On probability measures arising from lattice points on circles

1.6 MB·English

by  Par Kurlberg

ON PROBABILITY MEASURES ARISING FROM
LATTICE POINTS ON CIRCLES
PA¨R KURLBERG AND IGOR WIGMAN
5
1
Abstract. Acircle,centeredattheoriginandwithradiuschosen
0
2 so that it has non-empty intersection with the integer lattice Z2,
gives rise to a probability measure on the unit circle in a natural
n
way. Suchmeasures,andtheirweaklimits,aresaidtobeattainable
a
J from lattice points on circles.

8 We investigate the set of attainable measures and show that it
contains all extreme points, in the sense of convex geometry, of

] the set of all probability measures that are invariant under some
T
natural symmetries. Further, the set of attainable measures is
N
closed under convolution, yet there exist symmetric probability
.
h measures that are not attainable. To show this, we study the
t geometryofprojectionsontoafinitenumberofFouriercoefficients
a
m andfindthatthesetofattainablemeasureshasmanysingularities
with a “fractal” structure. This complicated structure in some
[
sense arises from prime powers — singularities do not occur for
√
1
circles of radius n if n is square free.
v
5
9
9
1 1. Introduction
0
. Let S be the set of nonzero integers expressible as a sum of two
1
0 integer squares. For n ∈ S, let
5
1 Λ := {(cid:126)λ = a+bi ∈ Z[i] : a2 +b2 = n}
n
:
v denote the intersection of the lattice Z[i] ⊂ C with a circle centered at
i √
X the origin and of radius n. For n ∈ S, let r (n) := |Λ | denote the
2 n
r cardinality of Λ ; for n (cid:54)∈ S it is convenient to define r (n) = 0. We
a n 2
define a probability measure µ on the unit circle
n
S1 := {z ∈ C : |z| = 1}
by letting
1 (cid:88)
µ := δ √ ,
n r (n) (cid:126)λ/ n
2
(cid:126)λ∈Λn
where δ denotes the Dirac delta function with support at z. The
z
measures µ are clearly invariant under multiplication by i and under
n
Date: January 12, 2015.
1
2 PA¨R KURLBERG AND IGOR WIGMAN
complex conjugation. We say that a measure on S1 is symmetric if it
is invariant under these symmetries.
Definition 1.1. A probability measure ν is said to be attainable from
lattice points on circles, or simply just attainable, if ν is a weak
limit point of the set {µ } .
n n∈S
We note that any attainable measure is automatically symmetric.
Now, if two integers m,n ∈ S are co-prime,
(1) µ = µ (cid:70)µ ,
mn m n
where (cid:70) denotes convolution on S1. Thus measures µ for n a prime
n
power are of particular interest. It turns out that the closure of the set
of measures given by µ for p ranging over all primes p ≡ 1 mod 4
pe
and exponents e ranging over integers e ≥ 1 contains µ , as well as
2k
µ for any prime q ≡ 3 mod 4, and any exponent k ≥ 0. (Note that
q2k
ql ∈ S forces l to be even.)
Motivated by the above, we say that a measure µ is prime power
attainable of µ is a weak limit point of the set {µ } . Sim-
pe p≡1mod4,e≥1
ilarly, we say that a measure µ is prime attainable if µ is a weak limit
point of the set {µ } .
p p≡1mod4
Proposition 1.2. The set of attainable measures is closed under con-
volution, and is generated by the set of prime power attainable mea-
sures.
Hence the set of attainable measures is the smallest closed w.r.t.
convolution set, containing all the prime power attainable measures.
The set of all symmetric probability measures is clearly a convex set,
hence equals the convex hull of its extreme points. Quite interestingly,
thesetofprimeattainablemeasuresisexactlythesetofextremepoints.
Now, since the set of attainable measures contains the extreme points,
and is closed under convolution one might wonder if all symmetric
probabilitymeasuresareattainable? BystudyingFouriercoefficientsof
attainable measures we shall show that not all symmetric measures
are attainable.
Given a measure µ on S and k ∈ Z, define the k-th Fourier coefficent
of µ by
(cid:90)
µˆ(k) := z−kdµ(z).
