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In the pre-MO era, I once realized that on the integers, the function $$ d(m, n) := \sqrt{\log \frac{\sqrt{mn}} {\text{gcd}(m,n)}}\ , $$ is a metric (all properties are easily verified; in fact ...

Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...