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Annals of Mathematics
A Mean Value Theorem in Geometry of Numbers
Author(s): Carl Ludwig Siegel
Source: The Annals of Mathematics, Second Series, Vol. 46, No. 2 (Apr., 1945), pp. 340-347
Published by: Annals of Mathematics
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ANNALS
OF MATHEMATICS
Vol. 45, No. 2, April, 1945
A MEAN VALUE THEOREM IN GEOMETRY OF NUMBERS
By
CARL LUDWIG SIEGEL
(Received December 8, 1944)
I. Let R be the space of the n-dimensionalreal vectorsx, withn > 1, denote
by dxj} the euclidean volume elementin R and considera bounded function
f(x) whichis integrablein the Riemannsense and vanisheseverywhereoutside
a boundeddomainin R. RecentlyE. Hlawka1provedthe followingremarkable
proposition:
For any arbitrarilysmall positive E thereexists a real n-rowed matrix A of determinant] A = 1 such that
Z
(1)
f(Ag) _
f
f(x) {dx } +
E,
where the summation is carried over all integral vectorsg - 0.
As a consequenceof his theoremHlawka deduced an assertionofMinkowski
whichhad remainedunprovedformore than fiftyyears:
If B is an n-dimensional star domain of volume < t(n), thenthereexists a lattice of determinant1 such thatB does not contain any latticepoint # 0.
This statementhad been announcedby Minkowskion severaloccasions,2and
eine arithmetischeTheorie
he observed: "Der Nachweis dieses Satzes erfordert
der Gruppe aus allen linearen Transformationen." Later this arithmetical
theorywas createdin the shape of Minkowski'smethodof reductionof positive
quadraticforms;but he did not come back to his assertionon star domains,except forthe special case connectedwith the closestpackingof spheres.
as one mightwish; however,
Hlawka's proofis as simpleand straightforward
it does not make clearthe relationto the fundamentaldomainofthe unimodular
group whichwas in Minkowski'smind. This relationwill become obvious in
the theoremwhich we are goingto state.
2. Let % denotethe multiplicativegroupof all real n-rowedA with I A I = 1.
The (n2 - 1) -dimensionalgroupspace Q1possessesan invariantvolumeelement
dw,unique up to a constantfactor. The properunimodulargroupFi is the subgroup consistingof all integralA in U1. We shall defineon Ui a fundamental
regionF with respectto P1, and we shall prove,as an immediateconsequence
of Minkowski'sreductiontheory,that the volume of F is finite. Now we determinethe arbitraryfactorin the definitionof dcoby the conditionthat F has
the volume 1. The connectionbetweenHlawka's theorem(1) and Minkowski's
reductiontheoryis providedby the following
1 EDMUND
2
HERMANN
HLAWKA,
Zur Geometrieder Zahlen, Math. Zeitschr. 49 (1944), pp. 285-312.
GesammelteAbhandlungen, vol. I, p. 265, p. 270, p. 277.
MINKOWSKI,
340
341
GEOMETRY OF NUMBERS
THEOREM: Let g run overall integralvectors-
f
(2)
Jf(Ag) dw =
0, then
ff(x) {dx}.
It followsimmediatelyfrom(2) that (1) holds fora suitablychosenA in F,
even withe = 0.
It is worthnotice that the proofof (2) also leads to the value of the volume
of F, in termsof an independentlydefinedvolume elementon i1. The result
is closely related to Minkowski'swell known formulafor the volume of the
domainof reducedpositivequadraticformswithdeterminant< 1; it seemsthat
our methodpresentsthe most satisfactoryway of provingthis formula.
real n-rowedmatricesY. The
3. Now considerthegroupU ofall non-singular
=
all mappings Y ->+ YC,
M
is
matrix
invariant
under
(dY)Y-'
differential
C e Q, of the groupspace U onto itself,and the positivedefinitequadratic differRiemannianmetric. Plainly
ential formu(M'M) defineson U a right-invariant
a
on
the
induces
(n2 - 1)-dimenright-invariant
subgroupspace U,
this metric
a
to
is
define
It
multiple
element.
certain
constant
volume
practical
sional
element
in
the
of
volume
following
way.
this
dw,
Let G be a subsetof %,whichis measurablein the Jordansense,and denoteby
O the cone overthe base G consistingof all matricesY = XA,where0 < X < 1
and A 6 G. If {dY} is the volume elementin the euclidean metricdefinedon
U by ds2= -(dY'dY), then
V(G)-L {dY}
(3)
Y -> YC has the
is the euclideanvolume of G. Since the lineartransformation
jacobian I C 1'nit followsthat V(GC) = V(G), forall C in UQ; consequentlythe
formula
dcw
V(G)-=
definesan invariantvolume elementon QN. If 61(A) is an integrablefunction
on Q, then we obtain
1(A) d
(4)
LI
(IYy I 1iny){dY}.
