The Euclidean Distortion of Flat Tori

We show that for every n-dimensional lattice L the torus Rn/L can be embedded with distortion O(n ·√log n) into a Hilbert space. This improves the exponential upper bound of O(n) due to Khot and Naor (FOCS 2005, Math. Annal. 2006) and gets close to their lower bound of Ω( √ n). We also obtain tight bounds for certain families of lattices. Our main new ingredient is an embedding that maps any point u ∈ Rn/L to a Gaussian function centered at u in the Hilbert space L2(R/L). The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine-Zolotarev bases.