The Euclidean Distortion of Flat Tori

Abstract

We show that for every n-dimensional lattice ${\mathcal L}$ the torus ${\mathbb{R}}^n / {\mathcal L}$ can be embedded with distortion $O(n \cdot \sqrt{\log{n}})$ into a Hilbert space. This improves the exponential upper bound of O(n ^3n/2) due to Khot and Naor (FOCS 2005, Math. Annal. 2006) and gets close to their lower bound of $\Omega(\sqrt{n})$. We also obtain tight bounds for certain families of lattices.

Our main new ingredient is an embedding that maps any point $u \in {\mathbb{R}}^n/{\mathcal L}$ to a Gaussian function centered at u in the Hilbert space $L_2({\mathbb{R}}^n/{\mathcal L})$. The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine-Zolotarev bases.