{"paper":{"title":"Nearly Optimal Embeddings of Flat Tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Ishan Agarwal, Oded Regev, Yi Tang","submitted_at":"2020-04-30T20:50:27Z","abstract_excerpt":"We show that for any $n$-dimensional lattice $\\mathcal{L} \\subseteq \\mathbb{R}^n$, the torus $\\mathbb{R}^n/\\mathcal{L}$ can be embedded into Hilbert space with $O(\\sqrt{n\\log n})$ distortion. This improves the previously best known upper bound of $O(n\\sqrt{\\log n})$ shown by Haviv and Regev (APPROX 2010) and approaches the lower bound of $\\Omega(\\sqrt{n})$ due to Khot and Naor (FOCS 2005, Math. Annal. 2006)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.00098","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2005.00098/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}