{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:5LA7NTHFDT5NEIECMB3SID3BRO","short_pith_number":"pith:5LA7NTHF","schema_version":"1.0","canonical_sha256":"eac1f6cce51cfad220826077240f618b9e8e4583037cac547bea5327c4d80008","source":{"kind":"arxiv","id":"1902.00876","version":2},"attestation_state":"computed","paper":{"title":"Spectrality of polytopes and equidecomposability by translations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.FA"],"primary_cat":"math.CA","authors_text":"Bochen Liu, Nir Lev","submitted_at":"2019-02-03T10:49:27Z","abstract_excerpt":"Let $A$ be a polytope in $\\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis (2002) asserts that if $A$ is a spectral polytope, then the total area of the $(d-1)$-dimensional faces of $A$ on which the outward normal is pointing at a given direction, must coincide with the total area of those $(d-1)$-dimensional faces on which the outward normal is pointing at the opposite direction. In this paper, we prove an extension of this result 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":"1902.00876","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-02-03T10:49:27Z","cross_cats_sorted":["math.CO","math.FA"],"title_canon_sha256":"1d9743f5da6e9f544785db1927f3f5da722d8d6dd2491f9f70d5552e2593671b","abstract_canon_sha256":"384f7a9ba403ceebc58f5b12dc56576c33c27227184999be7d72ce5f7da2a384"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:16:32.598168Z","signature_b64":"o4dsAmF1HjziO339lVYncmxEm4OO0WtN6CtV+984xFXZHabLfnm1fjvJ8jZZITEipwUJZ/3g2s5moRd6qOAKCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eac1f6cce51cfad220826077240f618b9e8e4583037cac547bea5327c4d80008","last_reissued_at":"2026-07-05T00:16:32.597768Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:16:32.597768Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectrality of polytopes and equidecomposability by translations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.FA"],"primary_cat":"math.CA","authors_text":"Bochen Liu, Nir Lev","submitted_at":"2019-02-03T10:49:27Z","abstract_excerpt":"Let $A$ be a polytope in $\\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis (2002) asserts that if $A$ is a spectral polytope, then the total area of the $(d-1)$-dimensional faces of $A$ on which the outward normal is pointing at a given direction, must coincide with the total area of those $(d-1)$-dimensional faces on which the outward normal is pointing at the opposite direction. In this paper, we prove an extension of this result t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.00876","kind":"arxiv","version":2},"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/1902.00876/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1902.00876","created_at":"2026-07-05T00:16:32.597826+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.00876v2","created_at":"2026-07-05T00:16:32.597826+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.00876","created_at":"2026-07-05T00:16:32.597826+00:00"},{"alias_kind":"pith_short_12","alias_value":"5LA7NTHFDT5N","created_at":"2026-07-05T00:16:32.597826+00:00"},{"alias_kind":"pith_short_16","alias_value":"5LA7NTHFDT5NEIEC","created_at":"2026-07-05T00:16:32.597826+00:00"},{"alias_kind":"pith_short_8","alias_value":"5LA7NTHF","created_at":"2026-07-05T00:16:32.597826+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/5LA7NTHFDT5NEIECMB3SID3BRO","json":"https://pith.science/pith/5LA7NTHFDT5NEIECMB3SID3BRO.json","graph_json":"https://pith.science/api/pith-number/5LA7NTHFDT5NEIECMB3SID3BRO/graph.json","events_json":"https://pith.science/api/pith-number/5LA7NTHFDT5NEIECMB3SID3BRO/events.json","paper":"https://pith.science/paper/5LA7NTHF"},"agent_actions":{"view_html":"https://pith.science/pith/5LA7NTHFDT5NEIECMB3SID3BRO","download_json":"https://pith.science/pith/5LA7NTHFDT5NEIECMB3SID3BRO.json","view_paper":"https://pith.science/paper/5LA7NTHF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.00876&json=true","fetch_graph":"https://pith.science/api/pith-number/5LA7NTHFDT5NEIECMB3SID3BRO/graph.json","fetch_events":"https://pith.science/api/pith-number/5LA7NTHFDT5NEIECMB3SID3BRO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5LA7NTHFDT5NEIECMB3SID3BRO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5LA7NTHFDT5NEIECMB3SID3BRO/action/storage_attestation","attest_author":"https://pith.science/pith/5LA7NTHFDT5NEIECMB3SID3BRO/action/author_attestation","sign_citation":"https://pith.science/pith/5LA7NTHFDT5NEIECMB3SID3BRO/action/citation_signature","submit_replication":"https://pith.science/pith/5LA7NTHFDT5NEIECMB3SID3BRO/action/replication_record"}},"created_at":"2026-07-05T00:16:32.597826+00:00","updated_at":"2026-07-05T00:16:32.597826+00:00"}