Minkowskis Theorem Chapter 2 Preface A lattice point is a point in R d with integer coordinates Later we will talk about general lattice point Lattice Point Let C R d be symmetric around the origin convex bounded and suppose that volumeCgt2 ID: 328842
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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)