A Measure for the Non-Orthogonality of a Lattice Basis

• Authors: M. Seysen
• Venue: Combinatorics, Probability and Computing
• Year: 1999

Let B = [b1, …, bn] (with column vectors bi) be a basis of ℝn. Then L = ∑biℤ is a lattice in ℝn and A = B⊤B is the Gram matrix of B. The reciprocal lattice L* of L has basis B* = (B−1)⊤ with Gram matrix A−1. For any nonsingular matrix A = (ai,j) with inverse A−1 = (a*i,j), let τ(A) = max1≤i≤n {∑nj =1∣ai,j ·a*j,j∣}. Then τ(A), τ(A−1)≥1 holds, with equality for an orthogonal basis. We will show that for any lattice L there is a basis with Gram matrix A such that τ(A), τ(A−1) = exp (O((ln n)2)). This generalizes a result in [8] and [20].For any basis transformation A→Ā with Ā = T⊤AT, T = (ti,j)∈SLn(ℤ), we will show ∣ti,j∣≤τ(A−1) ·τ(Ā). This implies that every integral matrix representation of a finite group is equivalent to a representation where the coefficients of the matrices representing group elements are bounded by exp (O((ln n)2)). This new bound is considerably smaller than the known (exponential) bounds for automorphisms of Minkowski-reduced lattice bases: see, for example, [6].The quantities τ(A), τ(A−1) have the following geometric interpretation. Let V(L) ∶= {x∈ℝn∣∀λ∈L ∶∣x∣≤∣x−λ∣} be the Voronoi cell (also called the Dirichlet region) of the lattice L. For a basis B of L, we call C(B) = {∑xibi, ∣xi∣≤1/2} the basis cell of B. Both cells define a lattice tiling of ℝn (see [6]); they coincide for an orthogonal basis. For a general basis B of L with Gram matrix A we will show V(L)≤τ(A−1)·C(B) and C(B)≤n·τ(A)·V(L).