Pair correlation for quadratic polynomials mod 1

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by  Jens Marklof

PAIR CORRELATION FOR QUADRATIC POLYNOMIALS
MOD 1
7 JENS MARKLOF AND NADAV YESHA
1
0
2 Abstract. It is an open question whether the fractional parts of non-
linear polynomials at integers have the same fine-scale statistics as a
r
a Poissonpointprocess. Mostresultstowardsanaffirmativeanswerhave
M
so far been restricted to almost sure convergence in the space of poly-
4 nomials of a given degree. We will here provide explicit Diophantine
2 conditions on the coefficients of polynomials of degree 2, under which

the convergence of an averaged pair correlation density can be estab-
]
T lished. The limit is consistent with the Poisson distribution. Since
N quadratic polynomials at integers represent the energy levels of a class
. ofintegrablequantumsystems,ourfindingsprovidefurtherevidencefor
h
t the Berry-Tabor conjecture in the theory of quantum chaos.
a
m
[

2
v 1. Introduction
3
6
Let ϕ(z) = α zq +α zq−1 +...+α be a polynomial of degree q. In
1 q q−1 0
9 his ground-breaking 1916 paper [22], H. Weyl proved that, if one of the
0
1. coefficients αq,...,α1 is irrational, then the sequence ϕ(1),ϕ(2),ϕ(3),... is
0 uniformly distributed mod 1. That is, for any interval A ⊂ R of length
7
|A| ≤ 1 we have
1
v: #{j ≤ N | ϕ(j) ∈ A+Z}
(1.1) lim = |A|.
Xi N→∞ N
r A century after Weyl’s work, it is remarkable how little is known on the
a
fine-scale statistics of ϕ(j) mod 1 in the case of polynomials ϕ of degree
q ≥ 2. (The linear case q = 1 is singular and related to the famous three
gap theorem; cf. [19, 2, 7, 14]). A popular example of such a statistics is the
gap distribution: for given N, order the values ϕ(1),...,ϕ(N) in [0,1) mod
1, and label them as
0 ≤ θ ≤ θ ≤ ... ≤ θ < 1.
1,N 2,N N,N
Date: March 27, 2017.
1
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 2
As we are working mod 1, it is convenient to set θ := θ −1. The gap
0,N N,N
distribution of {ϕ(j)}N is then given by the probability measure P (·;ϕ)
j=1 N
defined, for any interval A ⊂ [0,∞), by
#(cid:8)j ≤ N | θ −θ ∈ N−1A(cid:9)
j,N j−1,N
(1.2) P (A;ϕ) = .
N
N
Rudnick and Sarnak [16] conjecture the following.
Conjecture 1.1. Let q ≥ 2. There is a set Q ⊂ R of full Lebesgue measure,
such that for α ∈ Q and any interval A ⊂ [0,∞),
q
ˆ
(1.3) lim P (A;ϕ) = e−sds.
N
N→∞ A
That is, the gap distribution of a degree q polynomial with typical lead-
ing coefficient convergences to the distribution of waiting times of a Poisson
processwithintensityone. Sinai[18]pointedoutthat, forquadraticpolyno-
mials(withrandomcoefficients), highmomentsofthefine-scaledistribution
do not converge; this is of course not in contradiction with Conjecture 1.1,
which only concerns weak convergence. Pellegrinotti on the other hand
pointed out that the first four moments do converge to the Poisson distri-
bution [15].
The validity of Conjecture 1.1 is completely open. It is known that the
conjecture cannot hold for every irrational α . The general belief, however,
q
is that Q includes every irrational number of Diophantine type 2 + (cid:15), for
any (cid:15) > 0 (see (1.10) below); this includes for instance all algebraic num-
bers [16]. The only result to-date on the gap distribution, due to Rudnick,
Sarnak and Zaharescu [17], holds for quadratic polynomials, and establishes
the convergence in (1.3) along subsequences for leading coefficients α that
2
are well approximable by rationals. Unfortunately, for these α we cannot
2
expect convergence along the full sequence, as they do not have the required
Diophantine type.
A more accessible fine-scale statistics is the pair correlation measure
R (·;ϕ) which, for any bounded interval A ⊂ R, is defined by
2,N
#(cid:8)(i,j) | i 6= j ≤ N, ϕ(i)−ϕ(j) ∈ N−1A+Z(cid:9)
(1.4) R (A;ϕ) = .
