332 VIII The Geometry of Numbers
As forS 1 , it may be seen thatSk+ 1 consists of all positive integer multiples ofμk+ 1.
Hence anyx∈Λ∩〈y 1 ,...,yk+ 1 〉has the form
x=ζ 1 x 1 +···+ζkxk+ζk+ 1 xk+ 1 ,whereζ 1 ,...,ζk∈Randζk+ 1 ∈Z.Since
x−ζk+ 1 xk+ 1 ∈Λ∩〈y 1 ,...,yk〉,we must actually haveζ 1 ,...,ζk∈Z.
By being more specific in the proof of Proposition 7 it may be shown that there is
auniquechoice ofx 1 ,...,xmsuch that
y 1 =p 11 x 1
y 2 =p 21 x 1 +p 22 x 2
···
ym=pm 1 x 1 +pm 2 x 2 +···+pmmxm,wherepij∈Z,pii>0, and 0≤pij<piiifj<i(Hermite’s normal form).
It is easily seen that in Proposition 7 we can choosexi=yi( 1 ≤i≤m)if and
only if, for anyx∈Λand any positive integerh,xis an integral linear combination
ofy 1 ,...,ymwheneverhxis.
By combining Propositions 6 and 7 we obtain
Proposition 8Fo r a s e tΛ⊆Rnthe following two conditions are equivalent:
(i)Λis a discrete subgroup ofRnand there exists R> 0 such that, for each y∈Rn,
there is some x∈Λwith|y−x|<R;
(ii)there exist n linearly independent vectors x 1 ,...,xninRnsuch that
Λ={ζ 1 x 1 +···+ζnxn:ζ 1 ,...,ζn∈Z}.Proof If (i) holds, then in the statement of Proposition 7 we must havem=n,i.e.
(ii) holds. On the other hand, if (ii) holds thenΛis a discrete subgroup ofRn,by
Proposition 6. Moreover, for anyy∈Rnwe can choosex∈Λso that
y−x=θ 1 x 1 +···+θnxn,where 0≤θj< 1 (j= 1 ,...,n), and hence
|y−x|<|x 1 |+···+|xn|. AsetΛ⊆Rnsatisfying either of the two equivalent conditions of Proposition 8
will be called alatticeand any element ofΛalattice point. The vectorsx 1 ,...,xn
in (ii) will be said to be abasisfor the lattice.
A lattice is sometimes defined to be any discrete subgroup ofRn, and what we
have called a lattice is then called a ‘nondegenerate’ lattice. Our definition is chosen
simply to avoid repetition of the word ‘nondegenerate’. We may occasionally use the