{"paper":{"title":"Structure results for multiple tilings in 3D","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Dmitry Shiryaev, Mihail Kolountzakis, Nick Gravin, Sinai Robins","submitted_at":"2012-08-07T15:01:03Z","abstract_excerpt":"We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $P$ is translated with a discrete multiset $\\Lambda$ in such a way that each point of the space gets covered exactly $k$ times, except perhaps the translated copies of the boundary of $P$. It is known that all possible multiple tilers in 3D are zonotopes. In 2D it was known by the work of M. Kolountzakis that, unless $P$ is a parallelogram, the multiset of translation vectors $\\Lambda$ must be a finite union of translated lattices (also known as quasi periodic sets). In that work [K"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1439","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"}