Abstract
We show that for every n-dimensional lattice [Formula: see text] the torus [Formula: see text] can be embedded with distortion [Formula: see text] into a Hilbert space. This improves the exponential upper bound of O(n3n/2) due to Khot and Naor (FOCS 2005, Math. Ann. 2006) and gets close to their lower bound of [Formula: see text]. We also obtain tight bounds for certain families of lattices. Our main new ingredient is an embedding that maps any point [Formula: see text] to a Gaussian function centered at u in the Hilbert space [Formula: see text]. The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine–Zolotarev bases.
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