Number Theory: An Introduction to Mathematics

334 VIII The Geometry of Numbers

Rn=∪

x∈Λ

(Π+x),

int(Π+x)∩int(Π+x′)=∅ ifx,x′∈Λandx=x′.

For any latticeΛ⊆Rn,thesetΛ∗of all vectorsy∈Rnsuch thatytx∈Zfor
everyx∈Λis again a lattice, thedual(or ‘polar’ or ‘reciprocal’) ofΛ. In fact,

ifΛ={Tz:z∈Zn}, thenΛ∗ ={(Tt)−^1 w:w∈Zn}.

HenceΛis the dual ofΛ∗and d(Λ)d(Λ∗)=1. A latticeΛisself-dualifΛ∗=Λ.

3 Proof of the Lattice Point Theorem; Other Results................

In this section we take up the proof of Minkowski’s lattice point theorem. The proof
will be based on a very general result, due to Blichfeldt (1914), which is not restricted
to convex sets.

Proposition 9Let S be a Lebesgue measurable subset ofRn,Λa lattice inRnwith
determinantd(Λ)and m a positive integer.
Ifλ(S)>md(Λ), or if S is compact andλ(S)=md(Λ), then there exist m+ 1
distinct points x 1 ,...,xm+ 1 of S such that the differences xj−xk( 1 ≤j,k≤m+ 1 )
all lie inΛ.

Proof Letb 1 ,...,bnbe a basis forΛand letP be the half-open parallelotope
consisting of all pointsx=θ 1 b 1 +···+θnbn,where0≤θi< 1 (i= 1 ,...,n).
Thenλ(P)=d(Λ)and

Rn=∪

z∈Λ

(P+z), (P+z)∩(P+z′)=∅ ifz=z′.

Suppose first thatλ(S)>md(Λ). If we put

Sz=S∩(P+z), Tz=Sz−z,

thenTz⊆P,λ(Tz)=λ(Sz)and

λ(S)=

∑

z∈Λ

λ(Sz).

Hence
∑

z∈Λ

λ(Tz)=λ(S)>md(Λ)=mλ(P).

SinceTz⊆Pfor everyz, it follows that some pointy∈Pis contained in at least
m+1setsTz. (In fact this must hold for allyin a subset ofPof positive measure.)
Thus there existm+1 distinct pointsz 1 ,...,zm+ 1 ofΛand pointsx 1 ,...,xm+ 1 ofS
such thaty=xj−zj(j= 1 ,...,m+ 1 ).Thenx 1 ,...,xm+ 1 are distinct and