{"paper":{"title":"Pairs of Full-Rank Lattices With Parallelepiped-Shaped Common Fundamental Domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"H. Burgiel, V. Oussa","submitted_at":"2015-07-15T03:16:34Z","abstract_excerpt":"We provide a complete characterization of pairs of full-rank lattices in $\\mathbb{R}^{d}$ which admit common connected fundamental domains of the type $N\\left[ 0,1\\right) ^{d}$ where $N$ is an invertible matrix of order $d.$ Using our characterization, we construct several pairs of lattices of the type $\\left( M\\mathbb{Z}^{d},\\mathbb{Z}^{d}\\right) $ which admit a common fundamental domain of the type $N\\left[ 0,1\\right) ^{d}.$ Moreover, we show that for $d=2,$ there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04671","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":""},"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"}