Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers

0.36 MB·

by  Tristan Freiberg

POISSON DISTRIBUTION FOR GAPS BETWEEN SUMS OF TWO
SQUARES AND LEVEL SPACINGS FOR TORAL POINT SCATTERERS
TRISTANFREIBERG,PA¨RKURLBERG,ANDLIORROSENZWEIG
Abstract. Weinvestigatethelevelspacingdistributionforthequantumspec-
trumofthesquarebilliard. ExtendingworkofConnors–Keating,andSmilan-
7
sky, we formulate an analog of the Hardy–Littlewood prime k-tuple conjec-
1
0 tureforsumsoftwosquares,andshowthatitimpliesthatthespectralgaps,
2 after removing degeneracies and rescaling, are Poisson distributed. Conse-

n quently,byworkofRudnickandUeberscha¨r,thelevelspacingsofarithmetic
a toralpointscatterers,intheweakcouplinglimit,arealsoPoissondistributed.
J
Wealsogivenumericalevidencefortheconjectureanditsimplications.
4

]
h
p 1. Introduction
-
h
t According to the Berry–Tabor conjecture [2], the energy levels for generic
a
m integrablesystemsshouldbePoissondistributedinthesemiclassicallimit. As
[ noted by Connors and Keating [5], the square billiard, though integrable, is

not generic: due to spectral degeneracies, the level spacing distribution tends
1
v toaδ-functionatzero. However,ifweremovethedegeneraciesandrescaleso
7
that the meanspacing isunity, numerics indicate Poisson spacings.
5
1
1 1
0
.
1
0.8
0
7
1
0.6
:
v
i
X
0.4
r
a
0.2
2 4 6 8
Figure 1. Rescaled gaps between consecutive energy levels in
1099,1099 110000 , after removing degeneracies. The rescaled
r ` s
gapshavemeanone;withoutrescalingthemeangapis19.42 .
¨¨¨
Number of gaps: 5663. We also plot the density function (red in
color printout) P x e´x, consistent with Poisson spacings.
p q “
The energy levels of the square billiard, say with side length 2π, are num-
ber theoretical in nature, and given by a2 b2 for a,b Z. After removing
` P
Date:January6,2017.
1
2 T.FREIBERG,P.KURLBERG,ANDL.ROSENZWEIG
degeneracies and rescaling, we are led to study the nearest neighbor spacing
distribution
1 E E
# E x : n`1 ´ n λ (1.1)
n
N x " ď x N x ă *
p q { p q
(asx ), where E denotesthe nth smallestelementofthe set
n
Ñ 8
E .. a2 b2 : a,b Z , and N x .. # E x : E E . (1.2)
n n
“ t ` P u p q “ t ď P u
(Inoursettingtheleadingorderofthedensityofstatesisasymptoticallyequal
to C ?logx as x [cf. (1.5)], and hence the spacing distribution of the
{ Ñ 8
unfoldedlevels CE ?logE hasthesameasymptoticdistributionasthe
n{ n ně1
gapsin (1.1).) ` ˘
Ratherthan studyingthe spacingdistribution directly, weshall proceed by
investigating unordered k-tuples of elements in E. Thus, given k 1 and h
ě “
h ,...,h Zwith #h k, consider the correlation function
1 k
t u Ď “
1
Rkph;xq ..“ x 1Epn`h1q¨¨¨1Epn`hkq, (1.3)
nÿďx
where1 denotestheindicatorfunctionofE. Ifh 0 ,thisistheleveldensity
E
“ t u
N x
R1 x .. p q. (1.4)
p q “ x
Bya classical resultof Landau[22],
C
R x x , (1.5)
1
p q „ ?logx p Ñ 8q
where C 0 isan explicitlygiven constant (see (2.1)). To formulate an analog
ą
of (1.5) for k 1 we need some further notation. Given a prime p 1 mod 4,
ą ı
define
# 0 a pα : h h,a h (cid:3) (cid:3) mod pα
δh p .. lim t ď ă @ P ` ” ` u. (1.6)
p q “ αÑ8 pα
(ThatthelimitexistsisshowninSection5,cf.Propositions5.3and5.2.) Further,
for k 1 and a set h h ,...,h Z with #h k, we define the singular
1 k
ě “ t u Ď “
h
seriesfor by
δ p
Sh .. hp q , (1.7)
“ δ p k
pı1źmod4 t0u
p q
with δt0u p and δh p as in (1.6). We note`that δt˘0u p 0 for all p 1 mod 4,
p q p q p q ą ı
andthattheproductconvergestoanonzerolimitifδh p 0forallp 1 mod 4
p q ą ı
(cf.Proposition5.4). Ifδh p 0forsomep 1 mod 4,wedefineShtobezero;
p q “ ı
it iseasyto seethat Rk h;x 0 forall xifSh 0.
