆

ˇˇ ě δN∆,

we conclude thatµhyppSδpN,∆qq ! δ´2M´η{2

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 5

uniformly in the region (1.3). Now let Sδ be the set of α such that there exists a pair ofsequences Nj ,∆j with Nj Ñ 8 and pNj ,∆jq satisfying (1.3) such that α P SδpNj,∆jqfor all j. For approximation purposes we now consider the special sequences N˚

m “p1 ` δ2qm, and ∆˚

n “ p1 ` δ2q´n, where m,n P N. If δ is sufficiently small, then foreach j there exists a pair m,n such |Nj ´ N˚

m| ! δ2Nj and |∆j ´ ∆˚n| ! δ2∆j and so

SδpNj ,∆jq Ď Sδ{2pN˚m,∆

˚nq. Here we have necessarily n ! logN˚

m. We conclude that

SδpNj ,∆jq Ďď

n!logN˚m

Sδ{2pN˚m,∆

˚nq “: S˚

δ{2pN˚mq,

and hence Sδ Ď lim supm S˚δ{2pN˚

mq. As

µhyppS˚δ{2pN˚

mqq ! pN˚mq´η{2 logN˚

m “ p1 ` δ2q´ηm{2 logp1 ` δ2qm

we conclude from the Borel-Cantelli lemma that µhyppSδq “ 0. Therefore, the set

ď

n

S1{n

has measure zero, and Theorem 1 follows.

3. Proof of Proposition 3

Let F be a smooth non-negative function with compact support in (2.1) withF pαq ě 1 for α P D˚. We write }F } :“

şF pαqdhypα. We continue to assume (1.3).

It suffices to estimate

ż

D˚

F pαqˇˇGαpM,T q ´ xV ˚p0qxW p0qM

2

T

ˇˇ2

d˚hypα !ε

M4´ 1

44`ε

T 2` M2`ε

T.

Opening the square, the proposition follows from the following two estimates

(3.1) I1pM,T q “ż

D˚

F pαqGαpM,T qd˚hypα “ }F }xV ˚p0qxW p0qM

2

T` O

´M3{2`ε

T` 1

¯

and

I2pM,T q “ż

D˚

F pαq|GαpM,T q|2d˚hypα

“ }F }xV ˚p0q2xW p0q2M4

T 2` O

´M4´ 1

44`ε

T 2` M2`ε

T

¯.

(3.2)

Before we proceed, we remark that without changing GαpM,T q we can (and will)insert a redundant function ψpx{M,y{Mq into the sum, where ψ a smooth functionthat is constantly 1 on some sufficiently large box in R

4 (depending on the supportof F ).

6 VALENTIN BLOMER AND MAKSYM RADZIWI L L

3.1. Proof of (3.1). Fourier-inverting V and W we obtain

I1pM,T q “1

4

ż

D˚

ż

R2

F pαqÿ

px1,y1q­“˘px2,y2q

ψ´ x

M,y

M

¯pV pzqxW puq

e´Tu ¨ qαpx1, y1q ´ qαpx2, y2q

Dpαq¯e´zqαpx1, y1qM2Dpαq

¯du dz

dα1 dα3

π3Dpαq3 .

Changing variables z Ð z{Dpαq, u Ð u{Dpαq and integrating over α we obtain

I1pM,T q “1

4

ż

R2

ÿ

px1,y1q­“˘px2,y2q

ψ´x

M,y

M

¯e´Tupx1y1 ´ x2y2q ` z

x1y1

M2

¯

pG´Tupx21 ´ x22q ` z

x21M2

, Tupy21 ´ y22q ` zy21M2

; z, u¯du dz

where

Gpα1, α3; z, uq “ F pαqπ3Dpαq

pV pzDpαqqxW puDpαqq

and the Fourier transform is taken with respect to the first two variables. Note thatpG is a Schwartz-class function in all variables. We change variables

(3.3) a1 “ y1 ´ y2, a2 “ y1 ` y2, b1 “ x1 ´ x2, b2 “ x1 ` x2,

so that

(3.4) a1 ” a2 pmod 2q, b1 ” b2 pmod 2q, pa1, b1q ­“ p0, 0q ­“ pa2, b2q.We obtain

(3.5) I1pM,T q “ 1

4

ÿ

a,b

˚ψ´ a

M,b

M

¯Hpa,bq

where the star indicates that the a,b-sum is subject to (3.4),

ψpa,bq “ ψ´b1 ` b2

2,b2 ´ b1

2,a1 ` a2

2,a2 ´ a1

2

¯

and

Hpa,bq “ż

R2

e´12Tupa1b2 ` a2b1q ` z

4M2pa1 ` a2qpb1 ` b2q

¯

pG´Tub1b2 ` z

4M2pb1 ` b2q2, Tua1a2 ` z

4M2pa1 ` a2q2; z, u

¯du dz.

(3.6)

Let

Cmax :“ maxp|a1|, |a2|, |b1|, |b2|q, Cmin :“ minp|a1|, |a2|, |b1|, |b2|q,P “ maxp|a1a2|, |b1b2|, |a1b2 ` a2b1|q.

By (3.4) we have Cmax, P ­“ 0. Integrating by parts with respect to u we find

(3.7) Hpa,bq !A

1

T p|a1a2| ` |b1b2|q ` 1

´1 ` |a1b2 ` a2b1|

|a1a2| ` |b1b2| ` 1{T¯´A

! 1

TP ` 1

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 7

for any A ą 0. (If a,b are integral vectors satisfying (3.4), we have TP ` 1 — TP ,but for arbitrary arguments it is important to keep the extra `1). This argumentshows that we can restrict the u-integral in (3.6) to

(3.8) TuP ! Mε

up to a negligible error.For some 1 ă C ă M to be determined later we first estimate the contribution

of tuples pa,bq with Cmin ď C to (3.5). Assume without loss of generality thatCmin “ |a1|. If a1 “ 0, we distinguish the cases b2 ­“ 0 (in which case b1b2 ­“ 0 by(3.4)) and b2 “ 0 (in which case Hpa,bq is negligible by (3.7)). It is then easy to seethat we get a contribution of

! MplogMq2T

to (3.5). If |a1| ą 0, then by (3.7) we obtain a contribution

ÿ

0­“|a1|ďC

ÿ

0­“a2,b1,b2!M

1

T |b1b2| ! M1`εC

T.

We now attach a smooth cut-off function Ψpa{C,b{Cq to the pa,bq-sum in (3.5)where Ψ is supported on p´8,´1{2s Y r1{2,8q in each variable and constantly 1 onp´8,´1s Y r1,8q in each variable. By the above remarks we thus obtain

I1pM,T q “ 1

4

ÿ

a,b

˚Ψ´a

M,b

M

¯ψ´a

M,b

M

¯Hpa,bq ` O

´M1`εC

T

¯.

Recalling (3.8) and (3.7), we conclude

›››∇´Ψ´ a

C,b

C

¯ψ´ a

M,b

M

¯Hpa,bq

¯››› !ż

u!Mε{TP

´T |u|Cmax ` Cmax

M2

¯du` 1

CTP

! MεCmax

P

1

TP` 1

CTP! Mε

CTP.

By the Euler-MacLaurin formula we obtain altogether

I1pM,T q “ 1

4¨ 14

ż

R4

Ψ´a

C,b

C

¯ψ´a

M,b

M

¯Hpa,bqdpa,bq ` O

´M2`ε

CT` M1`εC

T

¯.

We can now remove Ψpa{C,b{Cq in the same way as we introduced it at the cost ofan error

!ż

|a1|ďC

ż

|a1|!a2,b1,b2!M

1

T |b1b2| ` 1dpa,bq ! M1`εC

T

by (3.7). Choosing C “ M1{2, we obtain

I1pM,T q “ 1

4¨ 14

ż

R4

ψ´a

M,b

M

¯Hpa,bqdpa,bq ` O

´M3{2`ε

T

¯.(3.9)

8 VALENTIN BLOMER AND MAKSYM RADZIWI L L

At this point we revert all transformations. We change variables pa,bq back to px,yqusing (3.3), undo the Fourier inversion with respect to α, the linear changes of vari-ables of pz, uq, and the Fourier inversion with respect to u, z. In this way the mainterm in (3.9) equals

1

4

ż

D˚

ż

R4

F pαqψ´ x

M,y

M

¯W

´T ¨ qαpx1, y1q ´ qαpx2, y2q

Dpαq¯V´qαpx1, y1qM2Dpαq

¯dx dy dhypα.

We drop ψ because it is redundant. Let

x1 “ x` y

2α1

, x2 “ x1α1{21

p4α1α3 ´ 1q1{4, y1 “ y

p4α1α3 ´ 1q1{4

p4α1q1{2.

Thenqαpx, yqDpαq “ π

α1px1q2 ` p4α1α3 ´ 1qy2{p4α1qp4α1α3 ´ 1q1{2

“ π`px2q2 ` py1q2

˘.

Changing variables, we obtain

}F }ż

R4

W´Tπpx21 ` y21 ´ x22 ´ y22q

¯V´π

px21 ´ y21qM2

¯dx dy.

Changing to polar coordinates and recalling (2.2), we get

}F }ż

r0,8q2W

´T pr1 ´ r2q

¯V´ r1

M2

¯dr2 dr1

“ }F }M2

T

ż 8

0

ż M2r1{T

´8

W pr2qV pr1qdr2 dr1 “ }F }M2

T

´xW p0qxV ˚p0q ` O

´ T

M2

¯¯.

Together with (3.9) this establishes (3.1).

3.2. Proof of (3.2). This follows to some extent the analysis in the proof of [ABR,Proposition 5]. By Fourier inversion we have

I2pT,Mq “ 1

16

ż

D˚

F pαqÿ

px1,y1q­“˘px2,y2qpx3,y3q­“˘px4,y4q

ψ´ x

M,y

M

¯ ż

R4

pV pz1qpV pz2qxW puqxW pvq

e

ˆTu

qαpx1, y1q ´ qαpx2, y2qDpαq

˙e

ˆTv

qαpx3, y3q ´ qαpx4, y4qDpαq

˙

e

ˆz1qαpx1, y1q ´ z2qαpx3, y3q

M2Dpαq

˙dpu, v, z1, z2q

dα

π3Dpαq3 ,

where ψ is a similar redundant function as before. We change variables zj Ð zj{Dpαq,u Ð u{Dpαq, v Ð v{Dpαq, integrate over α and introduce the variables

a1 “ y1 ´ y2, a2 “ y1 ` y2, a3 “ x3 ´ x4, a4 “ x3 ` x4,

b1 “ x1 ´ x2, b2 “ x1 ` x2, b3 “ y3 ´ y4, b4 “ y3 ` y4,

obtaining

(3.10) I2pT,Mq “ 1

16

ÿ 1

a,b

ψ

ˆa

M,b

M

˙Hpa,bq

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 9

whereř1 indicates the conditions

(3.11) a1 ” a2 pmod 2q, b1 ” b2 pmod 2q, a3 ” a4 pmod 2q, b3 ” b4 pmod 2q

as well as

(3.12) pa1, b1q ­“ p0, 0q ­“ pa2, b2q, pa3, b3q ­“ p0, 0q ­“ pa4, b4q.

Moreover,

ψpa,bq :“ ψ´b1 ` b2

2,b2 ´ b1

2,a3 ` a4

2,a4 ´ a3

2,a1 ` a2

2,a2 ´ a1

2,b3 ` b4

2,b4 ´ b3

2

¯“ ψpx,yq

and Hpa,bq is defined by

ż

R4

e´Tu

2pa1b2 ` a2b1q ` Tv

2pa3b4 ` a4b3q ` z1pa1 ` a2qpb1 ` b2q ´ z2pa3 ` a4qpb3 ` b4q

4M2

¯

pG´Tub1b2 ` Tva3a4 ` z1pb1 ` b2q2 ´ z2pa3 ` a4q2

4M2,

Tua1a2 ` Tvb3b4 ` z1pa1 ` a2q2 ´ z2pb3 ` b4q24M2

; z1, z2, u, v¯dpu, v, z1, z2q

with

Gpα1, α3; z1, z2, u, vq “ π´3DpαqF pαqpV pz1DpαqqpV pz2DpαqqxW puDpαqqxW pvDpαqq

and the Fourier transform pG of G is taken with respect to the first two variablesα1, α3. We have

D pGpU, V ; z1, z2, u, vq!A,D

`p1 ` |U |qp1 ` |V |qp1 ` |z1|qp1 ` |z2|qp1 ` |u|qp1 ` |v|q

˘´A(3.13)

for all A ą 0 and any differential operator D with constant coefficients. Put

P “ maxp|a1a2|, |a3a4|, |b1b2|, |b3b4|q,∆ “ a1a2a3a4 ´ b1b2b3b4,

∆1 “ a1a2b3a4 ` a1a2a3b4 ´ a1b2b3b4 ´ b1a2b3b4,

∆2 “ a1b2a3a4 ` b1a2a3a4 ´ b1b2a3b4 ´ b1b2b3a4.

