NISTIR 88-3777

Finite Unions of

Closed Subgroups of

the n-Dimensional

Torus

Jim Lawrence

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

Center for Computing and Applied Mathematics

Mathematical Analysis Division

Gaithersburg, MD 20899

August 1988

National Bureau of Standards became the

National Institute of Standards and Technology

on August 23, 1988, when the Omnibus Trade and

Competitiveness Act was signed. NIST retains

all NBS functions. Its new programs will encourage

improved use of technology by U.S. industry.

U.S. DEPARTMENT OF COMMERCE

C. William Verity, Secretary

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

Ernest Ambler, Director

NISTIR 88-3777

Finite Unions of

Closed Subgroups of

the n-Dimensional

Torus

Jim Lawrence

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

Center for Computing and Applied Mathematics

Mathematical Analysis Division

Gaithersburg, MD 20899

August 1988

75 Ymt» SamUeflr g Amrrtc* s Program
.

.
FINITE UNIONS OF CLOSED SUBGROUPS

OF THE N-DIMENSIONAL TORUS

by Jim Lawrence

Center for Applied Mathematics

National Bureau of Standards

and

George Mason University

Abstract

Let U be an open subset of the torus group T

n

. We

show that the set of maximal subgroups of T

n which miss U

is of finite cardinality. This result is applied to show

that the lattice of finite unions of closed subgroups of T

n

is a complete distributive lattice, and to show that, up to

unimodular equivalence, there are only finitely many convex

polytopes P c IR n having vertices in 7L n but no interior

points in Z

n and such that each subgroup G of the

additive group IR n which properly contains Z

n does have

points in common with the interior of P.

1

FINITE UNIONS OF CLOSED SUBGROUPS

OF THE N-DIMENSIONAL TORUS

by Jim Lawrence

1 . Introduction . Let x = (x.^, . . x^) be an element of IR n and let

U C IR n be an open neighborhood of 0. A classical theorem

of Dirichlet asserts that there exist a positive integer m

and a point z = (z^, . . z^) € Z

n such that mx - z € u.

The numbers x^, . . . , x

n

and 1 are independent over the

rational numbers if there is no w a Z

n ~ {0} such that

<w , x> £ Z . A classical theorem of Kronecker asserts that

the numbers x^, . . . , x

n

, and 1 are independent over the

rational numbers if and only if for every open set U c !R n

there exist a positive integer m and z e Z

n such that

mx - z e U . (These are Theorems 201 and 442 of [4]. See

also Chapter VII of [1].)

In this paper we consider, for open sets U C IR n

, the

nature of the sets t (U) = (x e IR n

: there exist m € Z and

n . ~

z e Z such that mx - z e U}. (Alternatively, t (U)

= {x e IR n

: the (additive) group generated by {x} U Z

n

intersects U}.) We show that IR n ~ t (U) is a finite union

of closed subgroups of IR n

; and moreover, the set M(IR

n

, U)

of maximal subgroups G of IR n such that G D U = 0 and

Z

n

c G, is finite. This is Corollary l.A, below.
2 As an example, let n = 2 and let U = { (x, y) € IR :

0 < x, 0 < y, and x + y < 1). Then the subgroups H of

[R 2 such that Z

2

c H and H fl U = 0 are precisely the

subgroups of the following four groups:

H

1 = { (x, y) € IR 2

: X e Z},

H

2 = { (X, y) € IR 2

: y £ Z) , H = {(x, y) € [R 2

: x + y e Z), and

H = {(x, y) € IR 2: 2x € Z and 2y e Z).

One of several interesting consequences of the general

finiteness result concerns subsets of the n-dimensional torus

group T

n

. It is obvious that these subsets form a finitely

distributive lattice under the operations of intersection and

union. It follows from the finiteness result that they

actually form a complete lattice: The intersection of an

arbitrary family of finite unions of closed subgroups of T

n

is again a finite union of closed subgroups of T

n

. (We will

have occasion in this paper to use the word "lattice" in two

different senses: We will use it as we have in this

paragraph, to mean a partially ordered set with certain

properties; we will also use it in its sense in the geometry

of numbers, to mean a discrete, full-dimensional subgroup of

[R n

. The useage must be ascertained from the context.)

In Section 3 we present some consequences of these

results concerning finiteness of certain sets of unimodular

equivalence classes of polytopes with integer vertices.

