Commun. Math. Phys. 194, 61 – 70 (1998)
Communications in
Mathematical
Physics
© Springer-Verlag 1998
The Pair Correlation Function of Fractional Parts of
Polynomials?
Zeév Rudnick1 , Peter Sarnak2
1 Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978,
Israel
2 Department of Mathematics, Princeton University, Princeton NJ 08544, USA
Received: 22 July 1997 / Accepted: 24 September 1997
Abstract: We investigate the pair correlation function of the sequence of fractional
parts of αnd , n = 1, 2, . . . , N , where d ≥ 2 is an integer and α an irrational. We
conjecture that for badly approximable α, the normalized spacings between elements of
this sequence have Poisson statistics as N → ∞.
We show that for almost all α (in the sense of measure theory), the pair correlation
of this sequence is Poissonian.
In the quadratic case d = 2, this implies a similar result for the energy levels of
the “boxed oscillator” in the high-energy limit. This is a simple integrable system in
2 degrees of freedom studied by Berry and Tabor as an example for their conjecture
that the energy levels of generic completely integrable systems have Poisson spacing
statistics.
1. Introduction
Hermann Weyl [11] proved that for an integer d ≥ 1 and an irrational α, the sequence
of fractional parts αnd mod 1, n = 1, 2, . . . is equidistributed in the unit interval. A
different aspect of the random behavior of the sequence has attracted attention recently:
Are the spacings between members of the sequence distributed like those between members of a sequence of random numbers in the unit interval (or as some would say, do
they have a “Poissonian” distribution)? This issue came up in the context of the distribution of spacings of the energy levels of integrable systems [1]. For the case d = 1 the
spacings between the fractional parts of αn are essentially those of the energy levels
of a two-dimensional harmonic oscillator [4, 2, 3]. For d = 2 the spacings are related
to the spacings between the energy levels of the “boxed oscillator” [1], a particle in a
? Supported in part by grants from the U.S.-Israel Binational Science Foundation, the Israel Science Foundation, and the NSF.
62
Z. Rudnick, P. Sarnak
2-dimensional potential well with hard walls in one direction and harmonic binding in
the other. The spacings of αn2 mod 1 were also investigated numerically in [5].
If d = 1 it is elementary that the consecutive spacings have at most 3 values [9, 10].
Hence the sequence is not random in this case. For d ≥ 2 the picture is very different.
To explain it, we recall a basic classification of real numbers with regards to their
Diophantine approximation properties: We say α is of type κ if there is c = c(α) > 0 so
that
|α − p/q| > c/q κ
for all integers p, q. For rational α, κ = 1 and α is irrational if and only if κ ≥ 2. It is
well known that almost all α (in the sense of measure theory) are of type κ = 2 + for
all > 0. We will call such α “Diophantine”. For instance, algebraic irrationals are of
this type (Roth’s theorem).
In [7] we establish some results towards the conjecture that αnd mod 1 is Poissonian
for any α of Diophantine type. In this note we examine the behavior for almost all α,
which according to the above should be Poissonian. The statistic we examine is the pair
correlation: The pair correlation density for a sequence of N numbers θ1 , . . . , θN ∈
[0, 1] which are equidistributed as N → ∞, measures the distribution of spacings
between the θj at distances of order of the mean spacing 1/N . Precisely, if kxk =
distance(x, Z) then for any interval [−s, s] set
1 n
so
.
(1.1)
R2 ([−s, s], {θn }, N ) = # 1 ≤ j 6= k ≤ N : kθj − θk k ≤
N
N
For random numbers θj chosen uniformly and independently,
R2 ([−s, s], {θn }, N ) → 2s
with probability tending to 1 as N → ∞. Our main result is that this holds for
the sequence of fractional parts {αnd mod 1} for almost every α: Denoting by
R2 ([−s, s], α, N ) the pair correlation sum (1.1) for this sequence, we show
Theorem 1. For d ≥ 2, there is a set P ⊂ R of full Lebesgue measure such that for
any α ∈ P , and any s ≥ 0,
R2 ([−s, s], α, N ) → 2s,
N → ∞.