2,N
N
RudnickandSarnakprovedthatthepaircorrelationmeasureofpolynomials
P(z) = αzq converges for almost every α to Lebesgue measure, the pair
correlation measure of a Poisson point process.
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 3
Theorem 1.2 (Rudnick-Sarnak [16]). Let q ≥ 2. There is a set Q ⊂ R of
full Lebesgue measure, such that for ϕ(z) = αzq, α ∈ Q, and any bounded
interval A ⊂ R,
(1.5) lim R (A;ϕ) = |A|.
2,N
N→∞
It is currently not known whether any Diophantine condition on α will
guarantee the convergence in (1.5). In the quadratic case q = 2, Heath-
Brown [8] gave an explicit construction of α’s such that (1.5) holds. The
question whether (1.5) holds for Diophantine α (which holds, e.g., for α =
√
2), however, remains open to this day. See Truelsen [21] for a conditional
result, and Marklof-Strömbergsson [13] for a derivation of (1.5) from a geo-
metric equidistribution result. Boca and Zaharescu [3] generalized Theorem
1.2 to ϕ(z) = αf(z), where f is any polynomial of degree q with integer
coefficients.
Zelditch proved an analogue of Theorem 1.2 for the sequence of polyno-
mials
(cid:18) z (cid:19)
(1.6) ϕ (z) = αNf +βz,
N
N
where f is a fixed polynomial satisfying f00 6= 0 on [−1,1]. Surprisingly,
the pair correlation problem for ϕ seems harder than the case of fixed
N
polynomials ϕ, and requires an additional Cesàro average:
Theorem 1.3 (Zelditch [23, add.]). There is a set Q ⊂ R2 of full Lebesgue
measure, such that for (α,β) ∈ Q, and any bounded interval A ⊂ R,
1 XM
(1.7) lim R (A;ϕ ) = |A|.
2,N N
M→∞ M
N=1
Slightly weaker results hold when f is only assumed to be a smooth func-
tion [23]. The motivation for the particular choice of ϕ is that the values
N
{ϕ (j)}N represent the eigenphases of quantized maps [23]; the under-
N j=1
standing of their statistical distribution is one of the central challenges in
quantum chaos [1, 11]. Zelditch and Zworski have found analogous results
for scattering phase shifts [24].
The aim of the present paper is to improve Theorem 1.3 for the specific
example
(z−α)2
(1.8) ϕ (z) = .
N
2N
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 4
Figure 1.1. Pair correlation of n(n−√2)2oN mod 1, with
2N n=1
N = 5000.
The corresponding triangular array {ϕ (j)} , is uniformly dis-
N 1≤j≤N,1≤N<∞
tributed mod 1: For any bounded interval A ⊂ R,
#{j ≤ N | ϕ (j) ∈ A+Z}
N
(1.9) lim = |A|.
N→∞ N
Appendix A comprises a precise bound on the discrepancy.
We will establish convergence of the pair correlation measure under an
explicit Diophantine condition on α, rather than convergence almost every-
where as in Theorem 1.3. Furthermore, we will shorten the Cesàro average
and provide a power-saving in the rate of convergence.
An irrational α ∈ R is said to be Diophantine, if there exists κ,C > 0
such that
(cid:12) p(cid:12) C
(cid:12) (cid:12)
(1.10) (cid:12)α− (cid:12) >
(cid:12) q(cid:12) qκ
for all p ∈ Z, q ∈ N. The constant κ is called the Diophantine type of α; by
Dirichlet’s theorem, κ ≥ 2. Every quadratic surd, and more generally every
α with bounded continued fraction expansion, is of Diophantine type κ = 2.
Theorem 1.4. Choose ϕ as in (1.8), with α Diophantine of type κ, and
N
fix
(cid:18) (cid:18)17 2 (cid:19) (cid:21)
η ∈ max ,1− ,1 .
18 9κ
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 5
Figure 1.2. Gap distribution of n(n−√2)2oN mod 1, with
2N n=1
N = 106.
There exists s = s(κ,η) > 0 such that for any bounded interval A ⊂ R
(1.11) 1 X R (A;ϕ ) = |A|+O(cid:0)M−s(cid:1)
Mη 2,N N
M≤N≤M+Mη
as M → ∞ (the implied constant in the remainder depends on α,η,A).
