The Euclidean distortion of flat tori

Ishay Haviv, Oded Regev

Research output: Contribution to journalArticle

Abstract

We show that for every n-dimensional lattice & scriptLsign the torus n/&scriptLsign can be embedded with distortion O(n√log n) 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 δ(√n). We also obtain tight bounds for certain families of lattices. Our main new ingredient is an embedding that maps any point u ε n/L to a Gaussian function centered at u in the Hilbert space L2(. n/L). The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine-Zolotarev bases.

Original languageEnglish (US)
Pages (from-to)205-223
Number of pages19
JournalJournal of Topology and Analysis
Volume5
Issue number2
DOIs
StatePublished - Jun 2013

Keywords

  • Lattices
  • embedding
  • flat tori

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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