Put Y'Y = S = (Ski); thisis a mappingof i intothespaceP ofall positive
real symmetricn-rowedmatrices. On the otherhand, the equation Y'Yj = S
has forany S e P a solution Y1 e Q, and the generalsolutionis Y = 0Y1, with
an arbitraryorthogonalmatrix0. We introducein P the euclidean volume
element {dS } = Hk < l dskl . Let Q be a measurableset in P, and Q* the set
in Q whichis mapped into Q. If h(S) is any integrablefunctionin P, then
(5)
h(Y'Y)
dY
= anJh(S) I KS I
{dS}.
an =
n
k12
k
342
CARL LUDWIG
SIEGEL
We denoteby D and T thediagonalmatrices[ti,
, to]withpositivediagonal
elementst1,
, t, and thetriangular
matrices(tl) withtkl 0 (1 ? 1 < k ? n),
1 (k- 1, *n),
tkk
tki real (1 ? k < I < n).
The Jacobitransformation
of quadratic formsleads to the decompositionS = D[T] = T'DT, and this definesa one-to-onemappingof P into the productspace of all D and T. Putting
{dDl = dt1... dtx {dt} = ilk<, dtkl, we obtain
...
,
{dS} = {dD} {dT}
ITtk
k=1
Instead of t,*
, t, we introducethe n- I ratios tk/tk+l
n - 1) and the determinantqn = IIk=1 tk-= S =
y
Y
tn
(6)
=
qk
qn
***qn-1
}
1S {d-=d1-
tn k=
=
2;
d(t,
ok
qk
then
tn)
(k
=
n tn
n-1
dqn
n {dT}Iqdqn
Il
k=1I
(q/k/2)(n-k)1
dqk)
We call qi,
, qn and tk1(I ?< k < 1 ? n) the normalcoordinatesof S. It is
clear that S and XS have the same normalcoordinates,withtheexceptionof qnX
for all positive scalar factors X.
4. The group F of all unimodularn-rowedmatricesU has in P the discontinuousrepresentation
S -* S[U]; plainly,U and - U definethe same mapping
in P. A well known result of Minkowski'sreductiontheorystates that this
representation
possessesin P a fundamentalregionK whichis a convexpyramid
withthe vertexin the point S = 0, and that the normalcoordinates,withthe exceptionof qnXare bounded in K. Now considerthe corresponding
domain K*
in U; this is a fundamentalregionin i forthe representation
Y -> i YU of the
factorgroupof F obtainedby identifying
U and - U. By the additionalcondition o-(Y) > 0 we defineone half of K* as a fundamentalregionH for r itself.
Finally,let F be the intersection
ofU,withH; thenF obviouslyis a fundamental
domain on ,1for the properunimodulargroup P1. On the cone F we have
SO also qn is bounded there. Since the exponents of q,
2<
qn =y
,
in
are
>
it followsfrom(3), (5), (6) that the volume
-1,
qn (6)
Vn = V(F)
=
f
dwl
is finite.
Let g run over all integralvectors3 0 and define
(7)
p(X,
A)
=Dn
E f(XAg),
gzo
0(X,A)
=
XnE abs f(XAg),
gto
where0 < X ? 1 and A e i1. The functionf(x) has the formermeaning,viz.,
it is bounded, integrablein the Riemann sense and 0 everywhereoutside a
boundeddomainin R; consequentlythe functionso(X,A) is integrablein U, .
343
OF NUMBERS
GEOMETRY
functionm(A), independentof X, so that
Thereexists an integrable
LEMMA:
in S and thattheintegral
40(X,A) < m(A), everywhere
IFm(A) dw,
J
converges.
