Document Open Access Logo

Nearly Optimal Embeddings of Flat Tori

Authors Ishan Agarwal, Oded Regev, Yi Tang

Thumbnail PDF


  • Filesize: 493 kB
  • 14 pages

Document Identifiers

Author Details

Ishan Agarwal
  • Courant Institute of Mathematical Sciences, New York University, NY, USA
Oded Regev
  • Courant Institute of Mathematical Sciences, New York University, NY, USA
Yi Tang
  • Courant Institute of Mathematical Sciences, New York University, NY, USA

Cite AsGet BibTex

Ishan Agarwal, Oded Regev, and Yi Tang. Nearly Optimal Embeddings of Flat Tori. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 43:1-43:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


We show that for any n-dimensional lattice ℒ ⊆ ℝⁿ, the torus ℝⁿ/ℒ can be embedded into Hilbert space with O(√{nlog n}) distortion. This improves the previously best known upper bound of O(n√{log n}) shown by Haviv and Regev (APPROX 2010, J. Topol. Anal. 2013) and approaches the lower bound of Ω(√n) due to Khot and Naor (FOCS 2005, Math. Ann. 2006).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Lattices
  • metric embeddings
  • flat torus


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Ishay Haviv and Oded Regev. The Euclidean distortion of flat tori. J. Topol. Anal., 5(2):205-223, 2013. Preliminary version in APPROX 2010. URL:
  2. Piotr Indyk. Algorithmic applications of low-distortion geometric embeddings. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), pages 10-33. IEEE Computer Soc., Los Alamitos, CA, 2001. Google Scholar
  3. Subhash Khot and Assaf Naor. Nonembeddability theorems via Fourier analysis. Math. Ann., 334(4):821-852, 2006. Preliminary version in FOCS 2005. URL:
  4. John Milnor and Dale Husemoller. Symmetric bilinear forms. Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail