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.