Math. Ann. 296, 625~535 (1993)

Springer-Verlag i 993

New bounds in some transference theorems

in the geometry of numbers

W. Banaszczyk

Institute of Mathematics, L6d~, University, Banacha 22, PL-90-238 L6d~, Poland

Received July 4, 1992

Mathematics Subject Classification (1991): 11H06, 11H60, 52C07

Introduction

The aim of this paper is to give new bounds in certain inequalities concerning

mutually reciprocal lattices in R". To formulate the problem, we have to introduce

some notation and terminology.

We shall treat IR" as an n-dimensional euclidean space with the norm II II and

metric d. The inner product of vectors u, v will be denoted by uv; we shall usually

write u 2 instead of uu. The closed and open unit balls in F," will be denoted by

B, and B'., respectively. If A c IR ", then span A and A z denote the linear subspace

spanned over A and the orthogonal complement of A in P~".

A lattice in IR" is an additive subgroup of IR" generated by n linearly indepen-

dent vectors. The family of all lattices in IR" will be denoted by A..

Given a lattice L e A., we define the polar (dual, reciprocal) lattice L* in the

usual way:

L* = {u ~ ~":uv~Z for each v~L} .

One has L**= L. By d(L) we shall denote the determinant of L, i.e. the n-

dimensional volume of a fundamental domain of L. Naturally, d(L*) = d(L)-a.

A convex body in/R" is a compact convex subset of IR" containing interior

points. The family of all o-symmetric convex bodies in ~" will be denoted by cg,.

Given a convex body U ~ cg., we define the polar body U ~ in the usual way:

U ~ = {u ~lRn:uv_-< 1 for each v~ U} .

By 1[ [Iu we shall denote the norm on 1R" induced by U (the Minkowski functional of

U). By dv we shall denote the metric induced by [I [Jr. Thus l[ 11 = 1[ IJn. and d = de..

Given a lattice L e A. and a convex body U ~ cg., we denote

p(L, U) = max{dr(u, L): u ~ R"}

= min{r > 0:L + rU = 1R"},

2t(L, U) = min{r > 0:di m span(Lc~rU) ~_ i} (i = 1 ..... n) .

626 W. Banaszczyk

The quantity #(L, U) is called the covering radius of L with respect to U. The

quantities hi(L, U) are called the successive minima of L with respect to U. To

simplify the notation, we shall write #(L) and 2i(L) instead of #(L, B,) and 2~ (L, B,),

respectively.

Given a convex body U ~ cg,, we define

~(U) = sup max 2~(L, U) 2.- ~+I( L*, U ~ ,

LeAn ].~i~n

r/(U) = sup #(L, U)21(L*, U ~ ,

L~An

~(U)= sup sup inf d(uv, 7Z) -1 dv(u,L)llvl[vo.

LeAn u~Rn\L v~L*

uvCZ

It is clear that the quantities ~(U), q(U) and ~(U) are affine invariants of U. The

obvious inequality/~(L, U) < n ,~,(L, U) implies that q(U) < n~(U). It is also

clear that ~/(U) < ~(U).

For each n = 1, 2 ..... we define

4, = sup r r/,--- sup r/(U), (, = sup ((U).

Ue~n Ue~dn Ue~gn

Then q. < and q, < Mahler [11] proved that 4, < (n!) 2 for n > 1. See

also [6], VIII, Sect. 5, Theorem VI, where (n!) 2 is replaced by n!. Recently,

Lagarias et al. [10] proved that ~(B,)< ~ n 2 for n > 7. Similar bounds were

obtained independently in [4], Theorem (2.1).

The inequalities ~/(B.) < ~2 n s/z for n > 5 follow from Lemma 1.4 of [3] (see also

[4], Lemma (1.4) and the subsequent remarks). Lagarias et al. [10] proved that

11n <= n a/2 for every n.

That (, < oo was proved by Khinchin [9]. Cassels [6], XI, Sect. 3, Theorem VI

gave the bound (, < 21 -"(n!) 2. Babai [1] proved that (. < C" for some universal

constant C. The inequalities ((B.) < 12n(n + 1) can be derived from Lemma 7 of

[2]; see also [4], Lemma (1.1) and the subsequent remarks. Hastad [7] proved that

((B,) < 6n 2 + 1.

In the present paper we show that there exists an universal constant C such that

~(B.), q(B,), ~(B.) < Cn for every n. Examples show that ~(B,), ~7(B,), ~(B,) > cn

where c is some other universal constant. For the numerical values of C and c, see

the final remarks in Sect. 2. The corresponding inequalities for convex bodies other

than B, are discussed in Sect, 3.

