Lattices and Minkowski’s Theorem

Chapter 2

Geometry of Numbers

Number Theory

Preface

A lattice point is a point in Rd with integer coordinates.

Later we will talk about general lattice point.

Lattice Point

Let C ⊆ Rd be symmetric around the origin, convex, bounded and suppose that volume(C)>2d. Then C contains at least one lattice point different from 0.

Minkowski’s Theorem

Definitions* A C set is convex whenever x,y∊C  implies

segment xy∊C .*  An object C  is centrally around the origin if

whenever (0,0) ∊ C and if x∊C then -x∊C.

Examples (d=2)Vol=2*2=4<22=4Vol=4*4=16>22=4

Proof}.:

21{

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Claim

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such

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Proof –Claim(1)

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:Q Define number.integer large a be MLet

false. is claim theSuppose ion.contradictBy

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C’

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2M

2M

C

Proof –Claim(2)

d

d

MDCvol

CvolMCvolQ

)12121()'(

)'()12()'(2D)(2Mvol(K)

:Hence C'. ofdiameter thedenotes D whereD]MD,-[-MK cube enlarged in the contained all are

They well.asdisjoint are s translate theseof every two thusand,assumption the toaccording C' fromdisjoint is latesuch transEach

d

d

Volume(cube) Possibilites of v in [-

M,M]d

K2M+2D

Upperbound

Proof –Claim(3)

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1 exceedingnumber fixed a is 1)(2)vol(C'hand,other On the

M. largely sufficientfor 1 toclose arbitrarly is side hand-right on the expression The

1)12121()'(

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MDCvol Md

Proof-Minkowski’s Theorem

theorem.sMinkowski' proves which C, that vmeans

'.21)(

21

21 have weso and too,C'in lies x)-x(v

segment theofmidpoint theconvex, is C' Since C'.x-obtain v wesymmetric, is C' since and ,C'v- xhave Then we

v).(C'C'point x a choose uslet claim, the toAccording

This

Cvxvx

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ExampleLet K be a circle of diameter 26 meters centered at the origin. Trees of diameter 0.16 grow at each lattice point within K except for the origin, which is where Shrek is standing. Prove Shrek can’t see outside this mini forest.

Proof

theorem.sMinkowski' scontradict which 4,16.40.16*26(C)But volume

origin. but the points lattice no containsconvex) itself is setsconvex ofon intersecti (The SKCset convex symetric theMeaning,

sight). thebolcked have would treea (otherwiseorigin for theexcept K in point lattice no contains line middle theas

l with 0.16 width of S strip that themeans This origin. ethrough thpassing l line some along outside see couldShrek than Suppose

K

D=26m

D=0.16m

Sl

PropositionApproximating an irrational number by a fraction

nN1

nm- and Nn

such that n m, numbers natural ofpair a exists Then there number. natural a N andnumber real a be (0,1)Let

Note: This proposition implies that there are infinitely many pairs m,n such that:

2

1||nm

m

Proof

.1nm- Meaning,

.m)y n,(x 1m-n and Nn gives C of definition The 0.n

assumemay wesymmetry,By m).(n,point latticeinteger nonzero some containsit

theorem,sMinkowski' toaccording thereforeand 4N21)(2N area ofset convex symetric a is This

}1,21

21-N- :Ry){(x,C

:Define

2

nN

N

NyxNx

General Lattices

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TheoremMinkowski’s theorem for general lattices

0. fromdifferent ofpoint a contains CThen .det2)with vol(C

setconvex symmetric a be RClet ,Rin lattice a be Let d

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and

Proof

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andsMinkowski

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Discrete subgroup of Rd

0.number real positive fixed somefor ,least at is of pointsdistinct any two of distance thesuch that and

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TheoremLattice basis theorem

).z,...,z,z(such that Rz,...,z,z

t vectoresindependenlinearly d exsits thereis, that basis; a has Then .R isspan linear whoseR of subgroup discrete a be R

d21d

d21

ddd

Let

Proof(1)

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all then ,z,...,z,zby spanned subspace ldimensiona 1)-(i thedenotes F If d.constructebeen already haveproperty following with thez,...,z,z t vectorsindependenlinearly that suppose 1,di1 i, someFor induction.

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Proof(2)

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Proof(3)

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. than F nearer to be would v'otherwisefor 0 havemust

weTherefore, P. ipedparallelep in the lies v'10 Since .in lies also '....v'

i.1,2,...,j , :Define .,..., numbers real some

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Question…

How efficiently can one actually compute a nonzero lattice point in a symmetric convex body?

hard.-NP be known to is problem theinput, theofpart a as considered is d If time.polynomialin solved becan problemsuch constant, a as considered is d If

An application in Number Theory

Theorem

.,,ap :squares twoof sum a as written becan 4) (mod 1p primeEach 22

Zbab

LemmaIf p is a prime with p ≡ 1(mod 4) then -1 is a quadric residue modulo p.

  For a given positive integer n, two integers a and b are called congruent modulo n, written a ≡ b (mod n) if a-b is divisible by n.For example, 37≡57(mod 10) since 37-57=-20 is a multiple of 10.

Definitions-Number Theory

.nonresidue quadratic a is a Otherwise,p). (mod with x*Fan x exists

thereif p modulo residue quadratic a called is *Faelement An {0}.\FF*let and number, prime a is p where

p, modulo classes residue of field for the stand GF(p)FLet

2 a

Example: 42≡6(mod 10) so 6 is a quadratic residue (mod 10).

Proof(Theorem)

.a Therefore

).(mod0)1(2)(a calculate We

).,z of definition(by jpiqb and ia that meanswhich Z,ji, somefor izb)(a, time,same At the .2a0 have We

{0}.\b)(a,point a contains C so and ,4det 4pp2 is C of area The2p}.yx:Ry){(x, Cdisk the

for lattices generalfor theoremsMinkowski' use Wep.det have Wep).(0,z and q)(1, z where),,(z lattice heConsider t

).(mod1qsuch that qnumber a choosecan welemma, By the

22

22222222222

21

2122

222

2121

2

pb

pqipjiqjpqiijiqib

zjzpb

zp

2pC

0≣q2≣-1(mod p)