Successive minima and Minkowski's second theoremsecond theorem

2 JAI,

Theorem 5.4 Let n > 2, and let

6.4 Successive minima and Minkowski's second theoremsecond theorem

180 6. Geometry of numbers

6.4 Successive minima and Minkowski's

and so

B(O,As)eA*K

for all A > 0. In particular, {a, , ... , a } C X* K for A sufficiently large. Since K is bounded and A is discrete, there exists A > 0 sufficiently small that (A * K) fl A = (0).

The successive minima of the convex body K with respect to the lattice A are the real numbers X1, A2, ... , A defined as follows:

Ak - inf(A > 0 : A * K contains k linearly independent elements of A).

It is easy to see that 0 < A, < A2 < ... < A,,, and that the definition of A, is

equivalent to (6.2).

Because K is an open set, it follows that Ak * K contains at most k - I linearly independent vectors in A and that Ak * K contains at least k linearly independent vectors.

There is an equivalent way to define the successive minima and at the same time identify a linearly independent set {b, , ... , b } of vectors in the lattice A such that every vector u E (Ak * K) fl A is a linear combination of b,, ... , bk_,. Let

A, - inf{A > 0 : A * K contains a nonzero vector b, E A}

and, for 2 < k < n,

Ak = inf{A > 0 : A * K contains a vector bk E A linearly independent of b,, b2, ... , bk_I }.

For example, let A - Z" and let K be the box

K = {(x,, E R" :

1x11 < r; fori = 1,...,n},

where 0 < r,, < r"_, _< ... < r2 < r, _< 1. Then ((1/r,) * K) fl v - {0} and

±e, E (A * K) fl Z" for all A > l/r,; hence A, - I/r,. Similarly, Ai = 11ri for

i=2,...,n.Since anddet(A)=1,weseethat

A, ... A" vol(K) = 2" det(A).

This simple example shows that the following theorem is best possible.

Theorem 6.6 (Minkowski's second theorem) Let K be a symmetric, convex body in R", and let A be a lattice in R". Let A, , ... , A,, be the successive minima of K with respect to the lattice A. Then

A, ... A vol(K) < 2"det(A).

Proof. Corresponding to the successive minima A, , ..., A,, are n linearly inde- pendent vectors b, , ... , b,, in the lattice A such that, fork = 1, ... , n, every vector in (Ak * K) fl A is a linear combination of b,, ... , bk_, and

{b1....,bk}cAk*K.

182 6. Geometry of numbers

We shall use the basis {b,, b2, ... , b } to construct a continuous map

rp : K K

such that

v(K) C X,, * K.

For j = 1, ..., n, let Vj be the subspace of R" spanned by {b,

, b2, ... , b_ ,

} and let Wj be the subspace of R" spanned by (bj , bj+, , ..., b,,}. Then

R"=Vj®Wj.

Let

irj:R"--W1

be the projection onto Wj. For every vector y e Wj,

it (Y)=Vj+{y}

is an affine subspace (or plane) of dimension j - 1. Let K' - 7rj (K). Then K' is a convex body in Wj, and K1 = trj(K) is a compact convex set in Wj.

Let cj : K -> K be the continuous function that maps x E K to the center of mass of of (x) = r 1 (trj (x)) fl K. We define the coordinate functions cij (x) by

it

cj(x) _ c;j(x)b;.

Ifx=Ell

x;b;,then

cij(x)=x;

for i

= j, j+I....,n

and c;j(x) is a continuous function of x 1 . .. ..

. x,, for i = 1, ... , j - 1.

Let X, be the successive minima of K, and let ,lo = 0. For x e K, we define

W(x) _ EUj -,Lj-,)cjW.

j-1

Let tj - (,lj - )/A for j = 1, ... , n. Then tj > 0 for all j and t, +

+ t,, - 1.

Since cj(x) E K for all x E K, it follows that

11

co(x) _ A E tjcj(x) a A,, * K,

j-1

and so o : K --+ A,, * K is a continuous function that satisfies W(K) c ,l * K.