This paper uses standard results concerning additive

subgroups of IR n

. The best reference for this topic for our

purposes is Chapter VII of [1].
Preliminaries

.

2 . Let ^ be the lattice of closed subgroups G of IR n

such that Z

n

C G. (We could equivalently work with closed

subgroups of the torus group T

n = [R n/Z n in view of the

bijective correspondence G -> tt(G) mapping the set of such

subgroups to the set of subgroups of T

n

, where it : lR n -» T

n

is the canonical map. We prefer to remain in IR n in order

to make easy use of results from the geometry of numbers.)

Let ^ be the lattice of closed subgroups of IR n

. For

^ G € <3, let G = ( x € lR n

: <x, u> € Z for each u e G}.

Then G is also in ^ and the map G -* G is an

anti-automorphism of . (See [1].)

. The lattice satisfies the descending chain

condition; that is, each non-empty subset of <3 possesses a

minimal element. Equivalently, any chain 3 H

2 3 . . . of distinct elements of <3 must be finite. To see this note

that c H

2 ... would be an ascending chain of

subgroups of (Z

n ) * = Zn

, which satisfies the ascending chain

condition, since it is a finitely generated abelian group.

For S C IR n

, let pol(S) = (x € lR n

: <x, y> < 1 for each

y € S). Then pol(S) is a closed, convex set which contains

the origin; pol(pol(S)) is the smallest closed, convex set

which contains S U (0); and pol is a dual automorphism of

the partially ordered set of closed, convex sets containing

the origin.
4

Our objective now is to establish a lemma (Lemma 3)

which will be used in the proof in the next section of the

main result.

LEMMA 1 . Suppose that U is a convex subset of IR n

and that p € U. if H is. a subgroup of IR n such that

H fl (1/n (U - p) ) contains a basis for IR n then

H + U = IR n. Proof. Let {b- ,

. . . b } be a basis for IR n contained in

1 n

H n (1/n (U - p) ) . Let P = { y a . b

.

: 0 < a. < 1 for

i = 1, . . . , n} . Then P C conv{0, nb^, * * •/ nt>

n

) C u - p. Any x e IR n

can be expressed in terms of the

basis: x = ) a.b. , i = 1, . . n. We then have:

L# .X. 1

X = ^ L^iJ }(a i )b

i € H + P,

so H + P = [R n

. (Here L a J denotes the greatest integer less

than or equal to a and (a) = a - [aj is the fractional

part of a.) It follows that H + (U - p) = [R n

; i.e. , H + U = IR n

.

In the proof of Lemma 2 we will use a result of Mahler

belonging to the theory of successive minima. Recall that

for a lattice L C IR n

, and a full-dimensional, compact,

convex set K symmetric about the origin, the successive

minima X ,

. . ., X

r

of L with respect to K are the

smallest real numbers such that (for each i) (X^K) fl L

contains a set of i linearly independent points.
o

with

Let X, ,

. . . , and X be the successive minima of L

1 n

respect to K (as above) and let X ,

. . and X^ be

the successive minima of L with respect to pol(K).

Mahler's result is that (for each i) one has

1 < X .X *

. _ < n! . 1 n-i+1 "

. 2

(In Mahler's original result, the right-hand bound was (n!) . The statement as we have it is Theorem VI of Chapter VIII,

Section 5, of [2]. The right-hand bound has been

spectacularly improved by Lagarias, Lenstra, and Schnorr in

[5] .)

LEMMA 2. Let K be a full - dimensional . convex . compact set

with K = - K. Let H be a closed subgroup of IR n such

that H D K does not contain a basis for IR n

. Then

* H fl (n! pol(K)) contains a non- zero element.

Proof. Suppose that there is a convex, full-dimensional,

compact set K symmetric about 0 and a closed subgroup H

such that H fl K contains no basis for IR n and

* H fl (n! pol(K)) = {0}. We may choose a basis

{x

x

,

. . ., x^} for IR n such that

n

H = ( ^a

i

x

i

: € Z for i = a + 1, . . . , b,

i=l

and a

^ = 0 for i = b + 1, . . . , n}.

Let L^ be the lattice generated by {x^/m, . . ., x /m,

xa+1 ,

. . ., x^, • • •/ mx

n

)* It i- s cl ear that we may
6

choose m sufficiently large that L fl K contains no basis

m

for IR n

, and L* n (n! pol(K)) = {0}. Let A^ .