p q “ “
We can now formulate an analog of the Hardy–Littlewood prime k-tuple
conjecture.
Conjecture1.1. Fixk 1,andaseth h1,...,hk Zwith#h k. IfSh 0,
ě “ t u Ď “ ą
then
Rk h;x Sh R1 x k x . (1.8)
p q „ p q p Ñ 8q
` ˘
POISSONSPACINGSBETWEENSUMSOFTWOSQUARES 3
Ourmainresult,Theorem1.2below,isconditionalonthehypothesisthat (1.8)
holds on average. To beprecise, letEh x be defined bythe relation
p q
Rk h;x .. Sh Eh x R1 x k. (1.9)
p q “ ` p q p q
` ˘` ˘
Further, let∆k be the region in Rk defined by
∆k .. x ,...,x Rk : 0 x x , (1.10)
1 k 1 k
“ tp q P ă ă ¨¨¨ ă u
and, given C ∆k and y R, letyC be the dilation ofC defined by
Ď P
yC .. yx1,...,yxk : x1,...,xk C .
“ tp q p q P u
OurhypothesisisthattheerrortermEh x issmallwhenaveragedoverdilates
p q
of certain bounded convex subsets.
Hypothesis (k,C,o). Fix an integer k 1 and a bounded convex set C ∆k. Set
ě Ď
o .. or set o .. 0 . Let x and y be real parameters tending to infinity in such a
“ H “ t u
way that yR x 1. There exists a function ε x , with ε x 0 as x , such
1
p q „ p q p q Ñ Ñ 8
thatforxsufficientlylargeintermsof k and C,
EoYh x ε x SoYh, (1.11)
ˇ p qˇ ď p q
ˇˇph1,...,hÿkqPyCXZk ˇˇ ph1,...,hÿkqPyCXZk
whereh h ,ˇ...,h inbothsummanˇds.
1 k
“ t u
Under the above hypothesis we find that the spacing distribution (1.1) is
indeedPoissonian. Moreover,thedistributionofthenumberofpointsininter-
valsofsizecomparabletothemeanspacingisconsistentwiththatofaPoisson
process. (We remark that our hypothesis can be weakened slightly — see Sec-
tion 4.)
Theorem 1.2. Let x and y be real parameters tending to infinity in such a way that
yR x 1. Fix integersm 0 and r 1, and fix λ,λ ,...,λ R`. Assume that
1 1 r
p q „ ě ě P
Hypothesis(k,C, 0 ))(respectively,Hypothesis(k,C, )holdsforallk 1,andall
t u H ě
bounded, convexsetsC ∆k. Then(a)(respectively,(b)) holds.