We see immediately that the contribution of P “ 0 to (3.10) is negligible by (3.12),since in this case a1a2 “ b1b2 “ a3a4 “ b3b4 “ 0, but a1b2 ` a2b1 ­“ 0 ­“ a3b4 ` a4b3, sothat repeated partial integration in u or v saves as many powers of T as we wish (andT Ñ 8 by (2.3)). From now on we restrict to P ­“ 0. If ∆ ­“ 0, we change variables

u “ a3a4V ´ b3b4U

∆, v “ a1a2U ´ b1b2V

∆

10 VALENTIN BLOMER AND MAKSYM RADZIWI L L

to see that Hpa,bq equals

1

T 2|∆|

ż

R4

e´U∆1

2∆` V∆2

2∆` z1pa1 ` a2qpb1 ` b2q ´ z2pa3 ` a4qpb3 ` b4q

4M2

¯

pG´U ` z1pb1 ` b2q2 ´ z2pa3 ` a4q2

4M2, V ` z1pa1 ` a2q2 ´ z2pb3 ` b4q2

4M2;

z1, z2,a3a4V ´ b3b4U

∆T,a1a2U ´ b1b2V

∆T

¯dpU, V, z1, z2q.

(3.14)

By (3.13) and repeated integration by parts we conclude

(3.15) Hpa,bq !A

1

T 2p|∆| ` P {T q´´

1 ` |∆1||∆| ` P {T

¯´1 ` |∆2|

|∆| ` P {T¯¯´A

for any A ě 0 and all a,b ! M satisfying P ­“ 0. This remains true for ∆ “ 0, inwhich case we change variables

ua1a2 ` vb1b2 “ U

so that ub1b2 ` va3a4 “ Ub1b2{a1a2 and apply the same argument, cf. also [ABR,(2.13)]. We also conclude that

(3.16)››∇Hpa,bq

›› ! 1

T 2|∆|´ 1

M` M3

∆

¯.

We finally observe the trivial bound

(3.17) Hpa,bq ! 1,

valid for all real a,b (even in the case P “ 0). Fix 0 ă δ ă 1{2. We claim thatthe error from dropping terms in I2pM,T q in (3.10) with ∆ ! M4´δ is small, moreprecisely (recall (3.15))

#tpa,bq satisfying (3.12) | a,b ! M,∆ ď M4´δ,∆1,∆2 ď Mεp|∆| ` P {T quT 2p|∆| ` P {T q

! Mε´M4´δ{5

T 2` M2

T

¯.

(3.18)

This is the analogue of [ABR, Proposition 5]. We postpone the proof of (3.18) to thenext subsection. Let φ be a smooth function with support on r1{2,8s that is 1 onr1,8s, and write

Φpa,bq :“ ψ

ˆa

M,b

M

˙φ

ˆ |∆|M4´δ

˙,

so that

I2pT,Mq “ 1

16

ÿ 1

a,b

Φpa,bqHpa,bq ` O´Mε

´M4´δ{5

T 2` M2

T

¯¯.

Note that the condition (3.12) is now void. From (3.15) and (3.16) we conclude

››∇Φpa,bqHpa,bq›› ! 1

T 2|∆|´ 1

M1´δ` M3

∆

¯! 1

T 2M5´2δ.

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 11

By the Euler-MacLaurin formula we conclude

ÿ 1

a,b

Φpa,bqHpa,bq “ 1

16

ż

R4

ż

R4

Φpa,bqHpa,bqda db ` O´M3`2δ

T 2

¯,

where the factor 1{16 comes from the congruences (3.11). We now re-insert thecontribution of the terms ∆ ! M4´δ into the integral by dropping the cut-off functionφp|∆|{M4´δq, and we claim that this introduces an error of at most

(3.19)

ż

∆!M4´δ

ψ

ˆa

M,b

M

˙Hpa,bqdpa,bq ! M4´δ{5`ε

T 2,

which is already present. Again we postpone the proof and revert all steps as inthe previous subsection and in the end of [ABR, Section 2], namely the change ofvariables px,yq ÞÑ pa,bq, the integration over α and the Fourier inversions. In thisway we finally obtain that I2pM,T q equals

1

16

ż

D˚

F pαqż

R4

ż

R4

ψ´ x

M,y

M

¯W

ˆqαpx1, y1q ´ qαpx2, y2q

Dpαq

˙

W

ˆqαpx3, y3q ´ qαpx4, y4q

Dpαq

˙V

ˆqαpx1, y1qM2Dpαq

˙V

ˆqαpx3, y3qM2Dpαq

˙dx dy dhypα

` O´Mε

´M4´δ{5

T 2` M2

T` M3`2δ

T 2

¯¯.

Here we can drop the function ψpx{M,y{Mq because it is redundant. We chooseδ “ 5{11. By the same change of variables as in the previous subsection we obtain

I2pT,Mq “ }F }ż

R8

W´Tπpx21 ` y21 ´ x22 ´ y22q

¯W

´Tπpx23 ` y23 ´ x24 ´ y42q

¯

V´π

px21 ´ y21qM2

¯V´π

px23 ´ y23qM2

¯dx dy ` O

´M43{11`ε

T 2` M2`ε

T

¯.

The main term equals

}F }M4

T 2

ż 8

0

ż M2r1{T

´8

ż 8

0

ż M2r3{T

´8

W pr2qV pr1qW pr4qV pr3qdr4 dr3 dr2 dr1

“}F }M4

T 2

´xW p0qxV ˚p0q ` O

´ T

M2

¯¯2

,

and (3.2) follows. It remains to prove (3.18) and (3.19) to which the following twosubsections are devoted.

3.3. Proof of (3.18). We use the notation X ď Y to mean X ! MεY . We put allvariables into dyadic intervals and suppose that

A1 ď |a1| ď 2A1, . . . , A4 ď |a4| ď 2A4, B1 ď |b1| ď 2B1, . . . , B4 ď |b4| ď 2B4

with 0 ď A1, . . . , B4 ! M . We also assume

(3.20) D ď |∆| ď 2D ! M4´δ.

12 VALENTIN BLOMER AND MAKSYM RADZIWI L L

We now count the number N pA,B, Dq of 8-tuples pa,bq subject to these size condi-tions as well as (3.12) and

(3.21) ∆1,∆2 ď |∆| ` P {T.We start with some degenerate cases and denote by N0pA,B, Dq the contribution ofA1 ¨ ¨ ¨A4B1 ¨ ¨ ¨B4D “ 0.Let us first assume that some of the a-variables vanish, but none of the b-variables.