. . , A ,

A., , . . . , and A be the successive minima for L with 1 '

' n m

* respect to K and for with respect to pol(K),

respectively. Since fl K contains no basis for IR n

, we

have A^ > 1. Also L* D (n! pol(K)) = {0}, so A* > n!

.

. . * This contradicts Mahler's Theorem, since then A A_ > n! . ' n 1

LEMMA 3. Let G be a closed subgroup of IR n. Let U be a

subset of G which contains a non- empty relatively open set . Then there is a bounded set X C IR n such that if H is. a

, * closed subgroup of G for which H + U G then H D X is

* not contained m G . Proof. It is clear that, if A : IR n -* IR n is a nonsingular

linear transformation, then the statement holds for a given

group G and open set U C G if and only if it holds for

the images A(G) and A (U)

.

We may therefore suppose that

G = { (x

1

, and

. . , x ) € IR

' n'

n

x , ,

. . . , x, € Z a+1, ' b

Xb+1 • • • x

n

°)»

where a and b are integers for which 0 < a < b < n.

Let

A = { (x lf

B = { (X

x

,

.

. x ) € [R n: x. = 0 unless i < a}, ' n

7 l -it

. , x ) € [R n

: x . = 0 unless a < i < b}, ' n' l

J '
/

and

C = { (x^, . . x^) € [R n

: x^ = 0 unless b < i};

and let a : IR n -* A, (3 : [R n -> B, and nr : [R n -» C be the

obvious projections. Then we may write

G = { x € IR n

: j3(x) € Z

n and -r (x) = 0}, and

G* = { x € IR n

: a(x) = 0 and (3 (x) € Z

n

}.

Let P = { x € IR n

: a (x) = nr (x) = 0 and 0 < (3 (x) < 1}.

* Note that P fl G = {0}. If G is a discrete group, so that

a = 0 , then we may take X = P . Otherwise, let W be the

unit ball in A: W = { x € A : llxll < 1}. Let p € U and

choose e sufficiently small that eW C - . Finally,

let X = (a!/e)W + P. Clearly X is bounded.

Suppose H is a closed subgroup of G such that

H + U G. We will show that (H* fl X) ~ G* ^ 0

.

Suppose (3 (H) is properly contained in (3(G) = Z

n

fl B.

It follows that H + A + C (= p

1

(j3 ( H) ) ) is properly

contained in G + A + C, so that H* D B = (H + A + C)

properly contains (G + A + C) * = G* D B = Z

n

fl B. Choose

x € (H fl B) ~ (G* fl B) ; say,

x = (0, . . ., 0, x . . ., x^, 0, . . ., 0). Then

X = (0, . . .,o, [xa+1 J ,

. . Lxb

J

,

o, . . 0) e z

n

n b

k ~

C H fl B, so x - x is a nonzero element of P which is in

H . Therefore x - x € (H* fl X) ~ G*.
o

Finally, suppose /3(H) = j3(G). If aeW + (H (1 A) = A

then aeW +H=G so U+HD (aeW + p) + H = G, contrary to

our assumption. Therefore aeW + (H fl A) ? A, and we see by

invoking Lemma 1 that aW fl H contains no basis for A. By

Lemma 2 applied to A there is a nonzero vector in

(n!/e)W n (H fl A)* = (n!/e)W D (H* + B + C) ; * i.e., we may find x € H such that a (x) € (nl/a)W, a(x)

0. Suppose x = (x

1

,

. . x

n

)

. Then x = (0, . . 0,

L xa+lJ '

• • • ' L xkJ / xb+1

A

contains G ) , and x - x

(H fl X)

* G .

* *

. . . , x ) € H (since H

is the required element of
9

3. Main Results and Corollaries . Let G be a closed subgroup of [R n

. Suppose U C G.

We shall call U full if its intersection with each closed

subgroup H of G is empty or contains a relatively open,

non-empty subset of H. In particular, open sets are full.