S
If µ is symmetric it is straightforward to see that µˆ(k) = 0 unless
4|k. Since µ is a probability measure, µˆ(0) = 1, hence the first two
informative Fourier coefficients are µˆ(4) and µˆ(8); note that µˆ(−k) =
LATTICE POINTS ON CIRCLES 3
µˆ(k) for all k since µ is both real and even (i.e. it is invariant under
complex conjugation).
Theorem 1.3. If µ is attainable and |µˆ(4)| > 1/3 then
(2) 2µˆ(4)2 −1 ≤ µˆ(8) ≤ M(µˆ(4)),
where
(cid:0) (cid:1)
(3) M(x) = max x4,(2|x|−1)2
denotes the “max curve”. Conversely, given x,y such that |x| ≤ 1 and
2x2 −1 ≤ y ≤ M(x),
there exists an attainable measure µ such that (µˆ(4),µˆ(8)) = (x,y).
For comparison, we note that the Fourier coefficients of the full set
of symmetric probability measures has the following quite simple de-
scription (see section 3.2 below):
{(µˆ(4),µˆ(8)) : µ is symmetric} = {(x,y) : |x| ≤ 1, 2x2 −1 ≤ y ≤ 1}.
As Figure 1 illustrates, the discrepancy between all symmetric mea-
sures and the attainable ones is fairly large. In particular, note that
the curves y = x4, y = 2x2−1, and (2|x|−1)2 all have the same tangent
at the two points (±1,1), consequently the set of attainable measures
has cusps near (±1,1). However, there are attainable measures corre-
sponding to points above the red curve for |x| ≤ 1/3.
1 1
0.5 0.5
-1 -0.5 0.5 1 -1 -0.5 0.5 1
-0.5 -0.5
2xx42−1
2x2−1 -1 (2|x|−1)2 -1
Figure 1. Left: {(µˆ(4),µˆ(8)) : µ is symmetric}. Right:
the region defined by the inequalities 2x2 − 1 ≤ y ≤
max(x4,(2|x|−1)2).
4 PA¨R KURLBERG AND IGOR WIGMAN
To give an indication of the rate at which the admissible region is
“filled out”, as well as illuminate what happens in the region |µˆ(4)| ≤
1/3, we next present the results of some numerical experiments in Fig-
ures 2 and 3.
1 1
0.5 0.5
-1 -0.5 0.5 1 -1 -0.5 0.5 1
-0.5 -0.5
yy==(x24|x|−1)2 yy==(x24|x|−1)2
-1 -1
Figure 2. Left: (µˆ (4),µˆ (8)) for n ∈ S, n ≤ 1000.
n n
Right: (µˆ (4),µˆ (8)) for n ∈ S,n ≤ 10000.
n n
Note that points lying clearly above the red curve, but below the
green one, are quite rare. However, “spikes” in the region |µˆ(n)| ≤ 1/3
are clearly present.
1
0.5
-1 -0.5 0.5 1
-0.5
yy==(x24|x|−1)2
-1
Figure 3. Left: (µˆ (4),µˆ (8)) for n ∈ S,n ≤ 100000.
n n
Right: (µˆ (4),µˆ (8)) for n ∈ S,n ≤ 1000000.
n n
1.1. Square free attainable measures. As we shall see, the spikes
in the region |µˆ(4)| ≤ 1/3 are limits of measures µ where n is divisible
n
by pe for e ≥ 2, but for measures arising from square free n ∈ S, the
structure is much simpler.
We say that a measure µ is square free attainable if µ is a limit point
of the set {µ : n ∈ S and n is square free}. The set of square free
n
attainable measures is also closed under convolution, and it is easy to
LATTICE POINTS ON CIRCLES 5
see that it is generated by the set {µ } , whose closure is the
p p≡1 mod4
set of prime attainable measures.
Theorem 1.4. If µ is square free attainable then
(4) 2µˆ(4)2 −1 ≤ µˆ(8) ≤ M(µˆ(4)).
Conversely, if 2x2−1 ≤ y ≤ M(x) there exists a square free attainable
measure µ such that (µˆ(4),µˆ(8)) = (x,y).