Then D ­“ 0, and by a divisor bound, this contribution is ď DM3. If some a-variablevanishes, say a1, and also some b-variable vanishes (which cannot be b1), say b2, then|b1a2| ě 1, ∆ “ 0, P “ maxp|a3a4|, |b3b4|q, ∆1 “ b1a2b3b4, ∆2 “ b1a2a3a4, which isimpossible by (2.3) and (3.21).So from now on we will assume a1 ¨ ¨ ¨ a4b1 ¨ ¨ ¨ b4 ­“ 0. Let us next assume D “ 0,

i.e. a1 ¨ ¨ ¨ a4 “ b1 ¨ ¨ ¨ b4. By a divisor bound contributes ď PM2. So from now on weassume D ­“ 0, and we have shown

(3.22) N0pA,B, Dq ď DM3 ` PM2.

Let N˚pA,B, Dq denote the contribution with A1 ¨ ¨ ¨A4B1 ¨ ¨ ¨B4D ­“ 0 and let uswrite minpA1, . . . , A4, B1, . . .B4q “ M1´η, say, with 0 ă η ă 1. We have trivially

N˚pA,B, Dq ď DminpA1A2A3A4, B1B2B3B4q ! DM4´η.

We consider another degenerate case, namely a2a4 “ b2b4, In this case let d “a1a3 ´ b1b3 “ ∆{pb2b4q ­“ 0. We have ď DM2 choices for pd, b2, b4, a1, a3q, and thenb1, b3 are determined by a divisor argument. This can be absorbed in the existingcount (3.22), so that from now we assume a2a4 ­“ b2b4 and similarly a2a3 ­“ b2b3.Substituting b1 “ pa1a2a3a4 ´ ∆q{pb2b3b4q into the definition of ∆1, we obtain

∆1 “ ´a1b2

pa2a3 ´ b2b3qpa2a4 ´ b2b4q ` a2

b2∆

so

(3.23) pa2a3 ´ b2b3qpa2a4 ´ b2b4q ďB2

A1

´D ` P

T

¯` A2

A1

!´D ` P

T

¯Mη.

By a standard lattice point argument [Sa, p. 200-201] the number of such (non-zero)6-tuples is

ď pD ` P {T qM2`η.

Having fixed a2, a3, a4, b2, b3, b4, we are left with pairs pa1, b1q satisfying

a1 “ b1b2b3b4

a2a3a4` O

´ D

A2A3A4

¯

so that we obtain in total

N pA,B, Dq ď DM3 ` PM2 ` min´DM4´η,

´D ` P

T

¯M3`η

´ D

M3´3η` 1

¯¯

ď DM3 ` PM2 `´D ` P

T

¯`M7{2 ` D1{5M16{5

˘

!´D ` P

T

¯`M4´δ{5 ` M2T

˘

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 13

by (3.20) (and since δ ă 1{2). The quantity on the left hand side of (3.18) is

ď maxA,B,D

N pA,B, DqT 2pD ` P {T q

and so (3.18) follows.

3.4. Proof of (3.19). Here we must estimateż

∆!M4´δ

ψ

ˆa

M,b

M

˙Hpa,bqdpa,bq

based on the bounds (3.15) and (3.17). This is the (much simpler) continuousanalogue of the previous subsection. We put all variables into dyadic intervalsAj ď |aj | ď 2Aj, Bj ď |bj | ď 2Bj , D ď |∆| ď 2D, ∆1,∆2 ď ∆ ` P . Thenumbers Aj, Bj , D run through logarithmically many positive and negative powersof 2 and are bounded by M´100 ď Aj, Bj ! M , M´100 ď D ! M4´δ. (If one ofthese is ! M´100, the coarsest trivial estimates suffice - this is the only point where(3.17) is needed.) We call the corresponding set SpA,B, Dq and as before we writeminpA1, . . . , A4, B1, . . . B4q “ M1´η. We have the trivial bound

volpSpA,B, Dqq ! DM4´η`ε

On the other hand, as in (3.23) we see that the integration condition implies

pa2a3 ´ b2b3qpa2a4 ´ b2b4q ďB2

A1

´D ` P

T

¯` A2

A1

!´D ` P

T

¯Mη

An easy computation shows that the volume of such 6-tuples is ! pD`P {T qM2`η`ε,so that we obtain the alternative bound

volpSpA,B, Dqq !´D ` P

T

¯M2`η`ε ¨M ¨ D

M3´3η!´D ` P

T

¯M4`4η´δ`ε.

Combining the two bounds we obtain

volpSpA,B, Dqq ! pD ` P {T qM4´δ{5`ε

and (3.19) follows from this and (3.15).

4. Proof of Theorem 2

As in the proof of Theorem 1 we can specialize α3 “ 1, and we can restrict ourselvesto a compact set of α P R0 Ď Rą0. Thus our quadratic forms have the shapeqαpm,nq “ αm2 ` n2, m ą 0, n ě 0 with α — 1. By the same argument as in theproof of Theorem 1 it suffices to show that the measure of α P R0 such that

(4.1)ˇ#tΛi,Λj ď N | 0 ď Λj ´ Λi ď ∆, i ­“ ju ´ N∆

ˇą δN∆

is !δ,η 1{plogNq2.5 (the exponent has to be larger than 2 for the Borel-Cantelli argu-ment to work), uniformly in the range (1.4). We define

(4.2) t1 “ m1 ´ m2, t2 “ m1 ` m2, t3 “ n2 ´ n1, t4 “ n2 ` n1,

14 VALENTIN BLOMER AND MAKSYM RADZIWI L L

so that #tΛi,Λj ď N | 0 ď Λj´Λi ď ∆, i ­“ ju equals the cardinality of all quadruplespt1, t2, t3, t4q P TαpN,∆q where TαpN,∆q is defined by

pt1, t3q ­“ p0, 0q, t1 ” t2 pmod 2q, t3 ” t4 pmod 2q, t2 ą |t1|, t4 ě |t3|,

α´t1 ˘ t2

2

¯2

`´t3 ˘ t4

2

¯2

ď 4?αN

π, 0 ď αt1t2 ´ t3t4 ď 4

?α∆

π.

For ρ ą 0 let

SρpNq “!n P N | n has a prime divisor in

`exppplogNqρq, exppplogNq1´ρq

‰).

Let T ρ,Aα pN,∆q be the set of pt1, . . . , t4q P TαpN,∆q such that

‚ at least one of t1, . . . , t3 is in SρpNq (we make no assumption on t4) and‚ all tj satisfy |tj | ě N1{2plogNq´A.

We claim that the contribution of pt1, . . . , t4q P TαpN,∆qzT ρ,Aα pN,∆q is negligible.