The determination of the quantities ~(U), r/(U) and ~(U) is a classical problem

in the geometry of numbers. The corresponding results belong to the so-called

transference theorems; a detailed description is given in Cassels' book [6].

Recently, these questions became a subject of intensive investigations in integer

programming; we refer the reader to the introductions in [1, 7, 10]. Another source

of motivation is commutative harmonic analysis, more precisely, the theory of

characters of additive subgroups and quotient groups of topological vector spaces.

This point of view is presented exhaustively in monograph [5], especially, in

Section 3. See also the survey article [4].

Papers [2-4, 7, 10] all found on the idea of Korkin-Zolotarev bases. The

method used in the present paper is entirely different. Taken from commutative

New bounds in the geometry of numbers 627

harmonic analysis, it consists in investigating certain probability measures on

lattices and Fourier transforms of such measures. The proofs are non-constructive.

1 Gaussian-like measures on lattices

Let # be a finite Borel measure on IR". For the purpose of the present paper, it is

convenient to define the Fourier transform p ^ of # by the formula

#^ (x) = S e2~ixr dll(y) (x ~ IR").

Nn

Similarly, given an integrable complex-valued function f on N", by the Fourier

transform f ^ of f we shall mean the function

/^( x) = f e:~xYf(y)dy ( xeR").

Rn

The symbol dy denotes integration with respect to the n-dimensional Lebesgue

measure on ~". Under such a definition, the function e -~x: is equal to its Fourier

transform.

Let L be a lattice in IR". By aL we shall denote the probability measure on

L given by the formula

aL(A)= E e-"X2 / Z e-'~2 (A = L).

x~A / xEL

By gOL we shall denote the function on IR" defined by the formula

0L(u) = Z / Y (u JR").

xcL+u / x~L

To simplify the notation, we shall write

p(A) = Z e-"X2 (A c IR").

x~A

Then aL(A) = p(A)/p(L) for A c L, and tpL(u) = p(L + u)/p(L) for u ~ IR".

In the remaining part of this section, n is a fixed positive integer, and L is an

arbitrary, but fixed lattice in N".

(1.1) Lemma, Let a, b be positive numbers such that ab = 7r. Let p be the measure on

L given by the formula p({x}) = e -"~2. Then

(i) ~, e2~i~re-axe=b"/ed(L)-i ~ e-e"iuze -"b(y+z)2 (u,y~]R");

xeL+u z~L*

(ii) p^ (y) = b n/2 d(L)-I ~, e-,btr+,)~ (y ~ IR").

zEL*

Proof Point (i) follows from the Poisson summation formula; see e.g. [8, (31.46)

(c)]. Point (ii) is a direct consequence of (i). []

(1.2) Corollary. One has try, = tpL* and of. = ~OL.

628 W. Banaszczyk

Proof It is enough to prove that ~ = ~ot:, Let # be the measure on L given by the

formul a #({x}) = e - ~. Then o'L = ll/#(L), so that ~2 = #~/#^ (0), and the result

follows from (1.1) (ii).

Let k = 1 ..... n and let x ~ IR". By Xk we shall denote the kth coordinate ofx. If

82

f is a function on IR", we shall write fkk = ~ f

(1.3) Lemma. Given arbitrary u ~ 1R", a > 0 and k = I ..... n, one has

1

I f u = O, then ~c ~ - -.

- 2a

E e-~

x~L+u J xsL -~ a

Proof Let # be the measure on L + u given by the formula p((x}) = e -"~, Then

#^( y) = ~ e z'~ixye-ax 2 (y ~"),

x~L+u

x~L+u

Setting here y = O, we get

x 2 e 2~xy e-": (y e IR ~) .

E x~ e-": = - (d~) -' ~(o).

xeL+u

(1)

Let us denote b = -~. It follows from (1.1) (ii) that

r

I~k(y ) = b ":2 d(L) -1 ~ e -2"i"" [ - 2rob + 4rceb2(yk + z~)23e -~b('+z?

for y s IR". Setting y = O, we get

It~(O) = b"/2 d(L) -1 ~ e -z"i"~ [ - 2rob + 4rt 2b2z~]e -"b~2 .

z~L*

Let v be the measure on L* given by the formul a v({z}) = e - ~:. Then

(2)

~_, e-Z"~U" e -~bz~ = v^( - u),

zeL*

(3)

z~ e- 2~i"~ e -'~b: = - (4rc2) -1 Vk~(-- U).