1.2. Prime power attainable measures. As mentioned before, the
spikes in the region |µˆ(4)| ≤ 1/3 are due to measures µ for which
n
n is divisible by a prime power pe, for e large. Recall that a mea-
sure µ is prime power attainable if µ is a weak limit point of the set
{µ } . If µ is a prime power attainable measure, then the
pe p≡1 mod4,e≥1
point (µˆ(4),µˆ(8)) can indeed lie above the curve max(x4,(2|x|−1)2) in
the region |µˆ(4)| ≤ 1/3, though this phenomenon only occurs for even
exponents (see Figure 4). In fact, we will show that for every k ∈ Z+
1
0.5
1
0.8
-1 -0.5 0.5 1
0.6
0.4
0.2 -0.5
-1 -0.5 0.5 1
yy==(x24|x|−1)2 -0.2 yy==(x24|x|−1)2 -1
Figure 4. Prime power attainable measures attainable
by pM, p ≡ 1(4) primes, M ≤ 19. Left picture: even M.
Right picture: odd M.
there exists prime power attainable µ such that
(cid:18) (cid:19)
1
(µˆ(4),µˆ(8)) = ,1 .
2k +1
1.3. Fractal structure for |µˆ(4)| ≤ 1. Let
3
(5) A := {(µˆ(4),µˆ(8)) : µ is attainable}
2
denote the projection of the set of attainable measures onto the first
two non-trivial Fourier coefficients. The intersection of A with the
2
verticalstrip{(x,y) : |x| ≤ 1/3}turnsouttohavearathercomplicated
fractal structure with infinitely many spikes — see Figure 5. Since A
2
6 PA¨R KURLBERG AND IGOR WIGMAN
is closed under multiplication and (−1,1) ∈ A it implies that it is
2
invariant w.r.t.
(6) (x,y) (cid:55)→ (−x,y),
and hence we may assume x ≥ 0.
1
0.5
-1 -0.5 0.5 1
-0.5
y=(2x 1)2
y=x4| |−
-1
Figure 5. Points (µˆ(4),µˆ(8)) for some attainable mea-
sures µ giving rise to spikes in the region |µˆ(4)| ≤ 1/3.
To be able to give a complete description of A we need a definition.
2
Definition 1.5. Let x ∈ [0,1] and a < x .
0 0
(1) We say that a pair of continuous functions
f ,f : (a,x ] → [0,1],
1 2 0
defines a cornered domain between a and x if for all x ∈
0
(a,x ] one has f (x) ≤ f (x), and f (x) = f (x) if and only if
0 1 2 1 2
x = x , whence f (x ) = f (x ) = 1.
0 1 0 2 0
(2) For a pair of functions f ,f as above the corresponding cor-
1 2
nered domain between a and x is
0
D (f ,f ) = {(x,y) ∈ R2 : x ∈ (a,x ], f (x) ≤ y ≤ f (x)}.
a,x0 1 2 0 1 2
The function f and f will be referred to as the “lower and upper”
1 2
bounds for D (f ,f ) respectively.
a,x0 1 2
LATTICE POINTS ON CIRCLES 7
Theorem 1.6. The intersection of the set A with the line y = 1 equals
2
(cid:26)(cid:18) (cid:19) (cid:27)
±1
,1 : k ≥ 1 ∪{(0,1)}∪{(±1,1)}.
2k +1
Further, for k ≥ 1, let x = 1 be the x-coordinate of a point of
k 2k+1
the intersection described above. Then, for every k ≥ 1 there exists
a pair of continuous piecewise analytic functions f , f defining a
1;k 2;k
cornered domain between 0 and x , so that A admits the following
k 2
global description:
(cid:26) (cid:27)
1
(7) A ∩ 0 < x <
2
3
(cid:32) (cid:33)
∞ (cid:26) (cid:27)
(cid:91) (cid:91) 1
= D (f ,f ) (x,y) : 0 < x < , y ≤ (2x−1)2 .