Moreover,

11

Ox) = E(kj -

j-1

Let

It

_ E(Aj -Xj-I)ECi.j(x)bi

j-I i-1

= E E(,kj - )Lj_I )ci.j

(x)bi

i-1

(j-1

It

(Ei

^{n 1}

E -,lj_i)xi + E(Xj -)j_1)ci,j(x))

bi

i-1

j-I

j-i+l

I, n

_

(x1+

(Aj - Aj-I)Ci.j(W

))

^bi.

i-I j-i+l

It

fi(x) = E (Aj - )Lj-1)ci. j(x)

j-i+l

Then Vi (x) _ Bpi (xi+1, .^,. , x,2) is a continuous, real-valued function of the n - i coefficients xi+1, ... , x of the vector x E K, and

-I

(P(x) (Aixi + (Pi (xi+l, ... , bi + A,,xnbr,. (6.5) i-1

This has two important consequences.

First, the function rp is one-to-one. Let x, x' E K, where x = x1 b1 + ... + x b and x' = xi bl + + x;,b,,. If V(x) = Q(x'), then

n-1

(Ai xi + ipi (xi+I , ... , X11)) bi + ;nxn '-I

n-1

(xix; bi +)L,,X11 At.

i-1

Equating the coefficients of b,,, we see that A,,x;, and so x,, = x;,. Equating the coefficients of we see that

A,,_1xn-1 +qn-1(xn) _ An-1X,-1 +(pn_1(xn) = A,,_1xn_1

and so x _I = x; _1. Applying this argument inductively, we obtain xi = x; for

i = 1,...,n,andsox=x'.

The second consequence is

vol(ip(K)) - Al A,1 vol(K).

In the special case when the functions B p i (x;, ...,x,,) are identically equal to zero for i - 1, ... , n, the function q is given by the simple formula

!p(x1....,xn)=A1x1b1 +...+Anxnb,,,

184 6. Geometry of numbers

and

vol(rp(K)) - Al ... A,, vol(K). (6.6) Since the functions B p i (x;+1, ..._,x,,) are continuous f o r i - 1, ... , no this formula for the volume of V(K) holds in general.

The set V(K) is not necessarily convex, but it is a bounded open subset of R"

to which we can apply Lemma 6.1 of Blichfeldt. Let K' - rp(K). If

vol(K') - I, ... X. vol(K) > 2" det(A),

then

vol((1/2) * K') > det(A),

and Lemma 6.1 implies that there exist vectors x,, x2 E K' such that

ie1-)e2

EA\(0).

2

Since K' - V (K), there exist vectors x1, X2 E K such that rp(xl) - x, and V(x2) -

x. Let

x1 - ^Ex;.1bi

and

x2 - xi.2bi

1_1

Since x, f x2, there exists k > 1 such that Xk.1 f xk.2 and xi. I - X;,2 for i - k + 1, ... , n. Recall that if x- E"_, x; b; , then the center of mass cj (xj , ... , x") E K f o r j - 1, ... , n. Since K is a symmetric convex body,

Cj(Xj+1.I,...,X,.l)-Cj(Xj+1.2,...,Xn.2)

_{E K,}

2 and so

Zk (C j (X j+1.1, ... , x,1 1) - Cj (X j+1.2, ... , X,,.2))

EAk*K

2

for j - 1, ..., n. Let tj - (A1 - Aj_1 /Ak for j - 1, ..., k. Then

X'1 -I e2 (P(xl)-(P(x2)

2

-

²

Cj(xl) - Cj(x2) j-1

k

/C1(XI) - Cj(x2)

E(Aj - A1_1) ²

j-1

1: tj Xk ₂

_ ^k

(CJ(X')_Cj(X2)\

j-1

E (ilk * K) n (A \ (0)).

Representing the map tp in the form (6.5), we see that

xI - x)

(P(x1) - (P(x2) 2

and

Since

2

2 x.2) + (Vi(xl) 2'Pi(x2)11 b.

(xi.l -

2 xi.2)

+

(P;(xl) -

2 ^(Pi(x2

a, )

(xi.1 -

2 xf.2)

+(rPf(xl) 2

(Pi(x2)) b;

+1k(xk.1 2

xk.2)_bk,

Ak ^(xk.1

2 Xk.2)

"''- E(kk*K)nA,

2

it follows from the definition of the successive minima),,,..., X. that (x, - x'2)/2 can be represented as a linear combination of the vectors b1,..., bk_1. On the other hand, we have just shown that (x1 - x'2)/2 is also a linear combination with real coefficients of the vectors b1, ... , b_1, bk and that the coefficient of bk in this representation is xk(xk,1 - xk.2)/2 f 0. This implies that bk is a linear combi- nation of the vectors b1... bk_1, which contradicts the linear independence of the vectors b1, ... , b,,. This completes the proof of Minkowski's second theorem.