Ď
(a)We have
1 r λj
# E x : j r,E E λ y e´tdt x .
N x t n ď @ ď n`j ´ n`j´1 ď j u „ Z p Ñ 8q
p q źj“1 0
(1.12)
(b)We have
1 λm
# n x : N n λy N n m e´λ x . (1.13)
x t ď p ` q´ p q “ u „ m! p Ñ 8q
In [26], Rudnick and Ueberscha¨r considered the spectrum of “toral point
scatterers”, namely the Laplace operator, perturbed by a delta potential, on
two dimensional tori. They showed that the level spacings of the perturbed
eigenvalues,intheweakcouplinglimit,havethesamedistributionasthelevel
spacingsoftheunperturbedeigenvalues(afterremovingmultiplicities). Anin-
terestingconsequenceofConjecture1.1(or(1.11))isthusthattheBerry–Tabor
4 T.FREIBERG,P.KURLBERG,ANDL.ROSENZWEIG
conjectureholdsfortoralpointscatterers, intheweakcouplinglimit,forarith-
metic tori ofthe form R2 Z2.
{
WeremarkthatGallagher[7]provedtheanalogofTheorem1.2(b)forprimes.
Justasinhisproof,akeytechnicalresultisthatthesingularseriesisofaverage
orderone, overcertain geometric regions.
Proposition 1.3. Fix an integer k 1, and a bounded convex set C ∆k. Set
ě Ď
o .. or seto .. 0 . Asy , wehave
“ H “ t u Ñ 8
SoYh yk vol C O y´2{3`op1q , (1.14)
“ p q`
ph1,...,hÿkqPyCXZk ´ ` ˘¯
whereh h ,...,h inthesummand,and vol stands forvolumeinRk.
1 k
“ t u
Acknowledgements. We thank Z. Rudnick for stimulating discussions on
the subject matter, and D. Koukoulopoulos for his comments on an early ver-
sion of the paper. T. F. was partially supported by a grant from the Go¨ran
Gustafsson Foundation for Research in Natural Sciences and Medicine. P. K.
andL.R.werepartiallysupportedbygrantsfromtheGo¨ranGustafssonFoun-
dation for Research in Natural Sciences and Medicine, and the Swedish Re-
search Council (621-2011-5498). L. R. wishes to thank and acknowledge the
Mathematics department at KTH, being his home institute during the period
where mostof the work on this paperwas done.
2. Discussion
Connorsand Keating[5]determinedthe singularseriesforshifted pairsof
sums of two squares and gave a probabilistic derivation of Conjecture 1.1 for
k 2,andfoundthatitmatchednumericsquitewell(towithin2%). Smilansky
“
[28]thenexpressedthesingularseriesforpairsasproductsofp-adicdensities,
andshowedthatitsmeanvalue(overshortintervalsofshifts)isconsistentwith
a Poisson distribution, and that the same is true for sums of two squares, on
assuming a uniform version of Conjecture 1.1 for k 2. He also determined
“
the singular seriesfortriples corresponding to the shifts h 0,1,2 .
“ t u
As already mentioned, the analog of Theorem 1.2 (b) for primes is due to
Gallagher;in[7]heshowedthatanappropriateformoftheHardy–Littlewood
prime k-tuples conjecture implies the prime analog of (1.12). (That it implies
theprimeanalogof (1.13)ismentionedinHooley’ssurveyarticle[12,p.137].)
To show that the singular series is one on average (i.e., the prime analog of
Proposition 1.3), Gallagher uses combinatorial identities for Stirling numbers
ofthesecondkind. In[18],Kowalskidevelopedanelegantprobabilisticframe-
work for evaluating averagesof singular series. Rather than using combinato-
rial identities, he showed that a certain duality between k-th moments of m-
tuplesandm-thmomentsofk-tuplesholds(cf.[18,Theorem1]). Thatthek-th
moment of 1-tuples equals one is essentially trivial; by duality he obtains the
non-trivial consequence that first moments of k-tuples also equals one. (Note
that (1.14)can be viewedasa first momentof k-tupleswhen o .)