Remark 1.1. The proof given below does not provide (and we do not know of) any
specific α which is provably in P .
Remark 1.2. Already with the pair correlation we see the necessity of a condition on
the type of α. For if there are arbitrarily large integers p, q so that
1
p
,
|α − | ≤
q
10q d+1
then R2 ([−s, s], α, N ) 6→ 2s. Indeed if we choose N = q, then for m 6= n ≤ N ,
d
d
d d
d
n α − md α = (n − m )p + t(n − m ) q
10q d+1 d
with
Hence
either n α − md α ≤ 1/10q = 1/10N if q divides nd − md , or
d |t| ≤ 1.
n α − md α ≥ 9/10q = 9/10N otherwise. Thus there are no normalized differences
N nd α − md α in the interval (1/10, 9/10) for this sequence of N = q.
Pair Correlation of Fractional Parts
63
The proof of the theorem follows the steps in [8] (where a similar assertion is proven
for the values of binary quadratic forms). We first establish that as a function of α ∈ [0, 1],
R2 ([−s, s], α, N )P→ 2s in L2 (0, 1). This together with standard bounds on the Weyl
sums S(n, N ) = x≤N e(nαxd ) allows us to pass to almost everywhere convergence.
In Sect. 4 we briefly discuss higher correlations and show that they do not converge in
L2 to the expected value. Thus our approach does not lend itself directly to establishing
almost everywhere convergence of the higher correlations.
2. Bounding the Variance
Let f ∈ Cc∞ (R) be a test function and set
R2 (f, {θn }, N ) :=
where
FN (y) =
X
1
N
FN (θj − θk ),
(2.1)
1≤j6=k≤N
X
f (N (y + m)).
(2.2)
m∈Z
The function FN (y) is periodic and has a Fourier expansion
1 X b n e(ny).
f
FN (y) =
N
N
(2.3)
n∈Z
Hence
R2 (f, {θn }, N ) =
1 X b n f
N2
N
n∈Z
X
e(n(θj − θk )).
(2.4)
1≤j6=k≤N
In particular, if θn = αnd mod 1, then the pair correlation function is given by
1 X b n sof f (n, N ),
f
R2 (f, α, N ) = 2
N
N
(2.5)
n∈Z
where
sof f (n, N ) :=
X
e nα(xd − y d ) .
(2.6)
1≤x6=y≤N
As a function of α, R2 (f, α, N ) is periodic and from (2.5) its Fourier expansion is
X
R2 (f, α, N ) =
bl (N )e(lα),
(2.7)
l∈Z
where for l 6= 0,
bl (N ) =
1 X
N2
n6=0
X
1≤x6=y≤N
n(xd −y d )=l
The mean of R2 (f, α, N ) over α ∈ [0, 1] is
n
.
fb
N
(2.8)
64
Z. Rudnick, P. Sarnak
1
hR2 i = b0 (N ) = 2
N
X
fb(0) =
1≤x6=y≤N
1
1−
N
fb(0),
(2.9)
so that
Z
hR2 i =
∞
−∞
f (x)dx + O
1
N
,
(2.10)
which is the expected value for a random sequence.
We next estimate the variance of R2 (f, α, N ) as a function of α:
Proposition 2. As a function of α ∈ [0, 1],
R2 (f, α, N ) − fb(0) N −1/2+
(2.11)
2
for any > 0, the implied constants depending on and f .
Proof. It is easy to see from (2.8) that since f ∈ Cc∞ (R), the Fourier coefficients bl (N )
are negligable for l ≥ N d+1+δ for any fixed δ > 0. Also from (2.8) we have for l 6= 0,
τ (|l|)2
,
N2
bl (N ) (2.12)
where τ (|l|) is the numbers of divisors of |l|. This is because the factors of l determine
n, x, y. We will use the well-known estimate
τ (m) m ,
for any > 0.