z~L*

Setting u = y = 0 in (1.1) (i), we see that

(4)

v ~ (0):= ~. e-~b'Z = d(L) b-"/2 ~ e-"x~ (5)

:t~L* xeL

New bounds in the geometry of numbers 629

Now, from (1)-(5) we derive

1 v ^( - u) l Vk~k(-u)

X- - 2a v ^(0~ +4a z v ^( 0~ (6)

Take an arbitrary v e IR". By (1.1) (i), we have

d(L)- 1 b.J2 v ^ (v) = d(L)- 1 b./2 ~, e2~i~ e-nbz2

zEL*

: Z e-aX2 1 [e_a(x+v)2

xeL+v = ~ xvL ~ + e- a( x- v) 2]

= e-aV= y' e-aX2 cosh 2axv > e-av2 2 e-aX2.

x~L xcL

Hence, by (5),

v ^ (v)/v ^ (0) >__ e-a~2 (v e F,"). (7)

The functions v ^ and - Vk~ are positive-definite, being the Fourier transforms

of the positive measures e -'~b~ and 4rc2z2ke -'~b~, respectively. Therefore

v^( - u) < v^(0) and vA( - u) <- V~k(O ). It follows directly from (7) that

1 1 1

^ ^ =-. If u = 0, then (6)

- Vkk(O)/v (0) < 2a. Now, (6) implies that ~c < 2a + 2a a

1

yields ~c < 2-aa' because Vk~k(0) < 0. []

(1.4) Lemma. For each a > 1, one has

(i) ~, e-~ta-I x2 ~_ a hI2 ~,, e-=~=;

xeL xeL

(ii) ~ e -'~"- ~ < 2a "/z ~_, e -'~= (u e IR") .

x~L+u xeL

Proof. Let us consider the function

f(a) = ~, e -"~-'x2 (a >= l ).

x~L

From (1.3) we obtain

7r 7[ n

if(a) = -~ ..LE x2 e-"~-'~'= = ~ ,=,E ,,eLE x~ e -'~-'x=

mz a x7 -=a-ix2 n

<~- ~_ ~ e =2af ( a) ( a>l ).

Ct ~Vt x~L

T/

Hence [logf(a)]' < 2a etc., which yieldsf(a) < a"/2f(1) for a > 1. This proves (i).

To prove (ii), take an arbitrary u e IR" and consider the function

g( a) = ~ e -=~-'*= ( a> 1).

xeL+u

630 W. Banaszczyk

Applying (t.3) and then (i), we may write

7~ n 1

?ITC a ~ ha-Ix2

<- 2~- 2 e- <=na-l+'/2f(1) ( a>l ).

a ~ xeL

Hence

a a

g(a) - - g(1) = f g'(t)dt < nf(1)j" t -' +'z dt

! 1

=2f ( 1) [ a "/2- 1] ( a>l ). (8)

By (1.2), the function q~L is positive-definite, being the Fourier transform of the

positive measure aL.. Thus

g(l) p(L + u)

- - - = ~OL(U) < q~L(O) = 1 ,

f(l ) p(L)

i.e. g(1) _<f(1). Now (8) yields

9(a) = 2f(l) [a "/2 - 1] + 0(1) < 2a"/~f(1) - f ( 1) < 2a"/2f(1). []

(1.5) Lenlma. For each c > (27z)-t/2, one has

(i) p( r\cx/n B',) < [c~e e-"r p(r),

(ii) p((L + u)\cx/-n B') < 2[c 2x/~-~ e-'~]" p(L) (u e IR") .

Proof. For each t e (0, 1), we have

Z e-~tx2= Z e~t(1-ox2 e-nX2

xeL xeL

> ~ e,.1-t)x: e-~X: > e~.-t)c2. ~ e-,,x2.

xeL xeL

x2~czn xZ>=c:n

On the other hand, (1.4) (i) says that

Thus

E e- r ex2< t-~/2 E e-~X2"

x~L xeL

E e-~X2 < t-n/2 e-~tl-t)czn Z e-nX2

x~L xeL

x2~_c2n

which can be written as

p(L\c /n B') < It e-"C2(1 - 0]" p(L).

Setting here t = (2~ce) - t we obtain (i). To prove (ii), it is enough to apply (1.4) (ii)

instead of (1.4) (i). []

New bounds in the geometry of numbers 631

2 Transference theorems

(2.1) Theorem. Let L be an arbitrary lattice in IR', n > 1. Then

2~(L)2,_~+1(L*)<n ( i = 1,...,n).

