Lattices and Minkowski’s Theorem

Lattices and Minkowski’s Theorem Chapter 2

Preface

Lattice Point A lattice point is a point in Rd with integer coordinates. Later we will talk about general lattice point.

Minkowski’s Theorem 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. 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=4 Vol=4*4=16>22=4

Proof

Claim C’+v C’

Proof –Claim(1) C’+v C’ 2M C 2M

Proof –Claim(2) Volume(cube) Upper bound Possibilites of v in [-M,M]d K 2M+2D

Proof –Claim(3)

Proof-Minkowski’s Theorem C’+v x C’

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.

Proof K D=26m D=0.16m l S

PropositionApproximating an irrational number by a fraction Note: This proposition implies that there are infinitely many pairs m,n such that:

Proof

General Lattices

TheoremMinkowski’s theorem for general lattices

Proof f

Discrete subgroup of Rd

TheoremLattice basis theorem

Proof(1)

Proof(2)

Proof(3) v v’

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

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.

Definitions-Number Theory 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. Example: 42≡6(mod 10) so 6 is a quadratic residue (mod 10).

Proof(Theorem) q2≣-1(mod p) 0≣ 2p C

Lattices and Minkowski’s Theorem