For any measurable fundamental domain K of a lattice Lambda, the flat torus spectral gap obeys lambda_SG(T_Lambda) >= pi^2/(3||Cov_K||_op), with sharp constants and an equivalence under a sectional tiling condition.
The second minimum of Barnes-Wall lattices
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The paper gives a recursive construction of the Barnes-Wall lattices as subdirect products. This is used to show that the Barnes-Wall lattices of minimum $d$ do not contain any vectors of norm $a$ with $d<a<3d/2$.
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Spectral and Isoperimetric Bounds on Flat Tori
For any measurable fundamental domain K of a lattice Lambda, the flat torus spectral gap obeys lambda_SG(T_Lambda) >= pi^2/(3||Cov_K||_op), with sharp constants and an equivalence under a sectional tiling condition.