Thus by Parseval
2
R2 (f, α, N ) − fb(0) =
fb(0)
N
2
!2
+
X N
l6=0
N2
|bl (N )|
06=|l|≤N d+1+δ
X
N
N2
N
N2
|bl (N )|2
l6=0
X
=
X
N
|bl (N )| + smaller order term
N2
|bl (N )|
l6=0
X
1≤x6=y≤N
n∈Z
1 b n | N −1+ .
|f
N2
N
(2.13)
Pair Correlation of Fractional Parts
65
3. Almost-Everywhere Convergence
3.1. Overview of the argument for Theorem 1. In order to prove Theorem 1 from the
decay of the variance of the pair correlation, we first show that for each f ∈ Cc∞ (R),
there is a set of full measure P (f ) ⊂ R so that for all α ∈ P (f ),
R2 (f, α, Nm ) → fb(0)
(3.1)
for a subsequence Nm which grows faster than m. Indeed, fix δ > 0, and let {Nm } be
a sequence of integers with
Nm ∼ m1+δ .
Set
XN (α) = R2 (f, α, N ) − fb(0).
By Proposition 2, kXN k2 N −1+ for all > 0 and so
2
∞ Z
X
m=1
1
|XNm (α)2 |dα < ∞.
0
Therefore (since |XNm |2 ≥ 0)
Z
0
and so
P
m
1
X
|XNm (α)| dα =
2
m
XZ
m
1
|XNm (α)|2 dα < ∞,
0
|XNm |2 ∈ L1 (0, 1). Thus the sum is finite almost everywhere:
X
|XNm (α)|2 < ∞,
for almost all α.
m
Therefore, XNm (α) → 0 as m → ∞ for almost all α, that is we have (3.1) on a set P (f )
of α’s which we may assume consists only of Diophantine numbers.
To go from almost everywhere convergence along a subsequence to almost everywhere convergence, we will show that as a function of N , R2 (f, N, α) does not oscillate
much for Diophantine α. More precisely, there is some ν > 0 so that if Nm ≤ n < Nm+1
then for Diophantine α, there is c(f, α) > 0 so that
−ν
|Xn (α) − XNm (α)| c(f, α)Nm
.
δ
, this estimate in turn will follow from:
Because 0 ≤ n − Nm ≤ Nm+1 − Nm Nm
Proposition 3. Let 0 < δ < 1/2d−1 . Then for all f ∈ Cc (R) and all α of Diophantine
type, there is some c(f, α) > 0 so that for all 0 ≤ k ≤ N δ ,
|XN +k (α) − XN (α)| ≤ c(f, α)N −ν ,
where ν < 1/2d−1 − δ.
66
Z. Rudnick, P. Sarnak
Since XNm (α) → 0 for all α ∈ P (f ), which by throwing out a measure-zero subset
we assumed consisted only of Diophantine α’s, Proposition 3 implies Xn (α) → 0 for all
α ∈ P (f ). We will prove this proposition after finishing the proof of Theorem 1. What
remains
to do is to find one subset P ⊂ R of full measure for which R2 (f, α, N ) →
R∞
f
(x)dx
for all α ∈ P and all f which are characteristic function of intervals [−s, s]
−∞
(or in Cc∞ (R)). To do this, pick a (countable) sequence of positive fi ∈ Cc∞ (R) so that for
eachRf ≥ 0 as above, there are subsequences {fi± } ⊂ {fi } which satisfy fi− ≤ f ≤ fi+
∞
and −∞ (fi+ − fi− )(x)dx → 0. Take P := ∩i P (fi ) which is still of full measure. For
every α we have
R2 (fi− , α, N ) ≤ R2 (f, α, N ) ≤ R2 (fi+ , α, N ),
R∞
R∞
and in addition for α ∈ P , we have R2 (fi± , α, N ) → −∞ fi± . Since −∞ fi± →
R∞
R∞
f , this shows that R2 (f, α, N ) → −∞ f for α ∈ P and gives Theorem 1.
−∞
The proof of Proposition 3 will occupy the rest of this section.
3.2. Estimates for Weyl sums. We
Pbegin with some consequences of Weyl’s estimates
for the “Weyl sums” S(n, N ) = x≤N e(nαxd ) which we will need. Throughout the
remainder of this section, we set D = 2d−1 .