Theorem 1.4 extends to more general pair correlation measures, see Ap-
pendix B for a discussion. Numerical experiments (see Fig. 1.1) seem to
suggest that the average over N in (1.11) might in fact not be necessary in
(1.11), and that furthermore the gap distribution is exponential (Fig. 1.2).
The proof of this statement reduces to a natural equidistribution the-
orem on a three-dimensional Heisenberg manifold, which we show derives
from Strömbergsson recent quantitative Ratner equidistribution result on
the space of affine lattices. Assuming that a more subtle equidistribution
result holds (Conjecture 2.4), we can remove the average over N in (1.11)
and obtain the limit R (A;ϕ ) → |A|; see Proposition 2.5.
2,N N
The fact that α is irrational and badly approximable is a crucial assump-
tion. In the case α = 0, the gap distribution reduces to the distribution of
spacing between quadratic residues mod 2N, which was proved by Kurlberg
and Rudnick [9] to be Poisson along odd, square-free and highly composite
N’s.
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 6
If we do not insist on a rate of convergence, and also allow for a large
average, Theorem 1.4 holds in fact for the more general polynomials
(z−α)2
(1.12) ϕ (z) = β ,
N
2N
where β 6= 0 is Diophantine and α ∈ R arbitrary, or β 6= 0 arbitrary and α ∈
R Diophantine. We will show in Appendix C that under these assumptions
1 X
(1.13) lim R (A;ϕ ) = |A|.
2,N N
N→∞ M
M≤N≤2M
This statement is an immediate corollary of the quantitative Oppenheim
conjecture for quadratic forms of signature (2,2), first proved by Eskin,
Margulis and Mozes [6] for homogeneous forms, and generalized to inhomo-
geneous forms by Margulis and Mohammadi [10].
We conclude this introduction by briefly describing the relevance of our
results to the theory of quantum chaos. The values {ϕ (j)}N for ϕ as in
N j=1 N
(1.8), or more generally (1.12), represent the eigenphases of a particularly
simple quantum map studied by De Bièvre, Degli Esposti and Giachetti
[4, §4B]: a quantized shear of a toroidal phase space with quasi-periodic
boundary conditions, where β quantifies the shear-strength and α is related
to the quasi-periodic boundary condition. Theorem 1.4 proves that the
average two-point spectral statistics of this quantum map are Poisson.
The spacings in the sequence {ϕ (j)} furthermore represent the quan-
N j
tum energy level spacings of a class of integrable Hamiltonian systems: a
particle on a ring with quasi-periodic boundary condition (representing a
magnetic flux line through its center), coupled to a harmonic oscillator. The
corresponding energy levels are (in suitable units) E ((cid:126)) = β(cid:126)2(j −α)2+
j,k 2
(cid:126)(k− 1), where j ∈ Z, k ∈ N and β > 0. It is a short exercise to show that,
2
after rescaling by (cid:126), the energy levels E ((cid:126)) ∈ [E−(cid:126),E−(cid:126)) have the same
j,k 2 2
(cid:16) (cid:17)1/2
spacing statistics as ϕ (j) mod 1 for (cid:126) = N−1 and |j −α| < 2E N.
N β
The index range is thus not j ≤ N as in (1.4); cf. Appendices B and C for
the relevant generalizations.
Berry and Tabor [1] conjectured that typical integrable quantum systems
should have Poisson statistics in the semiclassical limit (cid:126) → 0, and the
results presented in this study may therefore be viewed as further evidence
for the truth of the Berry-Tabor conjecture. Other instances in which the
conjecture could be rigorously established are reviewed in [11].
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 7
Acknowledgements. We thank Zeév Rudnick for helpful comments. The
researchleadingtotheseresultshasreceivedfundingfromtheEuropeanRe-
searchCouncilundertheEuropeanUnion’sSeventhFrameworkProgramme
(FP/2007-2013) / ERC Grant Agreement n. 291147.