PROOF: Since f(x) is bounded and f(x) = 0 outsidea certainspherex'x < r2,
it sufficesto prove the assertionforthe characteristicfunctionof this sphere,
namely
r2).
f(x) = 0 (x'x
f(x) = 1 (x'x < r2),
Put A'A = S = D[T] and considerthe integralsolutionsg of the inequality
0 < S[g] < p2, forany givenpositivep. If gi,
, gnare the coordinatesof g,
then
_
S[g] = D[Tg]
n
n
Z
=
tk gk +
k= I
g1g);
\2
tkl
E
l =k+l
hence gk lies in an intervalof length2pk , and the numberof solutionsghas the
value
a(p, A)
<]I
n
k= 1
(1 + 2ptk).
This estimateimpliesthat, for0 < X < 1,
(8)
n
A) K TI (X + 2rtj1) <
=
+(X, A) X-a(X-'r,
k=1
n
II
k=1
(1 + 2rtk*)=
say. Plainlythe functionm(A) dependsonlyupon S = A'A and r.
For any A in Si, thereexistsa uniquelydeterminedintegerv = 0, 1, * , n
such that tk < (l(k= 1, ... , v) and t,+1 _ 1; this means in case v = 0 that
ti > 1, and in case v = n that tk <1 (k = 1, ... , n). Let F, be the set of all
A in F withgiven v, and put
J=
I|
v = 0*
m(A) dw,
n)
+ J.,X and it remainsto prove the convergenceof the
then J = Jo +
integralsJ,.
Since S = A'A lies in the reduceddomainK, forall A in F, it followsthat the
, n - 1) are bounded; hence ti is bounded in F,
ratiosqk = tk/tk+l(k = 1,
fork = v + 1,
,n,and
(9)
n-I
m(A) < c Ijtjk = cI
k-1
k-i
(qk
qn-i)'
][I7qkp2n
k=1
by (8), wherec depends only on n and r. Now we change the notationand
defineA = i Y I 'Y, S = Y'Y; this does not affectthe coordinatesql, **
If Y lies in the cone F,, then (6) and (9) lead to the inequality
qn-1.
(10)
m( I Y I
Y) I S
I
{dS} <
?
n-1
{dT}qn'I
dqn kT
qk4
dqk,
344
CARL LUDWIG SIEGEL
withak
ak =
=
k (n
2
k(n-k
k - 1 + v/2n) > 0, for 1 _ k < min(v, n - 1), and
-
+ v/2n) - v/2 > 0, forv < k _ n -1.
Formulae(4), (5), (10)
implythe convergenceof J,; q.e.d.
Put
Lf(x) {dx}
=
then
(11)
nlimo(X,A)
ff(Ax) {dx}
E f(XAg)
g
rn
lim
X-0X-O
R
=
y,
ofthe integral. On the otherhand,we inferfromthe
by virtueofthe definition
lemmathat the integral
(12)
I(X)
=
f
Xn f(XAg)dw1
p(X,A) dw, =I
0
gFF
converges,that
(X)=
(13)
# dJ f(XAg)
and that, by (11),
lim p(X, A) dw, =
lim 4t(X)=
(14)
V,.
5. In this sectionwe investigatethe sum
(15)
x(X) =
dw,
Jf(g)
extendedover all primitiveg, i.e., over all integralg withthe greatestcommon
divisor (g9,
, g,9)= 1.
We completethe primitiveg to a properunimodularmatrix U = U, with
the firstcolumng; then
.
(16)
Lf(Ag) dw,
^p
=
ff(AU-'g) do,1
F~~U
ff(I Y I
FU
1'x) {dY},
wherex denotesthe firstcolumnof the variablematrixY in the cone FU.
unimodularmatricesof the particularform
U,=
The
U)'
with an arbitrary(n - 1)-dimensional integralvectoru and an arbitraryproper
unimodular (n - 1)-rowedmatrix C, constitutea subgroup A of it. The
GEOMETRY
345
OF NUMBERS
leftcosets of A, relativeto
are U0A,whereg runsexactlyover all primitive
IF,
n-dimensionalvectors;consequentlythe union of all FU, is a fundamentaldomain F(A) for A on Q1, and
(17)
x(1)
]
=
F(A)
f(I Y I x) {dY},
by (15), (16).
Completingx to a matrixWxin i1 withthe firstcolumnx, we obtain the decomposition
;)
Y1 = Qj
Y = WwY1,
(18)
with a real (n - 1)-dimensionalvectory and a real non-singular(n
matrix Y; plainly,
I Y1I,
}Y I=
(19)
dY}
-
1)-rowed
{dx}{dy}{dY.