Proof. If n = 1, then )~l (L) 21(L* ) = 1. If n = 2, then it is not hard to verify that

2t(L)22(L*) < 2"3 -1/:. So, we may assume that n > 3.

Let us suppose the contrary, that there is an index i = 1 ..... n with

2i(L) 4,_ i+ ~ (L*) > n. Replacing L by sL for a suitably chosen coefficient s, we may

assume that

3

2~(L) > ~ x/n, (9)

4

2._,+1(L*) >

(10)

Let K be the subgroup of ]R" generated by L n ~ B',, and let H be the

subgroup generated by L* c~{.v/nB',. Denot e M = span K and N = span H.

Fr om (9) and (10) we get dim M<i - 1 and dim N<n- i. Hence

di mM + di mN > n + 1, and we can find a vector ueM J- with [lull

= 3- 1/2.

As u e M we have uv = 0 for each v s K. Hence

a~ (u)= Y. ~rL({v))cos 2~zuv = Z + Z ~r,({v})cos 2auv

v~L vEK veL\K

>=

Now, (1.5) (i) implies

( rt ( L\K) <=

so that

Y'. ~L({V})-- Z ~L( {V}) =I - - 2r

v~K v~L\K

that

~L(t ~N//-~B'n)<I~N~e-9ml 613 <0.15,

r163 (u) > 0.7. (11)

Since Hull = 3-1J2 andn > 3,we have B, u c 3 B,, which means that

Hence, by (1.5) (ii),

< 2[ 2~e- ~] 3 p( L*). (12)

Next, as u e N we have

p( n + u) = e -""2 a( n) =< e -"/3 p( L*). (13)

632 W. Banaszczyk

From (12) and (13) we derive

p(L* + u) = p( ( L*\n) + u) + p(H + u) < 0.4p(L*),

i.e. ~oL,(u) < 0.4. In view of (1.2), this means that o-~ (u) < 0.4, which contradicts

(11). []

(2.2) Theorem. Let L be an arbitrary lattice in IR", n > 1. Then 21(L)#(L*) < n.

Proof If n = 1, then 21(L)#(L*) = If n = 2, then it is not hard to verify that

21(L)/~(L*) < 2 -1:2. Therefore we may assume that n > 3.

Suppose the contrary, that :~t(L)#(L*) > n. Replacing L by sL for a suitably

chosen coefficient s, we may assume that

21 (L) > 2-1/2 x/~, (14)

)~I(L*) > 2-1/z x/-n (15)

Condition (15) means that there is some u elR" such that (L* +u)c~

2- 1:2/~ B, = ~. Then, by (1.5)(ii),

p(L* + u) = p(iL* + u)\2 -l/z x/n B.) < 2[2 -1/2 2x/~e-"/233 p(L*),

whence

1

cpL.(u ) < ~. (16)

On the other hand, (14) means that L n 2-1/2 ~ B. = {0}, and then

e~( u) = Z + Z aL({V})COS2~UV

VEL v~L

v 2 < n/2 v 2 >= hi2

>- Y- E

v~L v~L

v2<n/2 v2>n[2

Now, (1.5) (i) implies that

aL({v}) = 1 - 2 eL (L\2-1/2x/~ B').

1

eL(L\2-1/2 v/~ B.) < [2-1/2 2x/~ e-~/213 < 4'

and therefore e~ (u) > In view of (1.2), this contradicts (16). []

(2,3) Theorem. Let L be an arbitrary lattice in R", n > 1, and let u ~ IRn\ L. Then

there exists some v ~ L* with cos 21ruv <= 0.1 and Ilvll d(u, L) <= n.

Proof. Let us suppose that n ~ 2, the case n = 1 being trivial. Replacing L by sL

and u by su for a suitably chosen coefficient s, we may assume that d(u, L) = x/~.

Then (1.5) (ii) yields

p(L + u) = p((L + u)\x/n B~,) < 2 [ 2x/'f~ e-~] 2 p(L).

Hence, by (1.2),

ab(u) = ~oL(u) < 2[V/2-~ e-~] 2 . (17)

New bounds in the geometry of numbers 633

On the other hand, (1.5) (i) implies that

Let us denote

Then we may write

x:= aL.(L*\x/nB',) < [ 2x/~e-"] 2 .

9=

~" e -~v2 cos 2nuv / ~ e -~vz .

vEL* l v~L*

v2<n v2<n

(18)

obt u) = Y + Z ~L.({v))cos2=u~

vsL* vEL*

u2<n u2>=n

> OaK. (L* n ~ B'n) -- ~L.(L*\v/n B'.)