Proposition 4. Let A ą 10 and ρ ă 1{20. Then the measure of all α P R0 such that

#`TαpN,∆qzT ρ,A

α pN,∆q˘

ě 1

2∆N

is ! plogNq´2.5.

We postpone the proof to Section 5.By symmetry we can assume that t1 ě 0 so that automatically t3 ě 0. Moreover,

we drop the conditions t2 ą t1, t4 ě t3. In this way we see that #T ρ,Aα pN,∆q is 1/2

times the cardinality of all pt1, . . . , t4q such that

t1 ” t2 pmod 2q, t3 ” t4 pmod 2q, tj ě N1{2plogNq´A

α´t1 ˘ t2

2

¯2

`´t3 ˘ t4

2

¯2

ď 4?αN

π, 0 ď αt1t2 ´ t3t4 ď 4

?α∆

π

and at least one of the t1, t2, t3 is in SρpNq.It is convenient to count a slightly smaller and a slightly large quantity. Fix some

δ1 ą 0. We restrict the ti to boxes Di ă ti ď p1 ` δ1qDi, where the Di run throughplog logXqOδ1p1q values of a sequence of the type Dp1 ` δ1qj . We can and will assume

(4.3)1

2ď αD1D2

D3D4

ď 2.

For the smaller count we only consider tuples pD1, . . . , D4q satisfying (4.3) suchthat the boxes lie completely inside the region

(4.4) α´t1 ˘ t2

2

¯2

`´t3 ˘ t4

2

¯2

ď 4?αX

π, t1, . . . , t4 ě 0.

We call the collection of such quadruples D´. For 0 ď θ ď 1 we note that

log´t3t4t1t2

` θ4?α∆

t1t2π

¯“ log

t3t4

t1t2` θ

4?α∆

t3t4π` O

´ θ2∆2

D23D

24

¯.

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 15

Therefore we sharpen the inequality

(4.5) 0 ď αt1t2 ´ t3t4 ď 4?α∆

π

to

0 ď logα ´ logt3t4

t1t2ď p1 ´ 3δ1q 4

?α∆

πD3D4

,

and we detect this with a smooth weight function

W´

´´logα ´ log

t3t4

t1t2

¯πD3D4

4?α∆

¯.

where W´ is constantly 1 on rδ1, 1 ´ 4δ1s and vanishes outside r0, 1 ´ 3δ1s.For the larger count we consider boxes satisfying (4.3) that intersect the part of

(4.4) where ti ě N1{2plogNq´A, calling this larger collection of quadruples D`. Werelax (4.5) to

0 ď logα ´ logt3t4

t1t2ď p1 ` δ1q 4

?α∆

πD3D4

and detect it with a factor

W`

´´logα ´ log

t3t4

t1t2

¯πD3D4

4?α∆

¯.

where W` is constantly 1 on r0, 1s and vanishes outside r´δ1, 1 ` δ1s. Eventually wewill choose δ1 “ cδ for some sufficiently small c. For notational simplicity we write

D1j “ Djp1 ` δ1q.

We summarize that

#T ρ,Aα pN,∆q ď 1

2

ÿ

pD1,...,D4qPD`

ÿ

tiPpDi,D1is

some t1,t2,t3PSρpNq2|t1´t2,t3´t4

W`

´´logα ´ log

t3t4

t1t2

¯πD3D4

4?α∆

¯

“ 1

2

ÿ

pD1,...,D4qPD`

4?α∆

πD3D4

ż

R

xW`

´4?α∆y

πD3D4

¯ ÿ

tiPpDi,D1is

some t1,t2,t3PSρpNq2|t1´t2,t3´t4

´αt1t2t3t4

¯2πiy

dy,

and we have a similar lower bound where the subscripts ` are replaced with thesubscripts ´. We consider only the ` case, the other one being identical. We extractthe main term from small values of y in the integral. Let V be a smooth non-negativefunction that is 1 on r´1, 1s and vanishes for |x| ą 2. Let B ą 3A, define

(4.6) Y :“ Y 1D3D4

N, Y 1 “ plogNqB

and decompose the previous y-integral as I1pαq`I2pαq where I1pαq contains the factorV py{Y q and I2pαq contains the factor 1 ´ V py{Y q. We claim

16 VALENTIN BLOMER AND MAKSYM RADZIWI L L

Proposition 5. We have

1

2

ÿ

pD1,...,D4qPD`

4?α∆

πD3D4

I1pαq “ p1 ` Opδ1qq∆N

where the implied constant depends only on ρ, A,B and R0.

Proposition 6. For suitable choices of A,B we haveż

R0

ˇˇ12

ÿ

pD1,...,D4qPD`

4?α∆

πD3D4

I2pαqˇˇ2dα

α! ∆2N2plogNq´50

where the implied constant depends only on ρ and R0.

We postpone the proofs to Sections 6 and 7 respectively. From these two proposi-tions we obtain that the measure of α P R0 such that

ˇ#T ρ,A

α pN,∆q ´ ∆Nˇ

" δ1∆N

is OpplogNq´50q. We choose δ1 “ δ{c where c is sufficiently large in terms of the im-plied constant in Propositon 5. Together with Proposition 4 we see that the measureof α P R0 satisfying (4.1) is indeed ! plogNq´2.5 which completes the proof.

5. Proof of Proposition 4

We recall that TαpN,∆qzT ρ,Aα pN,∆q is contained in the set of all pt1, . . . , t4q such

that

pt1, t3q ­“ p0, 0q, t2 ą |t1|, t4 ě |t3|,

α´t1 ˘ t2

2

¯2

`´t3 ˘ t4

2

¯2

! N, 0 ď αt1t2 ´ t3t4 ! ∆(5.1)

and none of t1, t2, t3 is in SδpNq or at least one tj satisfies |tj | ď N1{2plogNq´A.Let us first consider the contribution T1pαq of those quadruples, where at least one

ti, say t1, satisfies |t1| ď N1{2plogNq´A. We observe directly that the contributionof t1t2t3t4 “ 0 is bounded by OpN1{2q if ∆ " 1 and vanishes otherwise, so it is! N1{2∆. We assume from now on that all tj are non-zero, and without loss ofgenerality positive. We localize each ti in a dyadic interval of the shape pDi, 4Dis;there are at most OpplogNq4q such boxes, and we must have

(5.2) D1D2 — D3D4 ! NplogNq´A

and moreover ˇˇ logα` log

t1t2

t3t4

ˇˇ ! ∆

D3D4

.

