A reverse Minkowski theorem

We prove a conjecture due to Dadush, showing that if ℒ⊂ ℝ n is a lattice such that det(ℒ′) 1 for all sublattices ℒ′ ⊆ ℒ, then $$\sum_{ y ∈ℒ}^e -t 2||y||2 ≤3/2,$$ where t := 10(logn + 2). From this we also derive bounds on the number of short lattice vectors and on the covering radius.

Authors

• Oded Regev 0001
• Noah Stephens-Davidowitz

Keywords

• Geometry of Numbers
• Lattices