The Euclidean distortion of flat tori

Authors:
Ishay Haviv;Oded Regev
Affiliations:
The Blavatnik School of Computer Science, Tel Aviv University, Israel;The Blavatnik School of Computer Science, Tel Aviv University, Israel
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

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Abstract

We show that for every n-dimensional lattice L the torus Rn/L can be embedded with distortion O(nċ√logn) into a Hilbert space. This improves the exponential upper bound of O(n3n/2) 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(Rn/L). The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine-Zolotarev bases.