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.
Autissier, Un lemme matriciel effectif,Math
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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.