“ H
POISSONSPACINGSBETWEENSUMSOFTWOSQUARES 5
Ourapproach originateswith techniquesdevelopedin[19,20], andfurther
refined in [9,21]. Loosely speaking, the singular series Sh is expanded into
local factors of the form 1 ǫh p , andthus
` p q
S 1 ǫ p ǫ d ,
h h h
“ p ` p qq “ p q
źp dÿě1
squarefree
where ǫh 1 1 and ǫh d .. ǫh p . Hence
p q “ p q “ p|d p q
ś
S ǫ d ,
h h
“ p q
ÿh dÿě1 ÿh
squarefree
and the main term is given by d 1. For d large, ǫh d can be shown to be
“ | p q|
smallonaverage. Fordsmall,weusethatǫh d (approximately)onlydepends
p q
on h mod d, together with complete cancellation when summing over the full
set of residuesmodulo d, i.e., hmoddǫhpdq “ 0. This follows, via the Chinese
remainder theorem, from locařl cancellations hmodpǫhppq “ 0, which in turn
can be deduced from the following easilyverřifiable identity: given any subset
X Z pZ, we have(cf. Lemma 6.3(b)and its proof formore details):
p
Ď {
k
# m Z pZ : m h ,m h ,...,m h X #X .
1 2 k p p
t P { ` ` ` P u “
ph1,h2,...,ÿhkqPpZ{pZqk ` ˘
However,unlikethesetupin[9,19,21],wherethelocalerrortermsǫh p are
p q
determined by h mod p, in the current setting the image of h mod pα, for any
fixed α, is not sufficient to determine ǫh p . On the other hand, the function
p q
h ǫh p has nice p-adic regularity properties, allowing us to approximate
Ñ p q
ǫh p by truncations ǫh pα such that ǫh pα only depends on h mod pα, and
p q p q p q
ǫh p ǫh pα 1 pα´1 forallh. Apartfrommakingtheargumentsmorecom-
p q´ p q ! {
plicated, we also get a weaker error term: if ǫh p only depended on h mod p,
p q
we would get a relative error of size y´1`op1q, rather than y´2{3`op1q. We also
note that David, Koukoulopoulos and Smith [6], in studying statistics of ellip-
tic curves, have developed quite general methods for finding asymptotics of
weighted sums hwhSh, provided that the local factors have p-adic regular-
itypropertiessimřilartotheonesabove. Infact,Proposition 1.3,thoughwitha
weakererror term, can be deducedfrom [6, Theorem 4.2].
Wefinallyremark that the corresponding question in the function field set-
ting is better understood — Bary–Soroker and Fehm [1] recently showed that
the sums of squares analog of the k-tuple conjecture holds in the large q-limit
for the function field setting (e.g., replacingZ byF T and Z i by F ? T ).
q q
r s r s r ´ s
2.1. EvidencetowardsConjecture1.1. Webeginbyformulatingaqualitative
version of Conjecture 1.1.
Conjecture2.1. Fixk 1,andaseth h1,...,hk Zwith#h k. IfSh 0,
ě “ t u Ď “ ą
thenthereexistinfinitelymanyintegersnsuchthatn h E.
` Ď
6 T.FREIBERG,P.KURLBERG,ANDL.ROSENZWEIG
WeremarkthatwhetherornotSh 0canbedeterminedbyafinitecomputa-
ą
h
tion: this follows from Propositions 5.2 and 5.3. Examples of sets for which
Sh 0 are 0,1,2,3 and 0,1,2,4,5,8,16,21 : any translate of 0,1,2,3 con-
“ t u t u t u
tains an integer congruent to 3 modulo 4, and hence δh 2 0; any translate
p q “
of 0,1,2,4,5,8,16,21 contains an integer congruent to 3 or 6 modulo 9, and
t u
hence δh 3 0.
p q “
ItispossibletoshowthatSh 0foranysethcontainingatmostthreeinte-
ą
gers. Thequestion ofwhether, foranyh ,h ,h Z, wehaven h ,h ,h
1 2 3 1 2 3
P `t u Ď
E for infinitely many n, was apparently raised by Littlewood: Hooley [13]
showed, using the theory of ternary quadratic forms, that Conjecture 2.1 in-
deedholds fork 3. The conjecture remainsopen fork 4.