=
The mapping Y -> YU, is the same as Y -* YU, y -> U y + u; this shows that
anotherfundamentaldomain G forA on i1 can be definedin the followingway:
Writethe generalelementY = A of Q, in the form(18), restrictY= A to the
fundamentalregionF of the group IN of all properunimodular(n -1)-rowed
matricesU, in the space i21of all (n -1)-rowed matricesA withI A- = 1, and
, Yn- of y to the (n - 1)-dimensional unit cube
restrict the coordinates yj,
In view of (17), (19), we obtain
, n -1).
0 < yY ? 1 (k = 1,
X(1)
=
f(f
f(
'x) {dx}) {dY} = y
I
J Ij{df}.
If y is any positivescalar factor,then
fin {dY}
,uF
=
,U(n-12)2
Jd~~~j
F
{dY
(n
1)
_
=
and partial integrationleads to the formula
ftYt {dY}
=
(n
-
u
1)
du
F
This provesthat
(20)
x(1)
=
n-i
Replacing f(x) by f(Xx), we inferthat
(21)
(X) =X-n(1)
V
-
-1V
346
CARL LUDWIG
SIEGEL
6. If g runs over all primitivevectorsand 1 over all naturalnumbers,then ig
runsexactlyover all integralvectors5 0. Therefore,by (13), (20), (21),
00
4/(X)=
(22)
XA
X(1X)=
x(l)r(n);
this showsthat ,6(X)is independentof X. From (14), (20), (22) we deduce the,
recursionformula
(23)
nVn = (n
-
)Vn-l
(n).
Since V1 = 1, it followsthat
n Vn
(24)
n
H
(k).
k=2
Minkowski'sformulaforthe volumeofthe domainofall reducedpositiveS with
S I < 1 is a simpleconsequenceof (5) and (24).
On the otherhand, by (12), (14),
+(1)= fEf(Ag)
Definingdc
=
dwi =
yV.
V-' dwi, we have
fdw=1,
f(x) {dxj,
f|f(Ag)d=
and this is the assertionof the theorem.
From (15), (20) and (23) we deduce the additionalresultthat
(25)
t(n) f
'f(Ag) dw=
ff(x){dxj.
Now let B be a stardomainin R, i.e., a pointset whichis measurablein the Jordan
sense and whichcontainswith any point x the whole segmentXx,0 < X < 1.
Suppose that foreach A in UNthe domainA-1Bcontainsan integralpointg $ 0;
then it containsalso a primitiveg. If f(x) denotesthe characteristicfunction
of the set B, then we obtain
>,'f(Ag)=
U
A' 1 > 1
ge A-1B
and
ff(x) fdxj ? t(n),
in virtueof (25); this is Minkowski'sassertionconcerningstar domains.
Our theoremmay be generalizedin various directions:
1) We may drop the restrictionthat the integrablefunctionf(x) vanishes
everywhereoutsidea bounded domainand replaceit, e.g., by the weakerconditionthat (x'x)8f(x)is boundedin R, forsome fixeds > n/2.
GEOMETRY
347
OF NUMBERS
2) Instead of the functionf(x) of a singlevectorwe may introducean inte, Xm)of m vectors,with 1 < m < n-1.
The corregrablefunctionf(xl,
spondinggeneralizationof (2) is the formula
v
'F
E
,~gmf(Agi,
,
Agm)dw=
ff(xl,
,
xm){dxi} - {dxm},
wherethe summationis carriedover all systemsoflinearlyindependentintegral
vectorsg9, * , go .
3) We may considercertainother discretesubgroupsof topologicalgroups,
instead of U, and rl, e.g., the real symplecticgroupand the modulargroup of
degreen. In my researcheson symplecticgeometry,I have already applied
the methodofthe presentpaper to the determination
ofthe volumeofthe fundamental domain of the modulargroup of degreen. Anotherand more general
example is providedby the group of units of the simpleorderJL, n > 1, consistingofall n-rowedmatricesA = (akl), wheretheelementsaki(k, 1 = 1, - *, n)
belongto a givenorderJ1in a divisionalgebrawhichis of finiterankin the field
of rationalnumbers;thiscomprisesin particularthe groupofn-rowedunimodular matricesin an arbitraryalgebraic numberfield.
THE INSTITUTE
FOR ADVANCED STUDY.