Lattices and - PowerPoint Presentation

Slide1

Lattices and Minkowski’s Theorem

Chapter 2Slide2

PrefaceSlide3

A lattice point is a point in R

d

with integer coordinates.

Later we will talk about general lattice point.

Lattice PointSlide4

Let C ⊆ R

d

be symmetric around the origin, convex, bounded and suppose that volume(C)>2

d

. Then C contains at least one lattice point different from 0.Minkowski’s TheoremDefinitions

* 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

.Slide5

Examples (d=2)

Vol

=2*2

=

4<22=4Vol=4*4=16>22=4Slide6

ProofSlide7

Claim

C’

C’+vSlide8

Proof –Claim(1)

C’

C’+v

2M

2M

CSlide9

Proof –Claim(2)

Volume(cube)

Possibilites

of v in [-M,M]

d

K

2M+2D

Upper

boundSlide10

Proof –Claim(3)Slide11

Proof-Minkowski’s Theorem

C’

C’+v

xSlide12

Example

Let 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.Slide13

Proof

K

D=26m

D=0.16m

S

lSlide14

Proposition

Approximating an irrational number by a fraction

Note

: This proposition implies that there are

infinitely many pairs m,n such that:Slide15

ProofSlide16

General LatticesSlide17

Theorem

Minkowski’s

theorem for general latticesSlide18

Proof

fSlide19

Discrete subgroup of RdSlide20

TheoremLattice basis theoremSlide21

Proof(1)Slide22

Proof(2)Slide23

Proof(3)

v

v’Slide24

Question…

How efficiently can one actually compute a nonzero lattice point in a symmetric convex body?Slide25

An application in Number Theory

Theorem

Lemma

If p is a prime with p ≡ 1(mod 4) then -1 is a quadric residue modulo p.Slide26

  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

Example: 42

≡6(mod 10) so 6 is a quadratic residue (mod 10).Slide27

Proof(Theorem)

2p

C

0≣

q

2

≣-1(mod p)