THEOREM 1. Suppose G is a closed subgroup of IR n and U

is a full subset of G. Let M(G, U) be the set of maximal

subgroups H C G such that Z

n

C H and H fl U = 0. Then

M (G, U) is of finite cardinality . Proof. Let T denote the set of all closed subgroups G of

[R n containing 2 n for which there exists a full subset

U C G M(G, U) is infinite. Suppose G € r and U is a

corresponding full subset. Clearly U j* 0 . By Lemma 3 there

is a bounded set X C lR n such that if H is a closed

subgroup of G such that H + U ^ G then H* fl X £ G* . If

H € M(G, U) then H + U ^ G (since 0 € H + U) , so for such

H there is b € (H fl X) ~ G*. It follows that

M(G, . U) C U M(Gb

, U

b

)

,

b

where the union is taken over b e (H fl X) ~ G , G^ = ( x

£ G : <x, b> € JL) , and = U fl G^. Notice that, for each

such b, G^ is a proper subgroup of G (since b € G )

.

Also, since Z

n

C H, it follows that H* C 7L n

, so H* D X is

finite. It follows that M (Gb

, U) is of infinite

cardinality for some b € (H D X) ~ G , so that G^ € F .

/“V>
We have shown that r has no minimal element. By the

descending chain condition on <3, r = 0.

We present some corollaries of Theorem 1.

COROLLARY l.A. If U is a full subset of T

n then there

are only finitely many maximal closed subgroups H of T

n

such that H fl U = 0

.

COROLLARY l.B. Let S be a closed subset of T

n such that

if x € s and m is a positive integer then mx € s . Then

S is a finite union of closed subgroups of T

n

.

We now consider an order relation on open subsets of the

torus T

n

. For open subsets U and V of T

n

we write

U < V if for each x € u there is a positive integer m

such that mx c V. We write U ^ V if U <. V and V U.

Then ~ is an equivalence relation on the set of open

subsets of T

n and induces a partial ordering on the set

£ of equivalence classes. We wish to study this partially

ordered set.
11

For open subsets U of T

n let t (U) denote the

complement of the union of the closed subgroups G of T

n

such that G fl U = 0. We see from Theorem 1 that t (U) is

open. Perhaps it is easier to derive this fact as a

consequence of the following lemma.

LEMMA 4. t (U) = { x € T

n

: there is m € Z, m > 0, such that

mx € u }

.

Proof. Certainly if there is a positive integer m such

that mx € u then each subgroup G C T

n such that x € G

intersects U nontrivially , so x € t (U)

.

Suppose no such

m exists. The closure of the set { mx : m € 2, m > 0} is

then a closed subgroup G of T

n which .misses U. Since

x € G, x € t (U) .

We see that t is an algebraic closure operator on the

collection of all open subsets of T

n

: U C t (U) for each

open set U; if U C V then t (U) C t (V)

;

and

t ( t (U) ) = t (U)

,

for each open set U. Also from the lemma it

is immediate that t (U) is the largest open set such that

t (U) < U. The following theorem, which is now immediate,

characterizes the partial ordering of 2 induced by ^ .
THEOREM 2 . lf_ U and V are open subsets of T then n

U -< V if; and only if t (U) C t (V) , and U ~ V if, and only

if t (U) = t (V)

.

The partially ordered set £ is dually

isomorphic to the partially ordered set of finite unions of

closed subgroups of T

n (under inclusion ) . This partially

ordered set is a finitely distributive complete lattice .

Finally we wish to establish a chain condition for this

lattice

.

THEOREM 3 . Let

open subsets of

T (U

M^ = T ^ U

M+1^

Proof. Let T

such that there

r(U

1

) C r(U

2

) C

t (U.

)

D T

n ~ G.

m

= n (T

n ~ g.)

-i=1 -I

U

1

C U

2

C . . . be an ascending seguence of

T

n

. Then there is an integer M such that

o c c • , denote the set of closed subgroups G of T

n

exists an infinite ascending chain

. . . of distinct T-closed open sets

We may write t(U

1

) = T

n ~ (G^ U . . . U G )

for some closed subgroups G^, . . G . m

,n Then r(U.) = r(U.) U r(U ) = fl (r (U. ) U (T

j=l J

clear that for some j the seguence of sets

G . )) • it is

t (U i

) U (T

n ~ Gj) 2 T

n

G. must contain an infinite

3

subsequence of distinct T-closed open sets. Since G.

properly contains G f we see that T contains no maximal

element. By the chain condition on the closed subgroups of

,n T , it follows that r = 0 .
13

4. Some Consequences and Related Results .

LEMMA 5. Let U

n

= { (x. ,

. . x ) € IR n: x. > 0 for

l n l

i = 1 /

• • . , n and x^j^ + . . + x < 1 ) . There is a

n

1 number >( < 1 such that if Gr is a arouo for which

l n

c G C IR n and G n U

n

0 then there is a ooint

(x

1

,

. • • r x ) € g n u

n

n

‘ for which x_ + . . . + x < x

1 n

- A

Proof

.

