proceedings of the london mathematical society volume s2-47 issue 1 1942 [doi...

8/14/2019 Proceedings of the London Mathematical Society Volume s2-47 issue 1 1942 [doi 10.1112%2Fplms%2Fs2-47.1.26

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268 H. WEYL [Dec. 15

ON GEOMETRY OF NUMBERS

By HERMANN WEYL.

[Received 27 November, 1939.Read 15 December, 1939.]

1. Minkowski stated his fundamental theorem concerning convex solidsin relationship to a lattice as follows. Let the lattice in the w-dimensionalspace of vectorsjr= {xv ...,xn)consist of all vectors owith integral compo-nents, and let/0?) be any gauge function , i.e.any continuous functionin vector space enjoying the following properties:

f(t) > 0 except for the origin jc= o= (0, ..., 0);/(ft) = |$|./0f) for any real factor t;/(*+*')-

Then/(]r)


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1939.] ONGEOMETRY OFNUMBERS. 269them, Minkowski arrives at thesharper inequality f(i.2) if 1 . . . i r l l F < i ,where the numbersMkmay be described thus: A bubble of shape$whilebeing blown updescribes theseries offigures &{q) with increasing q\q= 2M= 2MXis thefirst moment whenalattice pointb j ^ oenters intothe bubble,2Mkthat stageatwhichfor thefirst timealattice pointbkenters off the (kl)-dimensional linear manifold [bl5...,&&_J spanned by&i> >&&-1- Clearly M1^.M2^i... k-i\frwhichf(a)assumestheleast possible value2Mk. So longasq2Mk their number is at least k.The vectors

t>JWXt t)2/2M2) .... bn/2Mn>and hence their convex closure,arecontained in Si. IfP =\b1...pn\ isthe volumeofthe parallelotope spannedbythe lattice vectors bl3..., &n,the volume of that closureis

2n p Pin\ 2M1...2M2~n\ Mx...M n'

pTherefore -^


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270 H. WEYL [Dec. 15,[AddedJan ua ry, 1940. Prof.L. J. Mordell, to whom I sent a copy of

the MS., kindly pointed out to me two papers which I had overlooked;one by H. Davenport, Quart. J. of Math., 10 (1939), 119-121, bearsclosely on Part I by containing another proof for the inequality (1.2),while the other by K. Mahler, Quart. J. of Math., 9 (1938), 259-262,actually anticipates the main result of Pa rt II , namely Theorem V. I amstill surprised th a t t he discovery of this fact in 1938 by D r. Mahler, an d ofthe simple rem ark on which it is based, had to wa it for forty-two yearsafter Minkowski established his inequality (1.2).]

I .2. A set in the affine space of pointsx= (xv ..., xn) may be describedby its characteristic function (x) which is equal to 1 inside the set andto 0 outside the set. The characteristic functionfor the union of two setswith the characteristic functions (f>v 2 is their "Boolean" sum

which reduces to the ordina ry sum only if the sets do no t overlap . An ynumberain the interval 0 a 1 is said to be aprobability. Ifaandbareprobabilities of two statistically independent events A and B, then theirBoolean sumc = av 6 = a+6 a b 1(1a)(l6),

which again satisfies t he ineq uality 0


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1939.] ON GEOMETRY OF NUMBERS. 271third line with regard to their order. Brackets as in (*, k, I) indicatethe first kind of summation.A function (x)in our space which satisfies the condition 0 1iscalled a probability function.

Let(z)now be an (integrable) probability function vanishing outsidea finite region # of the aj-space which may be fixed as a parallelotope:

Af


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272 H. WEYL p ec . 16,

In substituting a+h,a+c, ... for b, c, ... we see thatift(x)may be writtenas an ordinary sumwhere

[b,c, ..., lattice vectors # o].This trick works because the lattice translations form a group. Whenwereturn to combinations without regard to order, the second factor onthe right in the last formula is(2.4) **(x)= 1 - Sftx, ) + J S {x,a)(x,b ) - . . . .