2. Outline of the proof: Smoothing and geometric
equidistribution
We will first prove a smooth version of Theorem 1.4. Let S(R) be the
Schwartz space of rapidly decreasing smooth functions on R, equipped with
the norms
kfk = X (cid:13)(cid:13)f(l)(cid:13)(cid:13) ,
Ck (cid:13) (cid:13)
∞
0≤l≤k
and denote ˆ
∞
fˆ(ξ) = f(x)e(−xξ)dx
−∞
the Fourier transform of f ∈ S(R), where we use the standard notation
e(x) = exp(2πix). For f,h ∈ S(R), define the smooth pair correlation
function
(2.1)
1 X X (cid:18)i−α(cid:19) (cid:18)j −α(cid:19)
R (f,h;ϕ) = h h f(N (ϕ(i)−ϕ(j)+m));
2,N
N N N
m∈Zi,j∈Z
for technical reasons, introducing the shifts by α inside h in (2.1) gives a
more convenient approximation to the sharp cutoff in (1.4).
Throughout the remainder of this paper we will use ϕ(z) = ϕ (z) =
N
β (z−α)2 with β = 1 as in (1.8), except for Appendix C where we consider
2N
general values of β.
We prove the following smoothed variant of Theorem 1.4:
(cid:16) (cid:17)
Theorem 2.1. Fix a Diophantine α of type κ, and let max 17,1− 2 <
18 9κ
η ≤ 1 be fixed. There exist s = s(κ,η) > 0,n = n(κ,η) ∈ N such that for
any ν ∈ C∞(R) supported on [−L,L] (L ≥ 1) and ψ,h ∈ C∞(R) real valued
and supported on [−1/2,3/2],
1 X ψ(cid:18)N −M(cid:19)R (νˆ,h;ϕ ) = ψˆ(0)(cid:18)ν(0)(cid:12)(cid:12)hˆ(0)(cid:12)(cid:12)2+νˆ(0)khk2(cid:19)
Mη Mη 2,N N (cid:12) (cid:12) 2
N∈Z
(cid:16) (cid:17)
+O Lkhk2 kνk kψk M−s
Cn C2 C8
as M → ∞ (the implied constant in the remainder depends on α,η).
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 8
(The specific choice of the interval [−1/2,3/2] in the above is simply for
convenience. The proof generalizes to general bounded intervals.)
Define
1 X (cid:18) k (cid:19)(cid:12)(cid:12)(cid:12) 1 X (cid:18)n−α(cid:19) (n−α)2!(cid:12)(cid:12)(cid:12)2
QN (ν,h) = ν (cid:12)√ h e k (cid:12) .
N N (cid:12) N N 2N (cid:12)
k6=0 (cid:12) n∈Z (cid:12)
Theorem 2.1 will follow from the following proposition, which is key in
this paper:
(cid:16) (cid:17)
Proposition2.2. FixaDiophantineαoftypeκ, andletmax 17,1− 2 <
18 9κ
η ≤ 1 be fixed. There exist s = s(κ,η) > 0,n = n(κ,η) ∈ N such that for
any ν ∈ C∞(R) supported on [−L,L] (L ≥ 1) and ψ,h ∈ C∞(R) real valued
and supported on [−1/2,3/2],
(2.2) 1 X ψ(cid:18)N −M(cid:19)Q (ν,h) = ψˆ(0)νˆ(0)khk2
Mη Mη N 2
N∈Z
(cid:16) (cid:17)
+O Lkhk2 kνk kψk M−s .
Cn C2 C8
as M → ∞.
ThestrategyoftheproofofProposition2.2whichwillbegiveninSection
5.3, is first to interpret the l.h.s. of (2.2) as a smooth sum of the absolute
square of the Jacobi theta sum. Then we establish Proposition 2.2 from
a geometric equidistribution result, which in turn follows from an effective
Ratner equidistribution result due to Strömbergsson [20].
In order to give the exact formulation of our equidistribution result, let
µ be the Haar measure on the Heisenberg group
H
  
H(R) = 10 u1 ξξ12 : u ∈ R,ξ = (ξ1,ξ2) ∈ R2,
0 0 1 
normalized such that it induces a probability measure on H(Z)\H(R); the
latter is also denoted by µ .
H
Denote the differential operators
∂ ∂ ∂ u ∂ 1 ∂
D = v2 ,D = v ,D = v ,D = + .