If W denotes a suitable non-negative smooth compactly supported function, we con-clude that

T1pαq !ÿ

D1,D2,D3,D4

ÿ

tiPpDi,4Dis

W´D3D4

∆

´logα ` log

t1t2

t3t4

¯¯` N1{2∆

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 17

where the outer sum runs over powers of 2 subject to (5.2). Therefore the measureof α P R0 such that T1pαq ą ∆N{2 is at most

ż

R0

1

∆N

ÿ

D1,D2,D3,D4

ÿ

tiPpDi,4Dis

W´D3D4

∆

´logα ` log

t1t2

t3t4

¯¯dαα.

We write β “ logα, insert the weight e´β2{2 and extend the integration to all of R.By Fourier inversion we bound the previous display by

ÿ

D1,D2,D3,D4

1

∆N

∆

D3D4

ż

R

e´u2{2xW´ ∆u

D3D4

¯F pu,D1qF pu,D2qF p´u,D3qF p´u,D4qdu

where

F pu,Dq “ÿ

Dătď4D

1

t2πiu.

By trivial estimates and (5.2), the above is

!ÿ

D1,D2,D3,D4

D1D2

N! plogNq4´A.

Let now T2pαq denote the set of all quadruples satisfying (5.1) with |tj | ě N1{2plogNq´A,but none of t1, t2, t3 is in SρpNq. Again we localize each ti in a dyadic interval ofthe shape pDi, 4Dis; by our current assumptions, there are at most OApplog logNq4qsuch boxes. By the same argument as before, the measure of α P R0 such thatT2pαq ą ∆N{2 is at most

ÿ

D1,D2,D3,D4

1

∆N

∆

D3D4

ż

R

e´u2{2xW´ ∆u

D3D4

¯

Fρ,Npu,D1qFρ,Npu,D2qFρ,Np´u,D3qF p´u,D4qdu(5.3)

where

Fρ,Npu,Dq “ÿ

Dătď4DtRSρpNq

1

t2πiu.

At this point we invoke [We, Corollary 1.2(iii)] with y “ exppplogNq1´ρq, z “exppplogNqρq, u “ plogNqρ, r “ plogNq2ρ´1 in the form

(5.4)ÿ

nďDnRSρpNq

1 “ DPN ` O`D expp´plogNqρ{2q

˘

where

PN “ź

exppplogNqρqăpďexppplogNq1´ρq

´1 ´ 1

p

¯! plogNq2ρ´1,

uniformly in N1{3 ! D ! N1{2, say. By trivial bounds, (5.3) is at most

! plog logNq4plogNq6ρ´3,

and the proof is complete.

18 VALENTIN BLOMER AND MAKSYM RADZIWI L L

6. Proof of Proposition 5

The aim of this section is an asymptotic evaluation of

(6.1)1

2

ÿ

pD1,...,D4qPD`

4?α∆

πD3D4

ż

R

xW`

´4?α∆y

πD3D4

¯V´ yY

¯ ÿ

tiPpDi,D1is

some tiPSρpNq2|t1´t2,t3´t4

´αt1t2t3t4

¯2πiy

dy

with Y as in (4.6). By a Taylor argument we have

xW`

´4?α∆y

πD3D4

¯“ xW`p0q ` Oδ1

´ ∆Y

D3D4

¯.

(The definition of W˘ depends on δ1.) By inclusion-exclusion we can detect thecondition that some ti P SρpNq by an alternating sum of terms where a certainsubset of the ti is in SρpNq while the complement is unrestricted. All of these termsare handled in the same way, so for notational simplicity let us focus on the caset1 P SρpNq, t2, t3, t4 unrestricted. We recall the notation D1 “ Dp1 ` δ1q and define

(6.2) Gpy,Dq “ÿ

DătďD1

1

t2πiy, Gρ,Npy,Dq “

ÿ

DătďD1

tPSρpNq

1

t2πiy.

By partial summation we have

Gpy,Dq “ G˚p1 ´ 2πiy,D1, Dq ` OpY q, G˚pz,D1, D2q “ Dz1 ´ Dz

2

z

and using (5.4), we also have

Gρ,Npy,Dq “ p1 ´ PNqG˚p1 ´ 2πiy,Dp1 ` η1q, Dq ` O`Y D expp´plogNq1´2ρq

˘

for y ! Y . Let

Φρ,Npy,D1, D2q “ Gρ,Np´y,D1qGp´y,D2q, Φpy,D3, D4q “ Gpy,D3qGpy,D4q.

Detecting the congruence condition modulo 2, we have

ÿ

tiPpDi,D1is

t1PSρpNq2|t1´t2,t3´t4

´t1t2t3t4

¯2πiy

“´Φρ,Xpy,D1, D2q ´ Φρ,Xpy, 1

2D1, D2q ` Φρ,Xpy,D1,

12D2q

2´2πiy` 2Φρ,Xpy, 1

2D1,

12D2q

4´2πiy

¯

´Φpy,D3, D4q ´ Φpy, 1

2D3, D4q ` Φpy,D3,

12D4q

22πiy` 2Φpy, 1

2D3,

12D4q

42πiy

¯.

(6.3)

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 19

Substituting all of this, we recast (6.1) as

1

2

ÿ

pD1,...,D4qPD`

t4?α∆

πD3D4

xW`p0qp1 ´ PNq

ż

R

V´ yY

¯α2πiy

2

2ź

j“1

G˚p1 ` 2πiy,D1j, Djq

4ź

j“3

G˚p1 ´ 2πiy,D1j, Djqdy

` O`Y 2∆N expp´plogNqρ{2q

˘.

(6.4)

DefineHpt, D1, D2q “ δlogD2ătďlogD1

et.

Then

pHpy,D1, D2q “ż

R

Hpt, D1, D2qeptyqdt “ G˚p1 ` 2πiy,D1, D2q.

Thus the main term in (6.4) becomes

ÿ

pD1,...,D4qPD`

?α∆

πD3D4

xW`p0qp1 ´ PNqż

R

V´ yY

¯ ż logD11

logD1

ż logD12

logD2

ż logD13

logD3

ż logD14

logD4

et1`t2`t3`t4epplogα ` t1 ` t2 ´ t3 ´ t4qyqdt1 dt2 dt3 dt4 dy.

Changing variables and recalling (4.6), this equals

ÿ

pD1,...,D4qPD`

?α∆Y 1

πNxW`p0qp1 ´ PNq

ż D11

D1

ż D12

D2

ż D13

D3

ż D14

D4

pV´Y log

αt1t2

t3t4

¯dt1 dt2 dt3 dt4.

