A Measure for the Non-Orthogonality of a Lattice Basis

@inproceedings{Seysen1999AMF,
  title={A Measure for the Non-Orthogonality of a Lattice Basis},
  author={Martin Seysen},
  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… CONTINUE READING

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