Lemma 4. For α Diophantine, and M ≥ 1, we have
X
|S(n, N )|D M 1+ N D−1+
1≤n≤M
for all > 0 (D = 2d−1 ).
Proof. This follows from proof of Weyl’s inequality (see [6], Lemma 3). We will outline
the steps. By repeated squaring, one finds that
|S(n, N )|
D
N
D−1
+N
D−d
N
X
y1 ,...,yd−1 =1
1
min N,
kd!nαy1 . . . yd−1 k
,
where k·k denotes the distance to the nearest integer. Now sum over n ≤ M , collecting
together terms with the product d!ny1 . . . yd−1 having a given value m. The number of
such terms is at most the divisor function τ (m) m . Since the maximal value of m is
d!M N d−1 , we find
X
X
1
.
|S(n, N )|D M N D−1 + M N D−d+
min N,
kmαk (3.2)
1≤n≤M
m≤d!M N d−1
Proceeding as in [6], we replace α by a rational approximation a/q with |α−a/q| ≤ 1/q 2 ,
and divide the range of summation into consecutive blocks of length q. This will give
X
1
M N d−1
+ 1 · (N + q log q).
min N,
kmαk
q
d−1
m≤d!M N
Inserting into (3.2) we get
Pair Correlation of Fractional Parts
X
67
|S(n, N )|
D
MN
D−1
+M N
1≤n≤M
D−d+
M N d−1
+ 1 · (N + q log q).
q
(3.3)
Now choose q ≤ M N d−1 with |α − a/q| ≤ 1/qM N d−1 (so certainly |α − a/q| ≤
1/q 2 so (3.3) holds). Since α is Diophantine, |α − a/q| 1/q 2+ which gives q (M N d−1 )1− . Therefore
M N d−1
+ 1 · (N + q log q) (M N d−1 )1+ ,
q
and consequently
X
|S(n, N )|D M 1+ N D−1+
1≤n≤M
as required.
As an immediate consequence of this lemma, we get on repeatedly using the CauchySchwarz inequality that
Corollary 5. For α Diophantine, and M ≥ 1,
X
|S(n, N )|2 M 1+ N 2−2/D+
(3.4)
1≤n≤M
and
X
|S(n, N )| M 1+ N 1−1/D+ .
(3.5)
1≤n≤M
3.3. Proof of Proposition 3. We first show
XN +k (α) − XN (α) =
1
N2
X
0<|n|≤M
n
{sof f (n, N + k) − sof f (n, N )}
fb
N
+ O M 2+ N −2+δ−2/D .
(3.6)
We use the representation (2.5),
XN (α) =
1 X b n sof f (n, N ).
f
N2
N
n6=0
Since f ∈ Cc∞ (R), its Fourier transform fb is rapidly decreasing and so on using the
trivial estimate |sof f (n, N )| ≤ N 2 we see that for any b > 0, M = N 1+b ,
n
X
1
sof f (n, N ) + rapidly decaying term.
XN (α) = 2
fb
N
N
06=|n|≤M
Next we use |sof f (n, N )| ≤ N + |S(n, N )|2 and Corollary 5 to deduce that
X
X
|sof f (n, N + k)| ≤ M N +
|S(n, N + k)|2 M 1+ N 2−2/D .
(3.7)
06=|n|≤M
06=|n|≤M
68
Z. Rudnick, P. Sarnak
Next we claim that
n X
1
1
sof f (n, N + k) = 2
fb
2
(N + k)
N +k
N
06=|n|≤M
n
sof f (n, N + k)
fb
N
X
06=|n|≤M
+ O(M 2+ N −2+δ−2/D ).
Indeed, write
1
1
= 2 +O
2
(N + k)
N
k
N3
1
+ O(N −3+δ )
N2
=
and
nk
n
n
n
=
+ O( 2 ) =
+O
N +k N
N
N
M
N 2−δ
,
so that for |n| ≤ M , k < N δ ,
fb
n b n =f
+O
N +k
N
M
N 2−δ
.