(6.5)

By the rapid decay of pV we can restrict to logαt1t2{t3t4 ! Y ´3{4 at the cost of a totalerror ∆XY ´100. By a Taylor argument we then have

Y logαt1t2

t3t4“ Y

t3t4pαt1t2 ´ t3t4q ` OpY ´1{2q

“ Y 1

N

D3D4

t3t4pαt1t2 ´ t3t4q ` OpY ´1{2q “ p1 ` Opδ1qqY

1

Npαt1t2 ´ t3t4q ` OpY ´1{2q.

Using also xW`p0qp1 ´ PXq “ 1 ` Opδ1q and defining

D` “ď

pD1,...,D4qPD`

4ą

i“1

rDi, D1is,

we recast the main term (6.5) as

p1 ` Opδ1qq?α∆Y 1

πN

ż

D`

pV´Y 1

Npαt1t2 ´ t3t4q

¯dt.

We now add to the integration domain D` those points of (4.4) with minpt1, . . . , t4q ďN1{2plogNq´A. It is easy to see (e.g. by putting the variables into dyadic boxes) that

20 VALENTIN BLOMER AND MAKSYM RADZIWI L L

this infers an error of at most Op∆NplogNq2´Aq. Next we replace the integrationdomain with the exact region (4.4), the error of which can be absorbed in the existingp1 ` Opδ1qq-term. By symmetry we can assume t2 ě t1 and t4 ě t3 after multiplyingby 4, and we drop the condition t1 ě 0 at the cost of dividing by 2 and note that fornegative t1 we automatically have t3 ď 0 up to a negligible error. Finally we reversethe change of variables (4.2) (the Jacobian infers a factor 2) getting

4?α

πp1 ` Opδ1qq∆Y

1

N

żpV´Y 1

Npαn2

1 ` m21 ´ pαn2

2 ` m22qq

¯dn1 dn2 dm1 dm2.

where the integration is taken over

αn2j ` m2

j ď 4?αN

π, n1, n2, m1, m2 ě 0.

Changing variables, this equals

p1 ` Opδ1qq∆Y1

N

ż N

0

ż N

0

pV´Y 1

Npr1 ´ r2q

¯dr2 dr1

“ p1 ` Opδ1qq∆Y1

N

ż X

0

ż N´r1

´r1

pV´

´ Y 1

Nr2

¯dr2 dr1.

The portion r1 ď NpY 1q´3{4 and r1 ě N ´ NpY 1q´3{4 is negligible, in the remaining

part we can extend the r2-integral to all of R by the rapid decay of pV and finallyobtain

p1 ` Opδ1qq∆ż N´NpY 1q´3{4

NpY 1q´3{4

V p0qdr1 “ p1 ` Opδ1qq∆N

as desired.

7. Proof of Proposition 6

We haveż

R0

ˇˇ12

ÿ

pD1,...,D4qPD`

4?α∆

πD3D4

I2pαqˇˇ2dα

α

! supDi

∆2plog logNq8D2

3D24

ż

R0

´ ż

|y|ěY

ˇˇxW`

´4?α∆y

πD3D4

¯ˇˇˇˇ

ÿ

tiPpDi,D1is

some tiPSρpNq2|t1´t2,t3´t4

´αt1t2t3t4

¯2πiy ˇˇdy

¯2dα

α.

Again we treat the case t1 P SρpNq and t2, t3, t4 unrestricted, the other cases beingsimilar, and we note that t4 is unrestricted in all cases. We can detect the parityconditions in the same way as in (6.3) which replaces potentially some Di by

12Di.

Finally we replace the function xW`

`4?α∆y{πD3D4

˘by a non-negative majorant

W ˚p∆y{D3D4q that is independent of α, where W ˚ is some suitable Schwartz class

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 21

function. Opening the square and integrating over α, it suffices to bound(7.1)

supDi

∆2plogNqεD2

3D24

ż

|y|ěY

ˇˇW ˚

´ ∆y

D3D4

¯ˇˇ2

|Gρ,N py,D1qGpy,D2qGpy,D3qGpy,D4q|2dy

using the notation (6.2). We now manipulate Gρ,Npy,D1q similarly as in [MR, Lemma12]. Let I “ pexppplogNqρqq, exppplogNq1´ρqs and ωIpmq be the number of primedivisors of m in I. We have

Gρ,Npy,Dq “ÿ

DănďD1

nPSρpNq

1

n2πiy“ÿ

pPI

ÿ

D{pămďD1{pmPSρpNq

pωIpmq ` δpp,mq“1q´1

ppmq2πiy

“ÿ

pPI

ÿ

D{pămďD1{pmPSρpNq

pωIpmq ` 1q´1

ppmq2πiy

`ÿ

pPI

ÿ

D{pămďD1{pmPSρpNq,p|m

ωIpmq´1

ppmq2πiy ´ÿ

pPI

ÿ

D{pămďD1{pmPSρpNq,p|m

pωIpmq ` 1q´1

ppmq2πiy

(this is often called Ramare’s identity). We split the first p-sum into Opκ´1 logNqintervals of the shape P ă p ď P p1 ` κq for

κ “ plogNq´C

and exppplogNqρqq ď P ď exppplogNq1´ρq. We argue as in [MR, Lemma 12] andwrite

ÿ

pPI

ÿ

D{pămďD1{pmPSρpNq

pωIpmq ` 1q´1

ppmq2πiy

“ÿ

P

ÿ

PăpďP p1`κqpPI

1

p2πiy

ÿ

D{pP p1`κqqămďD1{PDďmpďD1

mPSρpNq

pωIpmq ` 1q´1

m2πiy

“ÿ

P

ÿ

PăpďP p1`κqpPI

1

p2πiy

ÿ

D{PămďD1{PmPSρpNq

pωIpmq ` 1q´1

m2πiy`

ÿ

mPJ

dm

m2πiy

22 VALENTIN BLOMER AND MAKSYM RADZIWI L L

for certain |dm| ď 1, where J “ rD{p1 ` κq, Dp1 ` κqs Y rD1, D1p1 ` κqs and P runsover a sequence of the type P0p1 ` κqj. We substitute this back into (7.1) getting

supDi

∆2plogNqεκ´1 logN

D23D

24

ÿ

P

ż

|y|ěY

ˇˇW ˚

´ ∆y

D3D4

¯ˇˇ2

|QP pyqRP py,D1qGpy,D2qGpy,D3qGpy,D4q|2dy

` supDi

∆2plogNqεD2

3D24

ż

R

ˇˇW ˚

´ ∆y

D3D4

¯V py,D1qGpy,D2qGpy,D3qGpy,D4q

ˇˇ2

dy

` supDi

∆2plogNqεD2

3D24

ż

R

ˇˇW ˚

´ ∆y

D3D4

¯Upy,D1qGpy,D2qGpy,D3qGpy,D4q

ˇˇ2

dy

(7.2)

where

QP pyq “ÿ

PăpďP p1`κqpPI

1

p2πiy, RP py,Dq “

ÿ

D{PămďD1{P

pωIpmq ` 1q´1

m2πiy,

V py,Dq “ÿ

mPJ

dm

m2πiy

Upy,Dq “ÿ

pPI

ÿ

D{pămďD1{pmPSρpNq,p|m

ωIpmq´1

ppmq2πiy ´ÿ

pPI

ÿ

D{pămďD1{pmPSρpNq,p|m

pωIpmq ` 1q´1

ppmq2πiy .