ď ě
Forfixed k 1 and h h ,...,h with #h k, the upperbound
1 k
ě “ t u “
x k
1 n h 1 n h 1 ,
Ep ` 1q¨¨¨ Ep ` kq !k logx k{2 ˆ ` p˙
nÿďx p q p”3źmod4
p|hj´hj
someiăj
can bededucedfrom Selberg’ssieve(see [27]),which isofthe correct orderof
magnitude, according to Conjecture 1.1. The special case h 0,1 is due to
“ t u
Rieger [25]; the special case h 0,1,2 is due to Cochrane and Dressler [4];
“ t u
the generalcase isdue to Nowak[24].
Lower bounds are more subtle. For k 2, Hooley [14] and Indlekofer [15]
“
showed that, foranynonzerointeger h,
x 1
1 n 1 n h 1 ,
Ep q Ep ` q " logx ˆ ` p˙
nÿďx źp|h
p”3mod4
but weare not awareofanysuch bounds fork 3.
ě
We remark that Iwaniec deduced the asymptotic 1 n 1 n 1
nďx Ep q Ep ` q „
3x 8logx , as x , from an analog of the Elliott–řHalberstam conjecture
{p q Ñ 8
for sums of two squares (cf. [16, Corollary 2, (2.3)]). However, note that the
leadingterm constant3 8disagreeswith the oneduetoConnorsandKeating
{
[5], namely 1 2. (We also obtain the constant 1 2; see Figure 2 below for a
{ {
numerical comparison.)
2.2. Numerical evidence. Using Propositions 5.2 (b), (c) and 5.3 (b), (c), we
S
cangive h explicitly,asinthefollowingexamples. Letusfirstrecordthatthe
constant C in (1.5) isthe Landau–Ramanujanconstant, given by
1 1 ´1{2
C .. 1 0.764223.... (2.1)
“ ?2 ˆ ´ p2˙ “
p”3źmod4
Itisstraightforward to verify that
1
S 0.856108.... (2.2)
t0,1u “ 2C2 “
POISSONSPACINGSBETWEENSUMSOFTWOSQUARES 7
If (1.8)holds with h 0,1 then, by (1.5)and (2.2),
“ t u
x x
N 0,1 ;x .. 1 n 1 n 1 R x 2 x .
pt u q “ Ep q Ep ` q „ 2C2 1p q „ 2logx p Ñ 8q
nÿďx ` ˘
The agreementwith numerics isquite good (to within 1%).
x Npt0,1u;xq xS pR pxqq2 Ratio
t0,1u 1
1000000000 25927011 25690391.1 1.00921
2000000000 50042411 49603435.5 1.00885
3000000000 73560246 72930222.0 1.00864
4000000000 96705170 95891759.7 1.00848
5000000000 119584162 118589346.3 1.00839
6000000000 142253331 141080935.2 1.00831
7000000000 164749254 163403937.1 1.00823
8000000000 187100631 185584673.5 1.00817
9000000000 209327440 207642640.3 1.00811
Figure2. Observed data vsprediction forh 0,1 .
