{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:ZOENGKYTMH3MEJVFNQK4OTJYFA","short_pith_number":"pith:ZOENGKYT","schema_version":"1.0","canonical_sha256":"cb88d32b1361f6c226a56c15c74d382814cb2765a078d74b22e003f5f7d0acc4","source":{"kind":"arxiv","id":"1103.3163","version":1},"attestation_state":"computed","paper":{"title":"Translational tilings by a polytope, with multiplicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dmitry Shiryaev, Nick Gravin, Sinai Robins","submitted_at":"2011-03-16T13:15:12Z","abstract_excerpt":"We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov and Minkowski. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski's conditions for 1-t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1103.3163","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-16T13:15:12Z","cross_cats_sorted":[],"title_canon_sha256":"71762429ff652a902ca491d3ef157df525a379c9a6f6ee1b846daa58057d1941","abstract_canon_sha256":"d1739cce96fbfdc1d1bc8b6aa742da3c9212d29c3accd053db86d0191441c2fb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:44.682278Z","signature_b64":"UdDpW8BdBIhA8PKa6UDhswtMo0igN8DVLAsfI4qys20Twx2xdUx9MCfLGB9VF6VfPRLJOF6PSkowtdNkiIzcBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb88d32b1361f6c226a56c15c74d382814cb2765a078d74b22e003f5f7d0acc4","last_reissued_at":"2026-05-18T04:26:44.681627Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:44.681627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Translational tilings by a polytope, with multiplicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dmitry Shiryaev, Nick Gravin, Sinai Robins","submitted_at":"2011-03-16T13:15:12Z","abstract_excerpt":"We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov and Minkowski. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski's conditions for 1-t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3163","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1103.3163","created_at":"2026-05-18T04:26:44.681726+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.3163v1","created_at":"2026-05-18T04:26:44.681726+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.3163","created_at":"2026-05-18T04:26:44.681726+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZOENGKYTMH3M","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZOENGKYTMH3MEJVF","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZOENGKYT","created_at":"2026-05-18T12:26:50.516681+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA","json":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA.json","graph_json":"https://pith.science/api/pith-number/ZOENGKYTMH3MEJVFNQK4OTJYFA/graph.json","events_json":"https://pith.science/api/pith-number/ZOENGKYTMH3MEJVFNQK4OTJYFA/events.json","paper":"https://pith.science/paper/ZOENGKYT"},"agent_actions":{"view_html":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA","download_json":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA.json","view_paper":"https://pith.science/paper/ZOENGKYT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.3163&json=true","fetch_graph":"https://pith.science/api/pith-number/ZOENGKYTMH3MEJVFNQK4OTJYFA/graph.json","fetch_events":"https://pith.science/api/pith-number/ZOENGKYTMH3MEJVFNQK4OTJYFA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA/action/storage_attestation","attest_author":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA/action/author_attestation","sign_citation":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA/action/citation_signature","submit_replication":"https://pith.science/pith/ZOENGKYTMH3MEJVFNQK4OTJYFA/action/replication_record"}},"created_at":"2026-05-18T04:26:44.681726+00:00","updated_at":"2026-05-18T04:26:44.681726+00:00"}