A Note on Sequences of Lattices

1 Introduction

A lattice is a discrete additive group of an euclidean space. Lattices are of great interest for discrete optimization, for example $Z^{n}$ , 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 ${(Λ_{k})}_{k \geq 1}$ is a sequence of full rank lattices from $R^{n}$ and $lim k \to \infty Λ_{k} = Λ$ - a full rank lattice also, then

¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ liminf k \to \infty V (Λ_{k}) = ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ limsup k \to \infty V (Λ_{k}) = V (Λ),

where, for a given sequence ${(A_{k})}_{k \geq 1}$ of sets from $R^{n}$

liminf k \to \infty V (A_{k}) = ⋃ h \geq 1 ⋂ j \geq h V (A_{j}), liminf k \to \infty V (A_{k}) = ⋂ h \geq 1 ⋃ j \geq h V (A_{j})

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 $Λ \subset R^{n}$ is a discrete additive group of $R^{n}$ , i. e., $x - y \in Λ$ , $\forall x, y \in Λ$ and there exists an $ϵ > 0$ , such that $B (0; ϵ) \cap Λ = \emptyset$ . Its rank, $r a n k (Λ)$ , is the dimension of its spanned subspace, $dim (span (Λ))$ .

For every lattice there exist $m$

independent vectors

v_{1}, v_{2}, \dots, v_{m}

Λ

, such that

Λ = {m \sum i = 1 α_{i} v_{i} : α_{1}, \dots, α_{m} \in Z},

where $m = r a n k (Λ)$ . This a basis of $Λ$ . For a given base, if $A$ is the matrix whose columns are the basis vectors, the determinant of $Λ$ is

det (Λ) = \sqrt{det (A^{T} A)} .

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

Π = {m \sum i = 1 a_{i} v_{i} : a_{i} \in [0, 1), \forall i = ¯ ¯¯¯¯¯¯ ¯ 1, n} .

The fundamental parallelipiped has the following property: every vector $x$ in $l i n (Λ)$ can be written uniquely as $x = v + y$ , where $v \in Λ$ and $y \in Π$ . 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 $Λ \subseteq R^{n}$ be a full rank lattice. Its Voronoi cell is $V (Λ) = {x \in R^{n} : ∥ x ∥ \leq ∥ x - v ∥, \forall v \in Λ}$ .

The Voronoi cell has a similar property of $span (Λ)$ tessellation:

span (Λ) = Λ + V (Λ),

where the intersection of different tiles (cells) occurs only on their boundaries (frontiers) and $det (Λ) = vol (V (Λ))$ .

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

ρ (Λ) = (min v \in Λ ∥ v ∥) / 2 = inradius [V (Λ)] .

The covering radius of $Λ$ is the smallest $μ$ such that spheres of radius $μ$ centered around all the points in $Λ$ cover the entire $l i n (Λ)$ :

μ (Λ) = max x \in l i n (Λ) d (x, Λ) = circumradius [V (Λ)] .

Definition 3.

Let $Λ$ and ${(Λ_{k})}_{k \geq 1}$ be full rank lattices in $R^{n}$ , we say that ${(Λ_{k})}_{k \geq 1}$ converges to $Λ$ , $lim k \to \infty Λ_{k} = Λ$ if, for each $k \geq 1$ , there exists a basis $v_{k 1}, v_{k 2}, \dots, v_{k n}$ of $Λ_{k}$ such that there exist the limits

lim k \to + \infty v_{k i} = v_{i}, \forall 1 \leq i \leq n,

and $v_{1}, v_{2}, \dots, v_{n}$ is a basis of $Λ$ .

3 Main result

Lemma 1.

Let $Λ \subseteq R^{n}$ be a full rank lattice. There exists $R \in R_{+}^{*}$ such that and $V (Λ)$ is a bounded set. (Consequence: $V (Λ)$ is a polytope.)

proof: We prove first that $V (Λ)$ is bounded; let $(u_{i})_{1 \leq i \leq n}$ be a basis of $Λ$ and $r = max 1 \leq i \leq n ∥ u_{i} ∥$ . If $x \in V (Λ)$ , then $x = n \sum i = 1 β_{i} u_{i}$ , for some $β_{i} \in R$ , $\forall 1 \leq i \leq n$ and

∥ x ∥ = ∥ ∥ ∥ ∥ n \sum i = 1 [β_{i}] u_{i} + n \sum i = 1 {β_{i}} u_{i} ∥ ∥ ∥ ∥ \leq ∥ ∥ ∥ ∥ n \sum i = 1 β_{i} u_{i} ∥ ∥ ∥ ∥ \leq n \sum i = 1 ∥ u_{i} ∥ \leq n \cdot max 1 \leq i \leq n ∥ u_{i} ∥ = n \cdot r .

