Sums of three squares and spatial
statistics on the sphere
Aisenstadt lecture
Zeev Rudnick Cissie & Aaron Beare Chair,
Tel Aviv University
joint with Jean Bourgain & Peter Sarnak (IAS)
Sums of 4 squares
Lagrange (1770): Every positive integer is a sum of 4 squares
n=x2+y2 +z2+w2
Jacobi (1834): The number of such representations
4
|0mod4
( ) 8d nd
N n d
Example: n=7 then we have N(7) =8(7+1)=64
( 2, 1, 1, 1 ), ( 1, 2, 1, 1), ( 1, 1, 2, 1), ( 1, 1, 1, 2)
Sums of two squares
Fermat: Primes are of the form p=x2+y2 ↔ p≠3 mod 4
( 50% of the primes)
An integer n>0 is a sum of two squares ↔ ordq(n) is even
for all primes q=3 mod 4
n=2c∏I pia(i)∏j qj
2b(j) pi =1 mod 4, qj =3 mod 4 primes,
e.g. 325=(3•2)2+(3•1)2 , but 33≠□+□
(0% of the integers !)
The number of representations
In terms of prime decomposition: if n=2c∏I pia(i)∏j qj
2b(j)
pi =1 mod 4, qj =3 mod 4 primes, then: N2(n) =4∏i (a(i)+1)
Example: n=21125=53 ∙132 → N2(n)=48=4∙(3+1)∙(2+1)
2 2 2
2 : # ( , ) :N n x y x y n Z
Upper bound: N2(n) << nε , for all ε>0.
This is the number of points on a circle of radius √n
Landau (1908): “on average” <N2(n)>= const∙√log(n) .
Sums of 3 squares
)78(4222 bnzyxn aLegendre/Gauss:
( n>0)
Primitive representation: gcd(x,y,z)=1
n is primitively represented as a sum of 3 squares ↔ n≠0,4,7 mod 8
Exercise: if n=4a then Nn=6
)2,1,0(),1,2,0(),2,0,1(
),1,0,2(),0,2,1(),0,1,2(
nzyxzyxNn 222:),,(#:
Example: n=5 then we have N5=24
The number of representations
The number of representations
Gauss’ formula : For squarefree n, Nn≈√n •L(1,χ-n)
1(1, ), 0nL
n
Siegel:
If n is primitively representable as a sum of three squares then Nn=n1/2±o(1)
1
1
( ) ( )( , ) 1
s sn p
n pL s
n p
( )n
np
p
Legendre symbol
Dirichlet L-function
GRH implies (1, ) 1/ lognL n
(1, ) lognL n Upper bound
Spatial distribution of solutions
Project the different representations of n
to the unit sphere S2:
2),,(1
),,( Szyxn
zyx
We get a set L(n) of Nn ≈√n points on S2
- call them “Linnik points”
Goal: are the point sets L(n) “random” or “rigid” ?
Random and rigid points sets
Binomial process: N independent points, each uniformly distributed on S2 :
kNk AAk
NkA
))(1()()(# Bin(N)Prob
Random sets:
“Rigid” sets: Want point sets which look like a lattice in the plane -
doesn’t exist on the sphere
Uniform distribution on S2
)(area
)(area
)(#
))((#2S
B
nE
BnEn
Definition: A collections of subsets E(n) in S2 become uniformly distributed if for any
nice set B in S2
Equivalently, for any continuous function fεC(S2),
2
)()(area
1)(
)(#
12
)( S
nnEP
dxxfS
PfnE
Linnik’s conjecture
Uniform distribution (Linnik’s conjecture)
As n→∞, n≠0,4,7 mod 8, the sets L(n) becomes uniformly distributed on S2 .
Proved by Linnik, partially assuming GRH, (1940),
Proved unconditionally by Duke, Golubeva-Fomenko (1988), (via Iwaniec).
A similar result holds in higher dimension
(Pommerenke, 1959)
Uniform distribution & its failure on
the circle Kátai-Környei (1977), Erdos-Hall (1999): For “almost all” n=□+□ ,
the projected lattice points (x,y)/√n εS1 are uniformly distributed
However:
Cillereuello: The is a subsequence of n’s so that the projected lattice points
converge to an average of 4 delta-masses
Kurlberg-Wigman (2014): Classified all possible limit measures on S1 .
2 2 12
1, ( )
( ) x y n S
x yf f z dz
N n n n
2 2
2
1 1, (1,0) ( 1,0) (0,1) (0, 1)
( ) 4x y n
x yf f f f f
N n n n
Beyond equidistribution :
randomness on smaller scales
Uniform distribution means randomness on scale of O(1) – subsets in S2 of fixed size.
Question: randomness on smaller scales?
The electrostatic energy
Visualization: Rob Womersley
N
i ij ji
NPP
PPEnergy1
1||
1:),,(
The electrostatic energy of N points on the sphere S2 is
Thomson’s question (1904): Find configurations of charges on
the sphere which minimize energy (stable configurations)
J.J. Thomson, Nobel prize 1903
The known minimal energy configurations
Minimum energy configurations have been rigorously identified only for N=2,3,4,5,6,12
the optimal configuration consists of electrons residing at vertices of
• N=2: antipodal points.