(a) (a, h) [a, b, ... lattice vectors ^ oTake a real parameter tranging from 0 to 1, and introduce the product

a 9*0 (a) (a ,b)Its integration from t= 0 to 1 leads to (2.4). Hence our final defini-tions are:(2.5)


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1939.] O N GEOMETRY OF NUMBERS. 273invariant under translation, and

if * i (*K*a() .Specilization of the theorem for the characteristic function (f> of an(open) set of definite Jordan volume V yields the simple Minkowski-Blichfeldt-Scherrer theorem: If 0 has no point in common with thesets a placed round the other lattice points a# 0, or, wha t is th e samething: if 0 never contains both end points x and + a of a lattice vectora^ 0, the n its volume is less th an or equ al to 1. I consider Theorem Ito be the natural general form of this principle, and our proof is nothingbut a neat analytical form of the simplest argument by which that prin-ciple has been provedf.[Our affine #-space could be replaced by any di ffer en tiate manifoldU of pointsxwhich carries an infinitesimal volum e measure, an d the groupof lattice tran slation s by any group of differentiable volume-preservingone-to-one transformations x-s-x8 of U which(1) is discrete in th e s trictest sense, so th a t t he set of po ints Xs,

equivalent to any given point x,never has a condensation point,(2) has a compact fundam ental d om ain; i.e., the manifold, arising fromU by identification of equivalent points, is supposed to be compact.]3. With SiegelJ we introduce the Fourier coefficients of the periodicfunction *fi(x):

J{= \ tfi(x).e(lxxx+...+ln xn) dxJEcorresponding to the various lattice-vectors 1= (lv ...,ln) in the dualspace, with e(x) as an abbreviation for e2frte, and formulate the Parsevalequation

12l lIThe zero coefficient Jo is our J = J[]. Making use of the equ ation

ip{x).a>(x)dx=\ *(x).a)(x)dx}E Joof G. Haj6s,Act.Litt.Set.,Szeged,6 (1934), 224-2 25. H . F . Bliohfeldt, Trans. American

Math. Soc, 16 (1914), 227-235. W . Scherrer, Math. Annalen, 86 (1922), 99-107.X Acta Math., 65 (1935), 307-323.8BR. 2. TOL. 47. N O . 2306. T


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274 H. W E Y L [Bee. 15,which holds forany periodic function co(x),we get( 3 . 1 ) JI = [ cf>*(x).e(llx1+...+lnxn)dxand [ [lp(x))2dx^[ *(x).ijj(x)dx.We find that

and thus** w i t n(3.2)


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1939.] ON GEOMETRY OF NUMBERS. 275it isthedifference between the tw o convex bodies r>and%r. Let(4.2) O = ro


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276 H . W EY L [Dec. 15,Sincewehave neither assumed(x)to be continuous nor the sets fto be

closed, the existence of a maximum is doubtful. But the following s lightmodification will do . Le t r{ in the finite sequence (4.2) be the highestlevel for which %r, is no t em pty . We cut off the top of the m oun tain atthe altitude r(, i.e. we form

and choose a;0 as a point in r.. Then we get

instead of (4.5), and hence

However, according to (2.8)

and by driving towards 0 the ine quality (4.4 ) is re-established.(Instead of the decapitated mountain, we might have used the terracedmo untain *=


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1939.] ON GEOMETRY OF NUMBERS. 277The integration is then conveniently carried out in two steps, first withrespecttoxv ..., xkfor any constantxk+1, ..., xn,and secondly with respectto xk+1, ...,xn. Thus we find that

implieswhile the inequality

holding under the condition 0 **+ , ..-,xn) Thus,by integrating (5. 2) with respect toxk+1, ..., xn,we get

The substitution(5.3) fperformed on the lastnkvariables results in the equation

Hence J(k)[q]> J(k)[]-H.


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278 H. W B Y L [Dec. 15,For k = 0 we define and therefore

Any ^-dimensional discrete lattice Zk may serve here instead of (5.1),since the firstkcoordinate vectorstx, ..., efcm ay be so chosen as to sp an th epre-assigned lattice (that is, so that the lattice consists of the vectorsa1t1-\-...-\-aktk with integral components ty).

THEOBEM IV . By means of a k-dimensionallattice kwedefine4f z)= x). f n { l - ( a > , a)} A,Jo a

the product extending to allvectors d ^ O of the lattice k, andj w M = f ^( a . )dx > jk { q )= jiW.

Then Jk(


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1939.] ON GEOMETRY OF NUMBERS. 279and ultimately

..Mn.V for q>Mn.Since J(q) always remains less than orequal to 1,Mmkowski's

inequality (1.2) is provedI I .