Consider the sequence

t (1/2 U

n

) C t (2/3 U

n

) C t (3/4 U

n

) C . . . . By Theorem 3 there is an m such that

T(m/(m + 1) U

n

) = T((m + l)/(m + 2) U

n

) = . . .. We may set x = m/(m + 1) .

In general it seems difficult to find a value for x*

We know that for n = 1 we can take x = 1/2; for n = 2,

X = 5/6. Any value for n = 3 must satisfy x > 41/42, but

we do not know a value even in this case.

Let x ndenote the least value for x satisfying Lemma

5. It is easy to see that x R

< x n+1 for n = 1, 2, . . ., for if G C IR n is a group such that G fl U

n

^ 0 and G D (a

U

n

) = 0 then G x [R has the analogous properties in IR n+1

. For S C IR n denote by S° its interior. For a convex

polytope K c IR n denote by vert(K) its vertex set.
-L H

LEMMA 6 . Let k = [ 1 . Suppose the convex

1 " X 2n-2

polytope K, having vert(K) C Z

n

, contains at least

(1 + k)

n

+ 1 points of Z

n

, and K° D Z

n = 0. Then there is

a linear function A : lR n -> IR n 1 such that A(Z n

) = Z

n 1

and A (K)

n Z n_1 = 0.

Proof. Clearly some pair of points of K fl Z

n must be

congruent modulo 1 + k; the line L containing these

satisfies |L fl K fl Zn

| > k + 2. Let u, w € Z n be such

that u, u + w, u + 2w, . . and u + (k+l)w are consective

points of L fl K fl Z

n

. We may choose a basis

{w, b

2

, b

3

, . . . , b^} for Z

n which contains w. For

x = a_w + a.b^ + . . . + a b € Z

n 12 , let

2 n n

A(x) = ( a

2

'

* • • / a

n

) € Then A : IR n -* ER n 1 is a

linear function such that A(Z n

) = Z

n 1

. We will complete the proof by showing that if

A (K) ° fl Z

n 1

? 0 then K° fl Z

n

^ 0. Suppose

p € A(K)° D Z

n 1

. Then by a theorem of Steinitz ([6]; see

also Exercise 2.3.5 of [3]) we may choose a set of

m < 2(n - 1) vertices of A(K)

,

say, (A(v

1

) ,

. . ., A(v

m

)},

where v- ,

. . . . and v are vertices of K, such that p

is in the interior of convfACv^, . . A(v

m

)}. We may

find a ,

. . ., a , and (3 , where > 0 (i = 1, . . . , m) , m m

( 1

a

i

) + P = 1, and p = ( J A(v

i

) ) + p (A(u)).

i=l i=l

P > o,
Id

Let G c !R m be the subgroup

m

G = { ( Ul ,

. . v m

) : l Ui (A(v.) - A (u) ) € Z

n-1

}.

i=l

Clearly G D Z

m

, and (a ,

. . . , a^) € G. By Lemma 5 it is

/V /V 'V possible to choose (a. ,

. . . , a ) € G such that a. v 1' ' irr l

> 0

for 1 < i < m and a. + . . . + a < x < x , > . 1 m m 2 ( n-l

)

Let

/Vp =

'V 1 - a

^ - . • • - ^ 1 - *2 (n-2)

> l/(k + 1) . Consider

m m

X =

) a.v. +

L ii pu and y = J a.v. + P ( u + (k + i) w )

Lmi J. 1. i=l i=l

= X + j3(k + l)w. Suppose cir x = t.w + T_b_ + . . . t b

1 2 2 n n

, so

that A (x) = (T 2' • • • f

• Since * * * r € G,

T

2

'

. . . , and T

n

are integers. Since p (k + 1) > 1, there

is an integer T

1

such that t < t < t + p (k + l)

.

Then

z = T

1

W + T

2

b

2

+ . . + t b € Z

n is in the relative

n n

interior of the line segment connecting x and y, so

z € K° fl Z

n

.