“ t u
Asthe simplestexamplewith k 3,we verify that
“
A 2
S , A .. 1 ,
t0,1,2u “ 4C2 “ ˆ ´ p p 1 ˙
p”3źmod4 p ´ q
so Conjecture 1.1impliesthat
Ax ACx
Npt0,1,2u;xq ..“ 1Epnq1Epn`1q1Epn`2q „ 4C2 R1pxq 3 „ 4 logx 3{2
nÿďx ` ˘ p q
asx . Here,theagreementbetweennumericsandmodelisonlytowithin
Ñ 8
10%.
x Npt0,1,2u;xq xS pR pxqq3 Ratio
t0,1,2u 1
1000000000 1490691 1362419.3 1.09415
2000000000 2818128 2584683.5 1.09032
3000000000 4093602 3762317.2 1.08805
4000000000 5338091 4912433.3 1.08665
5000000000 6560430 6042800.3 1.08566
6000000000 7764604 7157833.6 1.08477
7000000000 8954282 8260369.7 1.08400
8000000000 10132295 9352396.2 1.08339
9000000000 11299877 10435380.5 1.08284
Figure3. Observed data vsprediction forh 0,1,2 .
“ t u
3. Notation
We define the set of natural numbers as N .. 1,2,... . The letter p stands
“ t u
for a prime, n for an integer. We let (cid:3) (cid:3) stand for a generic element of E,
`
possibly a different elementeach time. Thus, for instance, a h (cid:3) (cid:3) mod
` ” `
8 T.FREIBERG,P.KURLBERG,ANDL.ROSENZWEIG
pα denotes that a h E mod pα for some E E. We view k as a fixed
` ” P
naturalnumber,andhasanonempty,finitesetofintegers,with#h kunless
“
otherwise indicated. We let n h .. n h : h h . For n N, ω n denotes
` “ t ` P u P p q
thenumberofdistinctprimedivisorsofn,ν n thep-adicvaluationofn. (We
p
p q
also define νp 0 .. .) That νp n α may also be denoted by pα n. The
p q “ 8 p q “ ||
radicalofnisrad n .. p,nottobeconfusedwiththesquarefreepartofn,
p q “ p|n
viz.sf n .. p. Bytśheleastresidueofanintegeramodulonwemeanthe
p q “ p||n
integer r sucśh that a r mod n and 0 r n. When written in an exponent,
” ď ă
α mod 2 is to be interpreted as the least residue of α modulo 2: for instance,
pαmod2 1ifα iseven.
“
We view x as a real parameter tending to infinity. Expressions of the form
A B denote that A B 1 as x . We also view y as real parameter
„ { Ñ Ñ 8
tending to infinity, typically in such a way that y x N x . We may assume
„ { p q
that x and y are sufficiently large in terms of any fixed quantity. Expressions
of the form A O B , A B and B A all denote that A c B , where
“ p q ! " | | ď | |
c is some positive constant, throughout the domain of the quantity A. The
constantcistoberegardedasindependentofanyparameterunlessindicated
otherwise by subscripts, as in A O B (c depends on k only), A B (c
k k,λ
“ p q !
depends on k and λ only), etc. By o 1 we mean a quantity that tends to zero
p q
asy .
Ñ 8
4. Deducing Theorem1.2 fromProposition1.3
Given~ι i ,...,i Nrsuchthati i k,andλ~ λ ,...,λ Rr,
1 r 1 r 1 r
“ p q P `¨¨¨` “ “ p q P
let
Θ~ι,λ~ ..“ tpx1,...,xkq P ∆k : xi1`¨¨¨`ij ´xi1`¨¨¨`ij´1 ď λj,j “ 1,...,ru, (4.1)
where forj “ 1 we letxi1`ij´1 “ x0 ..“ 0. In the case where r “ 1 andλ~ “ pλq,
Θ~ι,λ~ “ Θk,λ ..“ tpx1,...,xkq P Rk : 0 ă x1 ă ¨¨¨ ă xk ď λu. (4.2)
The following proof shows that Theorem 1.2 (a) and (b) hold under slightly
weaker hypotheses than the ones stated: for (a), it is enough to assume that
Hypothesis (k,Θ , ), where~ι i ,...,i and λ~ λ ,...,λ , holds for
~ι,λ~ H “ p 1 rq “ p 1 rq
allk r,andall~ι Nr satisfyingi i k;for(b),itisenoughtoassume
1 r
ě P `¨¨¨` “
that Hypothesis (k,Θ , )holdsforall k 1.