1 2 3 4
∂u ∂v ∂ξ v ∂ξ v∂ξ
1 1 2
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 9
For a compactly supported f = f(v,u,ξ) ∈ Ck((0,∞)×H(Z)\H(R)),
define the norm
X
kfk = kDfk
Ck ∞
ordD≤k
summing over all monomials in D (1 ≤ i ≤ 4) of degree ≤ k.
i
Recall that a vector η ∈ R2 is called Diophantine of type κ, if there exists
C > 0 such that
(cid:13) m(cid:13) C
(cid:13) (cid:13)
(cid:13)η− (cid:13) >
(cid:13) q (cid:13) qκ
for all m ∈ Z2, q ∈ N. By Dirichlet’s theorem, κ ≥ 3/2.
Proposition 2.3. Fix a Diophantine vector η of type κ, δ > 0. Then
for any f ∈ C8((0,∞)×H(Z)\H(R)) compactly supported on [1/4,∞)×
H(Z)\H(R) and ν ∈ C∞(R) supported on [−L,L] (L ≥ 1), we have
1 X (cid:18)d(cid:19) M a a b ! !
ν f , , η
M2 c c c c d
c,d
(c,d)=1
ˆ ˆ
6 ∞
= νˆ(0) f(v,u,ξ)v−3dµ dv
π2 H
0 H(Z)\H(R)
(cid:16) (cid:17)
+O Lkνk kfk M−min(1/2,2/κ)+δ
C2 C8
as M → ∞, where (a,b) are any integers that solve the equation ad−bc = 1.
To give the strategy behind the proof of Proposition 2.3, let G be the
semi-direct product Lie group G = SL(2,R)(cid:110)R2 with the multiplication
law
(g,ξ)(cid:0)g0,ξ0(cid:1) = (cid:0)gg0,gξ0+ξ(cid:1),
where the elements of R2 are viewed as column vectors, and let Γ be the
subgroup Γ = SL(2,Z)(cid:110)Z2. Denote
! ! ! !
1 x y 0
n (x) = ,0 ,a(y) = ,0 .
+ 0 1 0 y−1
We use the fact that, for every 0 < v < v , the set
0 1
Γ\Γ{(1,ξ)n (u)a(v) : v < v < v ,(u,ξ) ∈ H(R)}
+ 0 1
is an embedding of (v ,v ) × H(Z)\H(R) as a submanifold of Γ\G. By
0 1
thickening this submanifold, we can use Strömbergsson’s result to obtain
equidistribution of our points.
PAIR CORRELATION FOR QUADRATIC POLYNOMIALS MOD 1 10
As for the pointwise limit of the pair correlation measure R , we show
2,N
in Appendix D that it also converges to Lebesgue measure (and is thus
consistent with the Poisson distribution), under the following equidistribu-
tion conjecture on the Heisenberg group H(Z)\H(R), which is a pointwise
version of Proposition 2.3 in the special case η = (1/2,α):
For f ∈ Ck(H(Z)\H(R)), define the norm
X
kfk = kDfk
Ck ∞
ordD≤k
summingoverallmonomialsinthestandardbasiselementsoftheLiealgebra
of H(R) (which correspond to left-invariant differential operators on H(R))
of degree ≤ k.
Conjecture 2.4. There exist k,l ∈ N,κ ≥ 2, such that for any fixed Dio-
0
phantine α of type κ ≤ κ , there exists s = s(κ) > 0 such that for any
0
f ∈ Ck(H(Z)\H(R)) and ν ∈ C∞(R) supported on [−L,L] (L ≥ 1), we
have
ˆ
1 X (cid:18)d(cid:19) (cid:18)a a c(cid:19)
ν f ,bα+ ,dα+ = νˆ(0) fdµ
H
φ(c) c c 2 2
d∈Z H(Z)\H(R)
(c,d)=1
+O(cid:0)Lkνk kfk c−s(cid:1)
Cl Ck
as c → ∞, where (a,b) are any integers that solve the equation ad−bc = 1
and φ is the Euler totient function.
We show in Appendix D:
Proposition 2.5. Assuming Conjecture 2.4, there exists κ ≥ 2, such that
0
for any fixed Diophantine α of type κ ≤ κ , there exists s = s(κ) > 0 such
0
that for any bounded interval A ⊂ R
R (A;ϕ ) = |A|+O(cid:0)N−s(cid:1)
2,N N
as N → ∞.
3. Background
3.1. Effective Ratner equidistribution theorem. Recall that G is the
semi-direct product Lie group G = SL(2,R)(cid:110)R2 with the multiplication
law
(g,ξ)(cid:0)g0,ξ0(cid:1) = (cid:0)gg0,gξ0+ξ(cid:1),