We estimate the three terms in (7.2) separately and recall the standard mean valueestimate [IK, Theorem 9.1] for Dirichlet polynomials

(7.3)

ż T

´T

ˇˇÿ

nďX

an

nit

ˇˇ2

dt ! pT ` Xqÿ

nďX

|an|2.

For the second term in (7.2) we apply (7.3) directly getting the bound

supDi

∆2plogNqεD2

3D24

´D3D4

∆` D1D2D3D4

¯ ÿ

m1PJ

ÿ

D2ăm2ďD12

ÿ

D3ăm2ďD13

ÿ

D4ăm2ďD14

τ4pm1m2m3m4q

! supDi

∆2plogNqεD2

3D24

´D3D4

∆` D1D2D3D4

¯pD1D2D3D4κq1{2

´ ÿ

n!D1D2D3D4

τ4pnqτ4pnq2¯1{2

! p∆N ` ∆2N2qplogNq32´C{2 ! ∆2N2plogNq32´C{2

(7.4)

where we used Cauchy-Schwarz,ř

nďX τ4pnq3 ! XplogXq63 and the assumption ∆ ěN´1`η.

SMALL GAPS IN THE SPECTRUM OF TORI: ASYMPTOTIC FORMULAE 23

For the third term in (7.2) we proceed similarly, applying (7.3) directly. This givesthe bound

supDi

∆2plogNqεD2

3D24

´D3D4

∆` D1D2D3D4

¯ ÿ

pPID1ămp2ăD1

1

ÿ

D2ăm2ďD12

D3ăm2ďD13

D4ăm2ďD14

τ4pm1p2m2m3m4q

! supDi

∆2plogNqεD2

3D24

´D3D4

∆` D1D2D3D4

¯pD1D2D3D4q expp´plogNqρ{2q

! p∆N ` ∆2N2q expp´plogNqρ{4q ! ∆2N2 expp´plogNqρ{4q.

(7.5)

In the first term in (7.2) we split the y-integral into two parts. For those y withQP pyq ď P plogNq´D we use (7.3) to bound their contribution by

supDi

∆2plogNqεpκ´1 logNq2P 2

D23D

24plogNq2D

´D3D4

∆` D1D2D3D4

P

¯ ÿ

n!D1D2D3D4{P

τ4pnq2

! p∆NP ` ∆2N2qplogNq2C`18´2D ! ∆2N2plogNq2C`18´2D

(7.6)

since ∆ " N´1`η and P ď Nη{10.Now we treat the integral over the remaining y where QP pyq ě P plogNq´D.

Here we can discretize the integral and estimate it by a sum over certain pointsof distance at most 1. From [MR, Lemma 8] with T “ N2, V “ plogNqD andexppplogNqρq ď P ď exppplogNq1´ρq we conclude the that number of such points isat most ! exppplogNq1´ρ{2q. This gives the bound(7.7)

supDi

∆2plogNqεκ´1 logN

D23D

24

ÿ

P

ÿ

j

|QP pyjqRP pyj, D1qGpyj, D2qGpyj, D3qGpyj, D4q|2

where P runs over Opκ´1 logNq numbers and j runs over OpN εq numbers with Y ď|yj| ! N2. We recall from (4.6) that Y " plogNqB´2A. A standard application ofPerron’s formula and the convexity bound for the Riemann zeta function shows

Gpy,Dq ! D logN

TpT ` |y|qε ` D

1 ` |y| ` D1{2pT ` |y|q1{4`ε

for any parameter T ą 1, so that under the present conditions Gpyj, D4q ! D4Y´1.

For the remaining sum over j we use the discrete mean value theorem [MR, Lemma9] (which is [IK, Theorem 9.6]) and bound (7.7) by

supDi

∆2plogNqεpκ´1 logNq2D2

3D24

D24

Y 2pD1D2D3 `

aD1D2N

εqÿ

n!D1D2D3

τ4pnq2

! ∆2N2plogNq18´2B`4A`2C .

(7.8)

Combining (7.4) – (7.6) and (7.8) and choosing A “ 20, B “ 400, C “ 200, D “ 300,we complete the proof.

24 VALENTIN BLOMER AND MAKSYM RADZIWI L L

References

[ABR] C. Aistleitner, V. Blomer, M. Radziwi l l, Triple correlation and long gaps in the spectrum

of flat tori, arxiv:1809.07881[AEM] C. Aistleitner, D. El-Baz, M. Munsch, Difference sets and the metric theory of small gaps,

arxiv:2108.02227

[BT] M. Berry, M. Tabor, Level clustering in the regular spectrum, Proc. Roy. Soc. London A356 (1977), 375-394.

[BBRR] V. Blomer, J. Bourgain, M. Radziwi l l, Z. Rudnick, Small gaps in the spectrum of the

rectangular billiard, Ann. Sci. Ecole Norm. Sup. 50 (2017), 1283-1300.[EMM] A. Eskin, G. Margulis, S. Mozes, Quadratic forms of signature p2, 2q and eigenvalue spacings

on rectangular 2-tori, Ann. of Math. 161 (2005), 679-725.[IK] H. Iwaniec, E. Kowalski, Analytic Number Theory, AMS Colloquium Publications 53, Prov-

idence 2004[MR] K. Matomaki, M. Radziwi l l, Multiplicative functions in short intervals, Ann. of Math. 183

(2016), 1015-1056[Sa] P. Sarnak, Values at integers of binary quadratic forms. Harmonic analysis and number

theory (Montreal 1996), 181-203, CMS Conf. Proc. 21, Amer. Math. Soc., Providence, RI,1997.

[We] A. Weingartner, Integers free of prime divisors from an interval, II, Acta Arith. 104 (2002),309-343

University of Bonn, Mathematisches Institut, Endenicher Allee 60, D-53115 Bonn,

Germany

Email address : blomer@math.uni-bonn.de

Caltech, Department of Mathematics, 1200 E California Blvd, Pasadena, CA,

91125

Email address : maksym.radziwill@gmail.com