k,λ
H ě
DeductionofTheorem1.2. Asthisargumenthasappearedmanytimesinthelit-
erature, we merely give an outline of it and provide references. (a) To ease
notation, we let~ι i ,...,i ,h~ h ,...,h , h h ,...,h , and
1 r 1 k 1 k
“ p q “ p q “ t u
N 0 h;x .. 1E n 1E n h1 1E n hk .
pt uY q “ p q p ` q¨¨¨ p ` q
nÿďx
POISSONSPACINGSBETWEENSUMSOFTWOSQUARES 9
Let ℓ 0 be an integer, arbitrarily large but fixed. An inclusion-exclusion
ě
argument (see [11], [19, AppendixA]or[17, KeyLemma2.4.12])showsthat
r`2ℓ`1
1 k´r N 0 h;x
p´ q pt uY q
kÿ“r i1`¨¨ÿ¨`ir“k h~PyΘÿ~ι,λ~XZk
r`2ℓ
1 1 k´r N 0 h;x ,
ď ď p´ q pt uY q
En`j´EEÿnn`ďjx´1ďλjy kÿ“r i1`¨ÿ¨¨`ir“k h~PyΘÿ~ι,λ~XZk
j“1,...,r
(4.3)
the sumsoveri i k,hereand below,beingoverall~ι Nr forwhich
1 r
`¨¨¨` “ P
i i k. We make the substitution (1.9), with 0 h and k 1 in
1 r
` ¨¨¨ ` “ t u Y `
place of h and k; we apply Hypothesis (k,Θ , 0 ) for all k and~ι satisfying
~ι,λ~ t u
r k r 2ℓ 1 and i i k; we use Proposition 1.3, and our
1 r
ď ď ` ` ` ¨¨¨ ` “
assumption thatyR x 1,i.e.y x N x , asx . Thus,wededucefrom
1
p q „ „ { p q Ñ 8
(4.3)that
r`2ℓ`1 1
1 k´r vol Θ liminf 1 (4.4)
p´ q p ~ι,λ~q ď xÑ8 N x
kÿ“r i1`¨ÿ¨¨`ir“k p q Eÿnďx
En`j´En`j´1ďλjy
j“1,...,r
and
1 r`2ℓ
limsup 1 1 k´r vol Θ . (4.5)
N x ď p´ q p ~ι,λ~q
xÑ8 p q Eÿnďx kÿ“r i1`¨¨ÿ¨`ir“k
En`j´En`j´1ďλjy
j“1,...,r
Since vol Θ λi1 λir i ! i ! , the sums on the left and right of (4.4)
p ~ι,λ~q “ 1 ¨¨¨ r {p 1 ¨¨¨ r q
and (4.5) are truncations of the Taylor series for 1 e´λ1 1 e´λr . We
p ´ q¨¨¨p ´ q
havechosenℓarbitrarilylarge,sowemayconcludethat(1.12)holds,provided
Hypothesis (k,Θ , 0 )doeswheneverk r andi i k.