6. Contrary to the procedure in I, yielding the vectors b&and ratios ofdilation2Mk, theproblemofreductionrequires the determination of vectorstk and ratiosNk according to the following inductive rule:Let arange over all lattice vectors outside the (ft1)-dimensionalmanifold [ex...efc_Jsuch that every lattioe vector in the ^-dimensionalmanifold [ex...eft_xa],

h-i i nOi has an integral a-componentx. a=c&is a vector within this range 9t&forwhich /(a) takes on the least possible valueNk.As the first step, we may take cr=bltso thatNt =2MX. By inductionwe readily conclude that e^ ..., efc span {i.e.form a basis of) the wholelattice in the manifold [tx... cfc]; hence cls..., enspan the whole n-dimen-sional lattice. Relative to this coordinate system, the lattice vectors

eB=(a;1, .... a>n)are again described byintegral componentsxi}and our reduction is equivalentto the following conditions:(6.1) /(a^afc ...,*) >/(8A ...,8n*)= ^which hold provided that xlfxz, ...,xn are integers and xk, ...,xn arewithout common divisor. T hefy* are the Kronecker 8'sand thusc&= (8r&, ...,Snfc), the k-th unit vector.This characterization (6.1) obviously entails the inequalities

We have normalized the basis of our ^-dimensional lattice relative to theconvex solid by means of then sets (6.1) of infinitely many inequalities,and this is what the problem of reduction demands. The question ariseswhether one can establish a universal upper bound forNx... Nn V. Inowprove

THEOREM V. Ifareduced lattice basisefcis so chosen as to satisfy theconditions(6.1) , then(6.2) Nx...N nV^n,


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280 H. WBYL [Deo. 15,where( 6 . 3 ) /aft= 2 . (

Proof. One of the vectors t>v..., bk, say 6, lies outside the linearmanifoldEk i= ih efc-i]-

In ^fc*=[e1...eA:_1)]wecanfindalattice vector e*which, together with ev ..., tk_v spansthewhole latticeinEk*. Out of the finite number of lattice vectorsofthe form

w i t h - * < * * < * ( =1 , . .. . fc-1) , 0),as one of the numbers

2M lt ...,2Mk>is certainly less than orequal to 2Mk, weinfer from these remarks and(6.4) that

if e = l ,w ) if e > 2 .

Hence Nk cannot exceed the larger of the twonumbers(6.5) Mk+t(Ni+...+N*-i) and 2Mk.This allowsus toestablish universal inequalitiesofthe kind(6.6) Ni^26 iM i.Indeed (6. 6)istruefori = 1with81=1. Onceitholds goodfor

t = l , . . . , * - 1 ,the bound (6. 5)forNk yieldsasimilar inequalityfori = kwith6kas thegreater of the two numbers

1 and $ ( 1 + ^ + . . . + ^ ) .This determinesthefactors6kbyrecursion,andwe readily verify that

0 i= A, 0*= (f)&-2 for fc>2.


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1939.] ON GEOMETRY OP NUMBERS. 281Therefore (1.2) implies

It would be an interesting problem to ascertain the lowest constantfx,n for which the relation (6.2)holds.Fuller exploitationofthe method yieldsthefollowingGENERALIZED THEOREM V. Let qv ...,qn begiven positive num bersgreaterthan or equal to1. If theinequality

f(xvx2,...,xn)^.f(81*,...}8n)holdswheneverxv ...,xn areintegersand xk, ...,xn havenocommon factor(k= 1, ...,n), then the values

satisfy the relations qkNk+X ^N k (k=l, ...,n 1)an d

The proof ispractically thesame asbefore. The recursive equationsfor the 8k are now,=


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282 H. W B Y L [Dec. 15,This constantpnis much lower th an t he one in TheoremV. It s increase

with n is best judged by the recurrence formulae

and Pn= 2+.'v)+\ Pn-ifor od d v a n d n=Proo/. Out of the vector basis bv ..., &, we construct a la ttice basisei*"- en* according to the recurrent equations

(7 .2 ) e&*= t>k+aktk^t>k_1+... +ak> 1 bxekwith the conditions

theekbeing positive integers. If ek= 1, all th e coefficients akth_lt ...,aktlvanish, and /(e&*) = 2Mk. How ever, if ek^2,M k* =/(efc*)< A *l

with ^= = i . +To cover both cases we put r\k = 2 for ek= 1. Prom (1.2 ) we now get

and it rem ains to discuss the prod uct ^ i . . . i?n.We put8 = 2 or hk=1 according as ef c =l or e f c ^ 2 ;

^ f c ' = = 2 i f 8 f c = 2, and Vk' = 8f c +i (8*_ 1 +. . .+8 l) if 8*= 1.Then ^fc


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1939.] O N GEOMETRY OF NUM BERS. 283(all other figures remaining una ltered) affects merely th e tw o factorsikVk+i m (7-3 ), turn ing their values