• N=3: equilateral triangle about a great circle (Foppl, 1912)
• N=4: regular tetrahedron.
• N=5: triangular dipyramid (rigorous proof: R. Schwartz, 2013)
• N=6: regular octahedron. (Yudin 1993)
• N=12: regular icosahedron. (Andreev 1996)
triangular dipyramid
Finding stable configurations is notoriously difficult; known numerically for n <112 .
N=7
Energy of stable configurations
for large N
Wagner (1992): The energy of stable configurations is ~ N2 :
22/3
1
2
1,,
~)(||
),,Energy(min2 2
21
NNOyx
dxdyNPP
S S
NSPP N
Question: What is the energy for Linnik points L(n)?
Peled (2010): for N “random” points, Energy~N2 almost surely
N
i ij ji
NPP
PPEnergy1
1||
1:),,(
The energy of Linnik points
Theorem (Bourgain, ZR, Sarnak): The energy of the Linnik points L(n) is close to minimal
)())(( 22 NONnLEnergy
Proof: equidistribution + control of # of close neighbours
)( ||
1:))((
nLP PQ QPnLEnergy
NxP
dxN
QPS
QPnLQ
2 ||
~||
1
)(
Would like to use uniform distribution to claim that for each P
2
)(
~))(( NNnLEnergynLP
Problem: The function Q → 1/|P-Q| is not continuous !
In fact a point Q with |P-Q|<1/N1-o(1) , gives a contribution bigger than main term N
Example: A close pair
)0,1,(1
)0,,1(1
,)1( 22 kkn
Qkkn
Pkkn
1 (1)
(1, )2 1| | n
o
LP Q
N Nn
A packing argument leads us to believe that typical nearest neighbor distance is 1/√N .
Packing argument
Claim: For any set X of N points on the sphere
Xx
x 16)(
Indeed, the (spherical) area of a disk of (euclidean) radius A on the sphere is πA2 ;
around each x draw a disk of radius 𝛿/2 , which has area πδ(x)/4; these are disjoint so
their total area is at most the area of the sphere, which is 4 π
Xx
Sx
4)(area4
)( 2
Xx
X Nx
Nx
16)(
1:
Squared nearest neighbor distance:
2||min:)( yxxxy
Controlling close pairs
hyxnyxyxhnA 2223 ||,|||:|,#:),( Z
Venkov 1931, Pall 1948 : Explicit computation of local factors (crucial).
2/1),gcd(),( hnnhnA
- allows to control contribution of “close” pairs and show Energy(L(n))~N2 . QED
Counting pairs of points at a given distance
2)2(|
2 ),(),(24),(
phnhp
p hnhnhnA Siegel’s mass formula:
h≠0
hyxnyxpyxp
hn k
kk
p
222
3||,|||:|mod,#
1),( lim
For random points
1) Mean value <δ> ~ 4/N
2) Distribution of normalized squared nearest neighbor distances δ(x)/<δ> is exponential
- in particular is unbounded
Nearest neighbor distances
( )
1( ) : ( )
nx E nn
x xN
Squared nearest neighbor distance:
A packing argument leads us to believe that typical nearest neighbor distance is 1/√N .
Dahlberg (1978): For least energy configurations, the nearest neighbor distances are
ALL commensurable to 1/√N : There are 0<c<C s.t. for all N, any stable configuration
S(N) of N points on the sphere
2||min:)( yxxxy
N
Cx
N
cNSx )(:)( rigidity
Distribution of nearest neighbor distances
Conjecture: For the Linnik points L(n), the squared n.n. distances behave like those of
random pts
1) Mean: <δ>n~ 4/Nn = random value
2) Distribution P(s) of δ(x)/<δ> is exponential: P(s)=exp(-s)
THM (BRS): Any possible limit P(s) is absolutely continuous (assuming GRH)
The least spacing statistic
For N random points, least spacing is a.s. 1/N1±o(1) (lower & upper bound) --
“birthday paradox”
However for “rigid” points, and for minimal energy configurations,
the least spacing is of order 1/√N (Dahlberg 1978)
||min:)(
)(,
min yxnd
nEyxyx
N
CNSd
N
c ))((min
Least spacing for Linnik points Because the point sets L(n) come from integer points, the least spacing is >1/√n
Theorem: For almost all n, dmin(L(n))≈n-1/2+o(1)=N-1+o(1)
Implied by: almost all n is a sum of two squares
and a “mini-square”
nzzyxn ||,222
Wooley (2013): For a.e. n, can make |z|<(log N)1+o(1)
(x,y,+z)
(x,y,-z)
2z
Linnik conjectured that this holds for ALL n
)1(1
||||)(,
min
1min||min:))((
22
o
QnP
QPnLyx
yxN
nyxnLd
i.e. random-like behaviour
Summary
We studied properties of the sets L(n) of points on the sphere arising from writing
n=x2+y2+z2 which go beyond uniform distribution:
•The electrostatic energy is close to minimal .
•Distribution of nearest neighbors seems random
•Least spacing is typically random
•Some theoretical results.