~ι,λ~ t u ě 1 `¨¨¨` r “
(b) We use an argument of Gallagher [7], who proved an analogous result
for primes. Letℓ 1 bean integer, arbitrarily large but fixed. Wehave
ě
ℓ
N n λy N n ℓ 1 n h
E
p ` q´ p q “ ˆ p ` q˙
nÿďx` ˘ nÿďx 0ăÿhďλy
1 n h 1 n h
E 1 E ℓ
“ p ` q¨¨¨ p ` q
nÿďx0ăh1,ÿ...,hℓďλy
ℓ
̺ ℓ,k 1 n h 1 n h ,
E 1 E k
“ p q p ` q¨¨¨ p ` q
kÿ“1 0ăh1ăÿ¨¨¨ăhkďλy nÿďx
where̺ ℓ,k denotesthenumberofmapsfrom 1,...,ℓ onto 1,...,k . Thus,
p q t u t u
1 ℓ N x k
N n λy N n ℓ p q ̺ ℓ,k Sh Eh x ,
x p ` q´ p q “ ˆ x ˙ p q ` p q
nÿďx` ˘ kÿ“1 0ăh1ăÿ¨¨¨ăhkďλy` ˘
10 T.FREIBERG,P.KURLBERG,ANDL.ROSENZWEIG
withh h ,...,h inthelastsummand. Tosumover0 h h λy
1 k 1 k
“ t u ă ă ¨¨¨ ă ď
is to sum over h ,...,h yΘ Zk (see (4.2)). If Hypothesis (k,Θ , )
1 k k,λ k,λ
p q P X H
holds then for some function ε x with ε x 0(x ),we have
p q p q Ñ Ñ 8
S E x 1 O ε x S .
h h λ,k h
` p q “ ` p p qq
0ăh1ăÿ¨¨¨ăhkďλy` ˘ ` ˘0ăh1ăÿ¨¨¨ăhkďλy
Applying Proposition 1.3 (noting that vol Θ λk k!), and our assumption
k,λ
p q “ {
thatyR x 1,i.e.y x N x ,asx ,weseethatifHypothesis(k,Θ , )
1 k,λ
p q „ „ { p q Ñ 8 H
holds for1 k ℓ, then
ď ď
1 ℓ λk
ℓ
N n λy N n ̺ ℓ,k x . (4.6)
x p ` q´ p q „ p qk! p Ñ 8q
nÿďx` ˘ kÿ“1
Gallagher’s calculation in [7, Section 3] shows that ℓ ̺ ℓ,k λk k! is the ℓth
k“1 p q {
momentofthePoissondistributionwithparameterλř,andthatthecorrespond-
ingmomentgeneratingfunctionisentire. SinceaPoissondistributionisdeter-
mined byitsmoments, itfollows(see [3,Section 30])that foranygiven m 0,
(1.13)holdsasx ,providedHypothesis(k,Θ , )holdsforallk 1.ě(cid:3)
k,λ
Ñ 8 H ě
5. Preliminaries
Apositive integer nisa sum oftwo squaresifand onlyif
n 2β2 pβp p2βp,
“
p”1źmod4 p”3źmod4
where β ,β denote nonnegative integers. (See [10, Theorem 366].) In view of
2 p
this and the next proposition, whose proof, being routine and elementary, is
omitted, we have E S , where S n Z : n (cid:3) (cid:3) mod pα .
p p p αě1
“X “X t P ” ` u
Further, asS Zforprimesp 1 mod 4,we maywrite E S .
p pı1mod4 p
“ ” “X
Proposition 5.1. Let n Z. We have n S if and only if eithern 0 or n 2βm
2
P P “ “
for some β 0 and m 1 mod 4. For p 3 mod 4, we have n S if and only if
p
ě ” ” P
either n 0 or n p2βm for some β 0 and m 0 mod p. For p 1 mod 4, we
“ “ ě ı ”
haveS Z.
p
“
Let us introduce some notation in order to state further results. Given a
nonempty, finite seth Z, let
Ď
det h .. h h1 0. (5.1)
p q “ p ´ q ą
hź,h1Ph
hąh1
Note that ifp k 1, wherek #h,then twoelementsofhoccupy the same
ď ´ “
congruence class modulo p, so p det h . In other words, if p ∤ det h then
| p q p q
k p.
ď
Let
hp .. h1 h : h1 h Sp . (5.2)
“ t P ´ ` Ď u
Note that h contains atmost one element, forifh,h1 h then h h1 S ,
2 2 2
P ˘p ´ q P
which by Proposition 5.1 holds only if h h1 0. Similarly, if k 1 or k 2,
´ “ “ “