-- ~9(1 -- x) - K. (19)

Now, (17)-(19) imply that ~9 < 0.1. Consequently, there must exist a vector v

L* c~ x/n B', with cos 2nuv < 0.1. []

Theorem (2,1) says that r < n for every n. Its proof, based on (1.5), allows

one to deduce that

~( B.) <~( l +O( n- 1;2) ) asn~oo.

On the other hand, there is a result of Conway and Thompson (see [12], Ch. II,

Theorem 9.5) which asserts that one can construct a sequence of lattices L, e A,

such that L* = L, for every n, and

as n --* oo. (20)

n

2~(L.) -> ~e (1 + o(1))

Consequently, one has

/,/

~(B.) __> ~ne (1 + o(1)) as n ~ oo.

As L*=L,, we have d( L,) = 1 for every n, which implies

#(L,)" vol(B,) > 1. Thus

( n~ 1/2

:(L.) > \F~eJ (1 + o(1)) as n ~ oo.

Theorem (2.2) says that q(B.) < n for every n. The proof shows that

asn~.

that

(21)

q(B.) ~ ~(1 + 0(n-1/2))

On the other hand, (20) and (21) imply that

> n

q(B.) = ~ (1 + o(1)) as n--, oo.

634 w. Banaszczyk

Let L be an arbitrary lattice in IR", n > 2. Choose any u ~ L with [lull = 21(L). It

is not hard to see that there exists a vector v eL* with uv = 1 and

v2<l +3~[ r/( B,_l ) ] 2 (cf. the proof of Theorem (3.3) in [4]). As

r/{B,_a) < - 1), it follows that I[vll < 3 -1/2 n.

From (2.3) we get ((B,) < 5n for every n. The constant 5 may be replaced by

a smaller one, perhaps even just by 1. From the proof of (2.3) one can deduce that

~(Bn) < 2n (1 + 0(n-1/2)) as n ~ oo

On the other hand, from (20) and (21) we obtain

( ( B.) > n (1 + o(1)) as n ---> ~.

7~e

3 Other convex bodies

Let U be a symmetric convex body in ~". According to the John theorem, there

exists an n-dimensional ellipsoid D such that D c U c v/~D. Thus, from

(2.1)-(2.3) it follows that ~(U), ~(U), ((U) < Cn 3/2 for some universal constant C.

The technique used in Sections 2 and 3 allows one to show that if U is symmetric

through each of the coordinate hyperplanes, then ~(U), q(U), ((U) < C1 n log n for

some other constant Ca. Moreover, if U is the unit ball in l~, 1 < p < 0% then ~(U),

q(U), ((U) < C2 n 1~ n. The proofs will be given in a separate paper. On the

other hand, the following fact is true:

(3.1) Theorem. There exists an universal constant c such that, given an arbitrary

convex body U ~ ~,, one can find a lattice L ~ An with 21(L, U)21(L*, U ~ > cn.

This implies that ~(U), q(U), ((U) > cl n for each U e r Theorem (3.1) follows

easily from Siegel's mean value theorem; the details of the proof will be given

elsewhere.

Let us close with the following remark: it follows easily from the Bourgain-

Milman theorem on the product of volumes of polar bodies and from the

Minkowski convex body theorem that

;q( L,U) ),I ( L*,U~ ( L~An, U~Cg,,n= I, 2 .... )

for some universal constant C.

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119-128 (1984)

3. Banaszczyk, W.: Pontryagin duality for subgroups and quotients of nuclear spaces. Math.

Ann. 273, 653-664 (1986)

4. Banaszczyk, W.: Polar lattices from the point of view of nuclear spaces. Rev. Mat. Univ.

Complutense Madr. 2 (special issue), 35-46 (1989)

New bounds in the geometry of numbers 635

5. Banaszczyk, W.: Additive subgroups of topological vector spaces. (Lect. Notes Math., vol

1466) Berlin Heidelberg New York: Springer 1991

6. Cassels, J.W.S.: An introduction to the geometry of numbers. Berlin G6ttingen Heidelberg:

Springer 1959

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torica 8, 75-81 (1988)

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Akad. Nauk SSSR, Ser. Mat. 12, 113-122 (1948)

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11. Mahler, K.: Ein Ubertragungsprinzip f/Jr konvexe K6rper. (~as. P6stov/ini Mat. Fys. 68,

93-102 (1939)

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