A Note on Sequences of Lattices

We investigate the relation between the convergence of a sequence of lattices and the set-theoretic convergence of their corresponding Voronoi cells sequence. We prove that if a sequence of full rank lattices converges to a full rank lattice, then the closures of the limit infimum and limit supremum of the Voronoi cells converges to the corresponding Voronoi cell. It remains an open question if the converse is also true.

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1 Introduction

A lattice is a discrete additive group of an euclidean space. Lattices are of great interest for discrete optimization, for example , the lattice of points of integral coordinates, is extensively used in integer programming. We analyze the relation between the convergence of a sequence of lattices and the set theoretic convergence of the corresponding Voronoi cells.

We prove that if is a sequence of full rank lattices from and - a full rank lattice also, then

where, for a given sequence of sets from

are the set theoretic limit inferior and limit superior.

2 Notations and definitions

For the following definitions one can consult [1] and [2].

Definition 1.

A lattice is a discrete additive group of , i. e., , and there exists an , such that . Its rank, , is the dimension of its spanned subspace, .

For every lattice there exist

independent vectors

in , such that

where . This a basis of . For a given base, if is the matrix whose columns are the basis vectors, the determinant of is

When the lattice has full rank, the determinant is the volume of the fundamental parallelepiped:

The fundamental parallelipiped has the following property: every vector in can be written uniquely as , where and . That is the spanned subspace can be tiled with copies of the fundamental parallelepiped centered in the vectors of the lattice. The fundamental parallelepiped depends on a particular basis but there is a similar polytope which can be uniquely associated with a lattice:

Definition 2.

Let be a full rank lattice. Its Voronoi cell is .

The Voronoi cell has a similar property of tessellation:

where the intersection of different tiles (cells) occurs only on their boundaries (frontiers) and .

The following two lattice parameters (see [2], [3]) illustrates the importance of the Voronoi cell. If is a lattice, the packing radius of , , is half the length of the shortest non-zero vector of , i. e.

The covering radius of is the smallest such that spheres of radius centered around all the points in cover the entire :

Definition 3.

Let and be full rank lattices in , we say that converges to , if, for each , there exists a basis of such that there exist the limits

and is a basis of .

3 Main result

Lemma 1.

Let be a full rank lattice. There exists such that and is a bounded set. (Consequence: is a polytope.)

proof: We prove first that is bounded; let be a basis of and . If , then , for some , and

Define ; obviously, . Let and , with . We have .

Lemma 2.

Let be a sequence of full rank lattices from . If , a full rank lattice from , then there exists such that and , .

proof: Define and ; we have . There exists such that , . Hence by choosing and , we have the desired property.

Theorem 1.

([2], Theorem 1, V.3) A necessary and sufficient condition that is that the following two conditions be both satisfied

  • if , there are points , for such that

  • if , there is a number and an integer , both depending on , such that

Where and are full rank lattices from .

Theorem 2.

Let be a sequence of full rank lattices from . If , a full rank lattice from , then .

proof: Suppose that is a basis of , , and is a basis of such that , . For every , there exists such that , and .

Let , i. e., there exists such that . Choose and ; we have and

hence .

We proved that . Now we prove that . Suppose not and let . It means that there exists an such that

Since , we have , with . By perturbing and decreasing (if necessary), we can suppose that

Using this property we can define an increasing sequence such that , for any ; hence there exist the points , such that , . Since , we can extract a convergent subsequence from , which, for the sake of simplicity, will be named in the same way: . Obviously , , and, by (ii) from Theorem 1, - a contradiction with the above property.

4 Conclusions

The converse of our main result it remains an open problem for now, but it can be proved (which is left for another technical report) that, if the extremal points of are contained in (when this happens, , starting from a certain ), then . It remains to see what happens when some of the extremal points of belong on the boundary of .

References