[ l + * ( 8 * - i + . + 8 i ) ] - 2 i n t o 2 . [ 2 + J ( 8 f r_ 1 + . . .+8 1 ) ] fand so increasing ( 7. 3) . Fo r a given num ber v of 2's in the sequence, themax imum value is thu s attaine d when they come first (8ls ..., ), and then(7 .3) is equal to 2H-2 2v+3i.e. 2a -*.(Le t v pass throug h th e values v= 0, 1, ... , w. Since

is on the upgrade (wv> wv_xor wy^ wv_x)so long asor v(

and then goes down. Hence the maximum is attaine d for the last v forwhich v{v\) does not exceed n-\-\.Here is a table for the lowest values of pn:n = l , 2 , 3 , 4 , 5, 6, 7

pn = 2, 4, 12, 42, 168, 756, 3960, ...,Higher bounds, for instance

can even more easily be obtained. Again we hav e the question of the be stvalue of pn in the inequality (7.1).

8. A positive-definite quadratic form8.1) / *) = S0is said to be reducedif it satisfies th e inequalities(8.2) / (*)^gkk for every x in 3tfc and k= 1, ..., n,where 0tfcis the set of all lattice poin tsx= (xv ..., xn) for whichxk, ..., a;nare w ithout common divisor. W hile for every positive form of discrim inantD = det (gik),


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284 H . W E Y L [Dec. 16,we have found that for a reduced form(8-3) K9n-9nn=.0 = l , a>1=2.Indeed, with V / a s ga u g e function, the convex body $: / 0with integ ral coefficients a^. W e expressly exclude from th e set ofinequalities (8.2) those for which all coefficients ai} vanish, namely (8.2)with x ={hxk, . . . ,8 ,*) .Each individual inequality (8.4) of defines ahalf-space boundedby a plane(8.5) i


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1939.] ON GEOMETRY OP NUM BERS. 285A substitution S carries the central cell Z into an equivalent cell Zs

whioh is also convex. The equ ivale nt cells join tly cover the whole spaco0. Zj coincides with Zfor any of the 2nsubstitutionsJ: x1-*x1, ..., xn->xn,

and 8 and SJ have the same effect upon Z*From (8 . 3) Minkowski deduced two im po rtan t theorem s of finiteness:I . A finite number of the inequalities3 suffices to define the cell Z.

I I . There is only a finite number of substitutions 8 capab le of carryinga point of Z into a point of Z (worded more precisely, the demand on theelement 8 of {8 } which limits it to a finite set is to the effect that thereshould exist two points /, / ' in Z such that / ' = 8f).Minkowski has a straightforward algebraic proof ofI I . in 7of his pap er" Disk ontinu itatsbereich fur arithm etische Aquivalenz "f. His proof ofI is involved because he establishes I simultaneously with an inequality ofthe natu re of (8 . 3) by a complicated induction. This entanglem ent was

dissolved in a joint paper by L. Bieberbach and I. SchurJ (oh tempipass ati ). They improve on I by breaking it up into the twotheorems:I * . There is only a finite number of lattice points x in 9tfcfor which thereexists a reduced form f0 = {g$}satisfying the eq uation

( 8 . 6 ) /(*) =


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286 H. WEYL p ec . 15,

I say that those points / (in 0) for which all the inequalities $ holdwith the sign > rather than > form thecoreofZ. Any diagonal form

...+gnxn* with 0


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1939.] O N GEOMETRY OF NUM BERS. 287lies on a w all separating Z from an adjacent cell Zs. W e deduce this fromthe fact, important initself, that the equivalent cells cluster only towardsthe boundary of 0.

On closer examination Minkowski's proof of II reveals that there is notmore than a finite number of 8 capable of carrying a po int of Z into apoint ofO(p, fi). Herepis any number greater th an 1and/xis any positivenumber. If Dk denotes the determinant of the form f(xx... xk0... 0), thesubcone Q(p, fi)of 0 is defined by the following simultaneous inequalities:

9kk > Qu-i, fc-i faoo =Pf{x lt ...,x v 1, 0, ... , 0)

for any integers xlt ..., xk-vAll points ofZ areinnerpoints of O(p,/x). Th is is more th an sufficientto settle our question. Choose a definite p> 1 and /J,> 0. Le t / be apoint on the boundary of Z and let flt ...,/*> ... be a sequence of po intsoutside Z approaching / . ,W e may assume th a t they lie in G(p, fi). Acertain substitution 8'1 of {8} will carry / into a point /,' = /Sf71/Hof Z.In consequence of our statem ent abo ut G(p, p),a certain8 recurs infinitelyoften among the 8V. This 8 is no J. The pointsfv of the correspondingselected sequence all lie inZs, and so does their li m it /. Consequently anyp o in t/ on the boundary ofZ belongs to the comm on boun dary ofZ and anequivalent Zs. The domain O(p,JJ.) increases whenpand //, increase , andwith p->oo, fx-^-oo exhausts the whole O. A t every stage G(p,yC) haspoints in common with only a finite number of cells Zs. An elementaryargument like the one applied before shows that among the commonboundaries Z\ZS (which exist in finite num ber only) none bu t the" w a l l s " , the (N l)-dimensional ones, matte r. The su bstitutions 8which carry Z into cellsZs bordering on Z along a wall, together with theJ, generate the entire group {8}.The Generalized Theorem V proves that all our statements aboutQ(p, fi) remain true if we define it by the following linearinequ alities:

f{xv ..., xn)>jgkkwhenever the xi are integers and xk, ..., xn have no common factor,


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288 H. W E Y L [Dec. 15,whenever the xt are integers and (xk, xk+ 1, ..., xn)=(l, 0, ..., 0).Incidentally the estimates may be improved considerably if we carry outthe comparison of the coefficients gkk , g'kkof two equivalent forms / , / ' inZ andO(p, fi) respectively for any convex bodies rather than ellipsoids bythe procedure of 6.Throughout we have studied Z relative to 0; this is really the morenatu ral standpoin t. However, the results I and I** are true even relativeto the space Rof all quadratic forms. Zis no t closed in R; there may beboundary points f of Z relative to R which are no t positive forms. Imaintain that for such a form the equation gn = 0 holds. Since / is non-negative, it can be brought into the form

by a transfo rm ation with real coefficients. We can determ ine a latticepoint x ^ 0 for which the m linear forms tjv ..., m are in absolute valueless th an any preassigned positive num ber e, either by Minkowski's theorem(1.1) for parallelotopes, or by a simple application of Dirichlet's principleof the distribution of v-f1 objects over v boxes. Hence gn > 0 wouldcontradict the inequality f(x) ^gn which holds for all latti ce po ints x^O,and Z relative to R is defined by the inequalities described under I* andI** together with gu ^ 0. But even the addition of glx ^ 0 becomessuperfluous if n ^ 2. The subs titution

Xx = 1, X% = *J 3 = . . . = Xn = 0yields the inequality

for any reduced form, or 2|gr12|


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1939.J ON GEOMETRY OF NUMBERS. 289inductive kink bywhichin hisfinal publication theestablishmentof theinequality (6 . 2)istiedto theselectionof afinite nu m berofinequalities $?I tisnow idletospeculate. But ofhimi tmightbesaidas ofSaul tha thewentout tolook after bisfathe r's assesandfoundakingdom.For quadratic forms there is a simple algebraic method of deriving(1 .2 ) from (1 .1 ). Minkowski describes it in GeometriederZahlen,51;and very likely this suggestedto him thegeneral inequality (1 .2) . Henceevery improvement ofthe constantynin theinequalityfor positive forms

in which M is the minimum off(x) for all lattice points x 0 entails aproportional improvement in (1.2) and for theconstantsfxnandpnin theestimates (6.2), (7.1) for ellipsoids. Our value was yn(2n/con)2.Blichfeldtf succeededinimprovingit by the factor

This leads to the following Xn:(8 .9) 1/An=Remak recently obtained a better estimate^ which differs from (8.9)essentially byhaving f insteadof f. However, the chief interest of ourconstant jxnin (6.2)liesin itsvalidityfor allconvex solids whatsoever.Theorem VI together with Blichfeldt's improved yn shows that everypositive form has an equivalent one / satisfying the relation

where 1/^*= ( 1 + W2- >-*pn*.Suchanestimateisbest suitedtoprove tha t thereisonlya finite numberof classesof equivalent positive forms with integral coefficients.

The Institute for Advanced Study,Princeton, N.J., U.S.A.

t Trans. American Math. Soc, 15 (1914) , 227-235; cf. R. R e m a k , Math. Zeitechrift,26 (1927) , 694-699, andBlichfeldt , Math. Annalen, 101 (1929) , 605-608. Oompoaitio Math., 5 (1938) , 368-391.

B B . 2. VOL.47. M O. 2307. U

proceedings of the london mathematical society volume s2-47 issue 1 1942 [doi...

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