Lattices without short characteristic vectors

Mark Gaulter
1998 Mathematical Research Letters  
All the lattices here under discussion here are understood to be integral unimodular Z -lattices in R n . A characteristic vector of a lattice L is a vector w ∈ L such that v · w ≡ |v| 2 (mod 2) for every v ∈ L. Elkies has considered the minimal (squared) norm of the characteristic vectors in a unimodular lattice. He showed that any unimodular Z -lattice in R n has characteristic vectors of norm ≤ n; he also proved that of all such lattices, only the standard lattice Z n has no characteristic
more » ... no characteristic vectors of norm < n (Math Research Letters 2, 321-326). He then asked "For any k > 0, is there N k such that every integral unimodular lattice all of whose characteristic vectors have norm ≥ n − 8k is of the form L 0 ⊥ Z r for some lattice L 0 of rank at most N k ?" (Math Research Letters 2, 643-651). He solved this question in the case k = 1, showing that N 1 = 23 suffices; here I determine values for N 2 and N 3 .
doi:10.4310/mrl.1998.v5.n3.a8 fatcat:d3mximco2feahelgf5gkfhyfx4