Define $R = 2 r n$ ; obviously, $V (Λ) \subseteq V_{R} (Λ)$ . Let $x \in V_{R} (Λ)$ and $v \in Λ$ , with $∥ v ∥ \geq R$ . We have $∥ x - v ∥ \geq ∥ v ∥ - ∥ x ∥ \geq R - R / 2 = R / 2 \geq ∥ x ∥$ . $□$

Lemma 2.

Let ${(Λ_{k})}_{k \geq 1}$ be a sequence of full rank lattices from $R^{n}$ . If $lim k \to \infty Λ_{k} = Λ$ , a full rank lattice from $R^{n}$ , then there exists $R^{'} > 0$ such that $V (Λ) = V_{R^{'}} (Λ)$ and $V (Λ_{k}) = V_{R^{'}} (Λ_{k})$ , $\forall k \geq 1$ .

proof: Define $R_{k} = 2 n \cdot max 1 \leq i \leq n ∥ u_{k i} ∥$ and $R = 2 n \cdot max 1 \leq i \leq n ∥ u_{i} ∥$ ; we have $lim k \to \infty R_{k} = R$ . There exists $k_{0} \in N$ such that $R_{k} \leq 2 R$ , $\forall k \geq k_{0}$ . Hence by choosing $R^{'} \geq 2 R$ and $R^{'} \geq R_{k}$ , $\forall 1 \leq k < k_{0}$ we have the desired property. $□$

Theorem 1.

([2], Theorem 1, V.3) A necessary and sufficient condition that $lim k \to + \infty Λ_{k} = Λ$ is that the following two conditions be both satisfied

if $u \in Λ$ , there are points $v_{k} \in Λ_{k}$ , for $k = 1, 2, \dots$ such that

$lim k \to + \infty v_{k} = u .$
if $x \notin Λ$ , there is a number $ϵ_{0} > 0$ and an integer $k_{0}$ , both depending on $x$ , such that

$∥ u - x ∥ > ϵ_{0}, \forall u \in Λ_{k}, \forall k \geq k_{0} .$

Where ${(Λ_{k})}_{k \geq 1}$ and $Λ$ are full rank lattices from $R^{n}$ .

Theorem 2.

Let ${(Λ_{k})}_{k \geq 1}$ be a sequence of full rank lattices from $R^{n}$ . If $lim k \to \infty Λ_{k} = Λ$ , a full rank lattice from $R^{n}$ , then $¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ liminf k \to \infty V (Λ_{k}) = ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ limsup k \to \infty V (Λ_{k}) = V (Λ)$ .

proof: Suppose that $(u_{k i})_{1 \leq i \leq n}$ is a basis of $Λ_{k}$ , $\forall k \geq 1$ , and $(u_{i})_{1 \leq i \leq n}$ is a basis of $Λ$ such that $lim j \to \infty u_{j i} = u_{i}$ , $\forall 1 \leq i \leq n$ . For every $ϵ > 0$ , there exists $k_{ϵ}$ such that $∥ u_{k i} - u_{i} ∥ \leq ϵ$ , $\forall k \geq k_{ϵ}$ and $\forall 1 \leq i \leq n$ .

Let $x \in liminf k \to \infty V (Λ_{k}) \subseteq limsup k \to \infty V (Λ_{k}) = ⋂ k \geq 1 ⋃ j \geq k V (Λ_{j})$ , i. e., $\forall k \geq 1$ there exists $j = j_{k} \geq k$ such that $x \in V (Λ_{j_{k}})$ . Choose ${(α_{i})}_{1 \leq i \leq n} \subseteq Z^{n}$ and $u = n \sum i = 1 α_{i} u_{i} \in Λ$ ; we have $u_{j_{k}} = n \sum i = 1 α_{i} u_{j_{k} i} \in Λ_{j_{k}}$ and

∥ x ∥ \leq ∥ ∥ ∥ ∥ x - n \sum i = 1 α_{i} u_{j_{k} i} ∥ ∥ ∥ ∥, \forall k \geq 1 k \to \infty ⟹ ∥ x ∥ \leq ∥ ∥ ∥ ∥ x - n \sum i = 1 α_{i} u_{i} ∥ ∥ ∥ ∥ = ∥ x - u ∥,

hence $x \in V (Λ)$ .