Therefore
n sof f (n, N + k)
N +k
06=|n|≤M
n
X
1
sof f (n, N + k)
fb
− 2
N
N
06=|n|≤M
X 1
M
1
b n +O
=
f
+
O(
)
sof f (n, N + k)
N2
N 3−δ
N
N 2−δ
06=|n|≤M
n
X
1
sof f (n, N + k)
fb
− 2
N
N
06=|n|≤M
X
1
M
+
|sof f (n, N + k)|
N 4−δ N 3−δ
X
1
(N + k)2
fb
06=|n|≤M
M
M 1+ N 2−2/D = M 2+ N −2+δ−2/D
N 4−δ
by (3.7)
as required. This proves (3.6).
Next we express the difference sof f (n, N + k) − sof f (n, N ) as
X
X
e(−nαy d )
e nαxd
sof f (n, N + k) − sof f (n, N ) = 2 Re
N +1≤y≤N +k
+
X
1≤x≤N
e nα(xd − y d ) .
N +1≤x6=y≤N +k
We estimate the second term trivially by k 2 :
|sof f (n, N + k) − sof f (n, N )| ≤ k|S(n, N + k)| + k 2 .
(3.8)
Pair Correlation of Fractional Parts
69
Then inserting this into (3.6) we get
XN +k (α) − XN (α) 1
N2
k
N2
X
k|S(n, N + k)| + k 2 + M 2+ N −2+δ−2/D
0<|n|≤M
X
|S(n, N )| +
0<|n|≤M
M k2
+ M 2+ N −2+δ−2/D
N2
k
M k2
1+ 1−1/D
M
N
+
+ M 2+ N −2+δ−2/D
N2
N2
by (3.5)
M 1+ kN −1−1/D N b+δ−1/D+ .
Since b > 0 can be made arbitrarily small, this proves our proposition.
4. Triple and Higher Correlations
The higher correlations run into some basic difficulties. For example, consider the triple
correlation for αn2 mod 1. For a test function f ∈ Cc∞ (R2 ), let
X
FN (y1 , y2 ) =
f (N (y1 + m1 ), N (y2 + m2 )) .
(4.1)
(m1 ,m2 )∈Z2
This function is periodic and has a Fourier expansion
1 X b k
e(k · y).
f
FN (y) = 2
N
N
2
(4.2)
k∈Z
The triple correlation function of the sequence αn2 mod 1 and for the test function
f is
R3 (f, α, N ) =
where the sum
of R3 is
P0
1
N
0
X
FN α(x2 − y 2 ), α(y 2 − z 2 ) ,
(4.3)
1≤x,y,z≤N
is over all triples of distinct integers x, y, z. The Fourier expansion
R3 (f, α, N ) =
X
cl (N )e(lα)
(4.4)
l
with
1
cl (N ) = 3
N
0
X
1,x,y,z≤N,
k1 ,k2
k1 (x2 −y 2 )+k2 (y 2 −z 2 )=l
k 1 k1
b
,
.
f
N N
(4.5)
There is no doubt that the mean hR3 i = c0 (N ) → fb(0, 0), as N → ∞, the expected
answer for random sequence, and that more generally cl (N ) → 0 if l 6= 0. That is to say
that R3 (f, α, N ) → fb(0, 0) in the weak sense. This can probably be proven. However,
a much greater difficulty appears and that is that if f (0) 6= 0 then
70
Z. Rudnick, P. Sarnak
2
R3 (f, N ) − fb(0) N.
(4.6)
2
This shows that the L2 approach to almost-everywhere convergence is problematic in
this case. In fact, this feature of the L2 -norm being as large a manifestation of R3 being
very large at rational α’s. For almost all α we still expect that R3 (f, α, N ) → fb(0, 0).
To prove (4.6) note that as N → ∞,
X
l
1
cl (N ) = 3
N
0
X
X
1≤x,y,x≤N k1 ,k2
k 1 k1
b
,
∼ N 2 f (0).
f
N N
Hence if f (0) 6= 0 then
N |
2
X
cl (N )| ≤
l
X
!1/2
|cl (N )|
!1/2
1
lN 3
l
= N 3/2
X
2
X
!1/2
|cl (N )|2
,
l
which gives (4.6).
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