0,x 1;k 2;k
k 3
k=1
Theorem 1.6 is a rigorous explanation of the thin strips or “spikes”
connecting all the reciprocals of odd numbers on y = 1, and the curve
y = (2|x|−1)2, as in Figure 5. We remark that the functions f and
1;k
f can with some effort be computed explicitly. The lower bound f
2;k 1;k
is given as the (component-wise) product of (x ,1) by the parabola
k
y = 2x2−1 mapping (1,1) (cid:55)→ (x ,1); we re-parameterize the resulting
k
curve (x·x ,2x2 −1) so that it corresponds to the function
k
2
(8) f (x) = x2 −1,
1;k x2
k
whose slope at x is f(cid:48)(x ) = 4(2k +1).
k 1 k
The upper bound f (x) is of a somewhat more complicated na-
2
ture, see Definition 6.3; it is analytic around the corner with the slope
f(cid:48)(x ) = 4(2k +1) (see the proof of Theorem 1.6 in section 6), and it
2 k 3
is plausible that it is (everywhere) analytic. It then follows that the set
A has a discontinuity, or a jump, at x = x (this is a by-product of
2 k
the fact that the slopes of both f and f at x are positive.)
1 2 k
1.4. Discussion. Our interest in attainable measures originates in the
study [5] of zero sets (“nodal lines”) of random Laplace eigenfunctions
on the standard torus T := R2/Z2. More precisely, for each n ∈ S there
is an associated Laplace eigenvalue given by 4π2n, with eigenspace
dimension equal to r (n). On each such eigenspace there is a natural
2
notion of a “random eigenfunction”, and the variance (apropriately
normalized) of the nodal line lengths of these random eigenfunctions
equals (1+µ (4)2)/512+o(1) as r (n) → ∞. It was thus of particular
(cid:99)n 2
interest to show that the accumulation points of µ (4)2, as n ∈ S tends
(cid:99)n
toinfinityinsuchawaythatalsotheeigenspacedimensionr (n) → ∞,
2
8 PA¨R KURLBERG AND IGOR WIGMAN
is maximal — namely the full interval [0,1]. This is indeed the case
(cf. [5, Section 1.4]), but a very natural question is: which measures
are attainable?
In order to obtain asymptotics for the above variance it is essential
to assume that the eigenspace dimension grows, and one might wonder
if “fewer” measures are attainable under this additional assumption.
However, as the following shows, this is not the case (the proof can be
found in section 4.4.)
Proposition 1.7. A measure µ ∈ P is attainable (i.e. µ ∈ A), if
and only if there exists a sequence {n } such that µ ⇒ µ with the
j nj
additional property that r (n ) → ∞.
2 j
1.5. Outline. For the convenience of the reader we briefly outline the
contents of the paper. In Section 2 we give some explicit examples of
attainable, and non-attainable measures, and describe our motivation
for studying the set of attainable measures. In Section 3 we give a
brief background on Fourier coefficients of probability measures, and
inSection4werecallsomeneededfactsfromnumbertheoryalongwith
proving the more basic results above. Section 5 contains the proof of
Theorem 1.3 (a complete classification of attainable measures in the
region |µˆ(4)| > 1/3), and Section 6 contains the proof of Theorem
1.6 (the complete classification of attainable measures in the region
|µˆ(4)| ≤ 1/3), postponing some required results of technical nature to
the appendix. Finally, in Section 7, we classify the set of square-free
attainable measures.
1.6. Acknowledgements. We would like to thank Ze´ev Rudnick and
Mikhail Sodin for raising the problem considered in this manuscript,
and the many fruitful discussions concerning various subjects related
to the presented research. We thanks Fedor Nazarov and Peter Sarnak
for many stimulating and fruitful discussions leading to some improve-
ments of our results.
P.K. was partially supported by grants from the Go¨ran Gustafs-
son Foundation for Research in Natural Sciences and Medicine, and
the Swedish Research Council (621-2011-5498). The research lead-
ing to these results has received funding from the European Research
Council under the European Union’s Seventh Framework Programme
(FP7/2007-2013), ERC grant agreement no 335141 (I.W.), and an EP-
SRC Grant EP/J004529/1 under the First Grant Scheme (I.W.).