We proved that $liminf k \to \infty V (Λ_{k}) \subseteq limsup k \to \infty V (Λ_{k}) \subseteq V (Λ)$ . Now we prove that $V (Λ) \subseteq ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ liminf k \to \infty V (Λ_{k})$ . Suppose not and let $x_{0} \in V (Λ) ∖ ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ liminf k \to \infty V (Λ_{k})$ . It means that there exists an $ϵ_{0} > 0$ such that

B (x_{0}; ϵ_{0}) \cap liminf k \to \infty V (Λ_{k}) = \emptyset \Leftrightarrow B (x_{0}; ϵ_{0}) \cap ⎛ ⎝ ⋂ j \geq k V (Λ_{j}) ⎞ ⎠ = \emptyset, \forall k \geq 1.

Since $x_{0} \in V_{R^{'}} (Λ)$ , we have $∥ x_{0} - u ∥ \geq ∥ x_{0} ∥$ , $\forall u \in Λ$ with $∥ u ∥ \leq R^{'}$ . By perturbing $x_{0}$ and decreasing $ϵ_{0}$ (if necessary), we can suppose that

x_{0} \notin ⋂ j \geq k V (Λ_{j}), \forall k \geq 1 % and ∥ x_{0} - u ∥ > ∥ x_{0} ∥, \forall u \in Λ, ∥ u ∥ \leq R^{'} .

Using this property we can define an increasing sequence $(j_{k})_{k \geq 1}$ such that $x_{0} \notin V (Λ_{j_{k}})$ , for any $k \geq 1$ ; hence there exist the points $u_{j_{k}} \in Λ_{j_{k}}$ , such that $∥ x_{0} - u_{j_{k}} ∥ < ∥ x_{0} ∥$ , $\forall k \geq 1$ . Since $(u_{j_{k}})_{k \geq 1} \subseteq B (0; R^{'})$ , we can extract a convergent subsequence from $(u_{j_{k}})_{k \geq 1}$ , which, for the sake of simplicity, will be named in the same way: $lim k \to \infty u_{j_{k}} = u^{'}$ . Obviously $∥ x_{0} - u^{'} ∥ \leq ∥ x_{0} ∥$ , $∥ u^{'} ∥ \leq R^{'}$ , and, by (ii) from Theorem 1, $u^{'} \in Λ$ - 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 $V (Λ)$ are contained in $liminf k \to \infty V (Λ_{k})$ (when this happens, $V (Λ) \subseteq V (Λ_{k})$ , starting from a certain $k_{0}$ ), then $lim k \to \infty Λ_{k} = Λ$ . It remains to see what happens when some of the extremal points of $V (Λ)$ belong on the boundary of $liminf k \to \infty V (Λ_{k})$ .

References

[1] A. Barvinok. Combinatorics, geometry and complexity of integer points. electronic lecture notes. http://www.math.lsa.umich.edu/~barvinok/latticenotes669.pdf.
[2] J. W. S. Cassels An introduction to the geometry of numbers. Springer Verlag, New York, 1997.
[3] M. D. Sikiric, A. Tarlecki. Complexity and algorithms for computing Voronoi cells of lattices, Mathematics of Computation 78(267), 1713–1731, 2009. doi:10.1090/S0025-5718-09-02224-8.

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1 Introduction

2 Notations and definitions

Definition 1.

Definition 2.

Definition 3.

3 Main result

Lemma 1.

Lemma 2.

Theorem 1.

Theorem 2.

4 Conclusions

References

A Note on Sequences of Lattices

Authors

Related Research

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This week in AI

1 Introduction

2 Notations and definitions

Definition 1.

Definition 2.

Definition 3.

3 Main result

Lemma 1.

Lemma 2.

Theorem 1.

Theorem 2.

4 Conclusions

References