THEOREM 4. There are . up to unimodular equivalence , only

finitely many convex polvtopes P satisfying :

(i) vert(P) C Z

n

;

(ii) P° fl Z

n = 0; and

(iii) P° fl G ^ 0 , for each group G C [R n which

properly contains Z

n

. Proof. After Lemma 6, we need only show that there are only

finitely many equivalence classes of such P for which
| p n Z

n

| < m (where m = (1 + k)

n

+ 1 as in Lemma 6)

.

Indeed , if |

P

n Z

n

| > m and if P satisfies (i) and (ii)

then A

1

(Z

n X

) is a subgroup G of IR n for which (iii)

fails, where A is the linear function guaranteed by the

lemma

.

Suppose that P and Q are convex polytopes, each

satisfying conditions (i), (ii), and (iii), and neither

having m or more elements in common with Z

n

. Let

u

n

= {(x

1

x

m_ 2

) e R

m~ 2

: > o

for 1 < i < m - 2 and x_ + . . . + x 0 < 1 }

. 1 m-2

Let B : IR m 2

-» IR n and C : IR m 2-* IR n be affine functions

mapping cl(Un

) onto P and Q respectively and mapping

Z

m 2 onto 'Zn

. The subgroups G = B

1

(Z

n

) and H = C

1

(Z

n

) then miss U, and are maximal such subgroups. If G = H

then there is an affine unimodular function D : IR n -» IR n

such that B = DC. In particular, DQ = P.

m. 2 By Theorem 1, the number of maximal subgroups G D Z

such that G fi U = 0 is finite. We see from the preceding

paragraph that this number is an upper bound on the

cardinality of any collection of unimodularly inequivalent

convex polytopes P satisfying (i), (ii), (iii), and -

| vert (P) | < m.
17

5 . Unanswered Questions .

In this final section we present some problems and

questions that seem natural but with which we have not dealt.

A. Is there a reasonable method for computing the

finitely many groups of Theorem 1 — say, when the dimension

n is small and the set U is the interior of a convex

polytope?

B. Compute x ; or at least find numbers that can serve

as the x's of Lemma 5. (We know x

1 = 1/2, x

2 = 5/ 6 /

X

> 4 1/42 ,

. . .

.

)

C. Find the convex polytopes P of Theorem 4, when

(say) n = 3. (For n = 1, there is, up to unimodular

equivalence, only the interval [0, 1]; for n = 2, only

0 2 0

. 0 .

t

. 0 .

!

. 2 .

Acknowledgement . We are indebted to Mark Hartmann for a

careful reading of and several improvements over a

preliminary version of this paper.
18

References .

1. Bourbaki, N. , General Topology , Part 2. Hermann, 1966.

2. Cassels, J. W. S., An Introduction to the Geometry of

Numbers , Springer-Verlag, 1971.

3. Grunbaum, B. , Convex Polytopes . Interscience, 1967.

4. Hardy, G. H. , and E. M. Wright, The Theory of Numbers . Oxford University Press, 1968.

5. Lagarias, J. C. , H. W. Lenstra, Jr., and C. P. Schnorr,

"Korkine-Zolotarev Bases and Successive Minima of a Lattice

and its Reciprocal Lattice." (Preprint.)

6. Steinitz, E., "Bedingt Konvergente Reihen und Konvexe

System." J. reine angew . Math . 143 (1913), 128-175; 144

(1914), 1-40; 146 (1916), 1-52.
NBS-H4A (REV. 2-80

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4. TITLE AND SUBTITLE

PUBLICATION OR

R °0 RT NO.

NISTIR 38-3777

2. Performing Organ. Report No. 3. Publication Date

AUGUST 1983

Finite Unions of Closed Subgroups of the n-Dimensional Torus

5. AUTHOR(S)

Jim Lawrence

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No.

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

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\ H Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. I f document includes a significant

bi bl iography or literature survey, mention it here)

Let U be an open subset of the torus group T n

. We show that the set of maximal

subgroups of Tn which miss U is of finite cardinality a This result is applied to

show that the lattice of finite unions of closed subgroups of Tn is a complete

distributive lattice, and to show that, up to unimodular equivalence, there are

only finitely many convex polytopes P Rn having vertices in Zn but no interior

points in Zn and such that each subgroup G of the additive group R

n which properly

contains Z n does have points in common with the interior of P 0

12. KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolon s) torus; lattice; convex polytope; successive minima

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