A Measure for the Non-Orthogonality of a Lattice Basis

Abstract

Let B = [b₁, …, b_n] (with column vectors b_i) be a basis of ℝⁿ. Then L = [sum ]b_iℤ is a lattice in ℝⁿ and A = B^{[top ]}B is the Gram matrix of B. The reciprocal lattice L* of L has basis B* = (B⁻¹)^{[top ]} with Gram matrix A⁻¹. For any nonsingular matrix A = (a_i,j) with inverse A⁻¹ = (a*_i,j), let τ(A) = max_{1[les ]i[les ]n} {[sum ]ⁿ_{j =1}[mid ]a_i,j ·a*_j,j[mid ]}. Then τ(A), τ(A⁻¹)[ges ]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^{[top ]}AT, T = (t_i,j)∈SL_n(ℤ), we will show [mid ]t_i,j[mid ][les ]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) [ratio ]= {x∈ℝⁿ[mid ]∀λ∈L [ratio ][mid ]x[mid ][les ][mid ]x−λ[mid ]} be the Voronoi cell (also called the Dirichlet region) of the lattice L. For a basis B of L, we call C(B) = {[sum ]x_ib_i, [mid ]x_i[mid ][les ]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)[les ]τ(A⁻¹)·C(B) and C(B)[les ]n·τ(A)·V(L).