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).
@InProceedings{agarwal_et_al:LIPIcs.APPROX/RANDOM.2020.43, author = {Agarwal, Ishan and Regev, Oded and Tang, Yi}, title = {{Nearly Optimal Embeddings of Flat Tori}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {43:1--43:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.43}, URN = {urn:nbn:de:0030-drops-126464}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.43}, annote = {Keywords: Lattices, metric embeddings, flat torus} }
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