Chapter 2

A lattice point is a point in R d with integer coordinates. Later we will talk about general lattice point.

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. * 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.

Vol=2*2 = 4<2 2 =4 Vol=4*4 = 16>2 2 =4

C’ C’+v

C’C’+v 2M C

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

C’ C’+v x

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.

K D=26m D=0.16m S l

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

f

v v’

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

Theorem If 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. Example: 4 2 ≡6(mod 10) so 6 is a quadratic residue (mod 10).

2p2p C 0≣0≣ q 2 ≣ -1(mod p)