352 VIII The Geometry of Numbers
6 Mahler’sCompactnessTheorem...............................
It is useful to study not only individual lattices, but also the familyLnof all lattices
inRn. A sequence of latticesΛk∈Lnwill be said toconvergeto a latticeΛ∈Ln,
in symbolsΛk→Λ, if there exist basesbk 1 ,...,bknofΛk(k= 1 , 2 ,...)and a basis
b 1 ,...,bnofΛsuch that
bkj→bjask→∞ (j= 1 ,...,n).Evidently this implies that d(Λk)→d(Λ)ask→∞.Also,foranyx∈Λthere
existxk∈Λksuch thatxk→xask→∞. In fact ifx=α 1 b 1 +···+αnbn,where
αi∈Z(i= 1 ,...,n), we can takexk=α 1 bk 1 +···+αnbkn.
It is not obvious from the definition that the limit of a sequence of lattices is
uniquely determined, but this follows at once from the next result.
Proposition 19LetΛbe a lattice inRnand let{Λk}be a sequence of lattices inRn
such thatΛk→Λas k→∞.Ifxk∈Λkand xk→xask→∞,thenx∈Λ.
Proof With the above notation,
x=α 1 b 1 +···+αnbn,whereαi∈R(i= 1 ,...,n), and similarly
xk=αk 1 b 1 +···+αknbn,whereαki∈Randαki→αiask→∞(i= 1 ,...,n).
The linear transformationTkofRnwhich mapsbitobki(i = 1 ,...,n)can be
written in the form
Tk=I−Ak,whereAk→Oask→∞. It follows that
Tk−^1 =(I−Ak)−^1 =I+Ak+A^2 k+···=I+Ck,where alsoCk→Oask→∞. Hence
xk=Tk−^1 (αk 1 bk 1 +···+αknbkn)
=(αk 1 +ηk 1 )bk 1 +···+(αkn+ηkn)bkn,whereηki→0ask→∞(i= 1 ,...,n).Butαki+ηki∈Zfor everyk. Letting
k→∞, we obtainαi∈Z.Thatis,x∈Λ.
It is natural to ask if the Voronoi cells of a convergent sequence of lattices also
converge in some sense. The required notion of convergence is in fact older than the
notion of convergence of lattices and applies to arbitrary compact subsets ofRn.
TheHausdorff distanceh(K,K′)between two compact subsetsK,K′ofRnis
defined to be the infimum of allρ>0 such that every point ofKis distant at most
ρfrom some point ofK′and every point ofK′is distant at mostρfrom some point