LATTICE POINTS ON CIRCLES 9
2. Examples of attainable and unattainable measures
2.1. Some conventions. Let
3
1 (cid:88)
˜
δ := δ
0 4 ik
k=0
be the atomic probability measure supported at the 4 symmetric points
±1, ±i (“Cilleruelo measure”). Given an angle θ ∈ [0,π/4], let
3
1 (cid:88)
(9) δ˜ := δ˜ (cid:70)(δ +δ )/2 = (δ +δ );
θ 0 eiθ e−iθ 8 ei(πk/2+θ) ei(πk/2−θ)
k=0
recall that (cid:70) denotes convolution on S1. For θ = 0,π/4 the measure
˜
δ is supported at 4 points whereas for all other values of θ the support
θ
consists of 8 points. Given an integer m ≥ 1 and θ ∈ [0,π/4], let
(cid:32) (cid:33)
l
1 (cid:88)
δ˜ := δ˜ (cid:70) δ .
θ,m 0 m+1 eiθ(m−2j)
j=0
˜ ˜
We note that δ = δ , and that µ is symmetric if and only if µ is
θ θ,1
˜ ˜
invariant under convolution with δ ; convolving with δ is a convenient
0 0
way to ensure that a measure is symmetric.
2.2. Some examples of attainable and unattainable measures.
Given θ ∈ [0,π/4] let τ denote the symmetric probability measure
θ
with uniform distribution on the four arcs given by
{z : |z| = 1,arg(z) ∈ ∪4 [kπ/2−θ,kπ/2+θ]}.
k=0
Using some well known number theory given below (cf. section 4) it
is straightforward to show that τ is attainable for all θ ∈ [0,π/4]. In
θ
particular, dµ = dτ , the Haar measure on S1 normalized to be a
Haar π/4
probability measure, is attainable. In fact, it is well known (see e.g. [2])
that there exists a density one subsequence {n } ⊆ S, for which the
j
corresponding lattice points Λ become equidistributed on the circle;
nj
this gives another construction of dµ as an attainable measure.
Haar
It is also possible to construct other singular measures. In Section 4
we will outline a construction of attainable measures, uniformly sup-
ported on Cantor sets. Moreover, if q is a prime congruent to 3 modulo
4 it is well known that the solutions to a2 + b2 = q2 are given by
˜
(a,b) = (0,±q), or (±q,0), thus δ is attainable. A subtler fact, due
0
to Cilleruelo, is that there exists sequences {n } for which Λ has
j j≥1 nj
very singular angular distribution even though the number of points
r (n ) tends to infinity. Namely, it is possible to force all angles to
2 j
10 PA¨R KURLBERG AND IGOR WIGMAN
be arbitrarily close to integer multiples of π/2, hence 1 (cid:80)3 δ is an
4 k=0 ik
accumulation point of dµ as n → ∞ in such a way that r (n ) → ∞.
nj j 2 j
We may also construct some explicit unattainable probability mea-
sures on S1 satisfying all the symmetries; in fact the following corollary
of Theorem 1.6 constructs explicit unattainable measures, remarkably
supported on 8 points only — the minimum possible for symmetric
unattainable measures.
Corollary 2.1 (Corollary from Theorem 1.6). The probability measure
˜ ˜
η := aδ +(1−a)δ
a 0 π/4
is attainable, if and only if a = 0, 1, 1 or a is of the form
2
1 1
a = ±
2 2(2k +1)
for some k ≥ 1.
3. Fourier analysis of probability measures
3.1. Some notation and de-symmetrization of probability mea-
sures. It is convenient to work with two models: either with the unit
circle embedded in C, or
T1 := R/2πZ.
Rather than working with {µ } and its weak partial limits, for nota-
n
tional convenience we work with their de-symmetrized variants, i.e.
(cid:18) (cid:19)
θ
(10) dν (θ) = dµ ,
n n
4
θ ∈ T1. The measures ν are invariant under complex conjugation
n
(where thought of S1 ⊆ C); equivalently, for θ ∈ T1,
dν (−θ) = dν (θ).
n n
Notation 3.1. Let P be the set of all probability measures µ on S1
satisfying for θ ∈ T1
(11) dµ(−θ) = dµ(θ).
Further, let A ⊆ P be the set of all weak partial limits of {ν } i.e.
n
all probability measures µ ∈ P such that there exists a sequence {n }
j
with
ν ⇒ µ.
nj