{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:WBPCPXTVQJVITPXQGZJTAWQ7MA","short_pith_number":"pith:WBPCPXTV","canonical_record":{"source":{"id":"1309.7947","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-30T18:11:35Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"6b6e21737d60fd79d67a263c91caa15382691b13a5d898260db501e02aa7672e","abstract_canon_sha256":"e11f4aafd92e0025f071f3159b607242e6cdbaf6e706a72cfb8e48a4881f535b"},"schema_version":"1.0"},"canonical_sha256":"b05e27de75826a89bef03653305a1f6019cbfa1fe66caf9afbba885befc42835","source":{"kind":"arxiv","id":"1309.7947","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7947","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7947v1","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7947","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"pith_short_12","alias_value":"WBPCPXTVQJVI","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"WBPCPXTVQJVITPXQ","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"WBPCPXTV","created_at":"2026-05-18T12:28:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:WBPCPXTVQJVITPXQGZJTAWQ7MA","target":"record","payload":{"canonical_record":{"source":{"id":"1309.7947","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-30T18:11:35Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"6b6e21737d60fd79d67a263c91caa15382691b13a5d898260db501e02aa7672e","abstract_canon_sha256":"e11f4aafd92e0025f071f3159b607242e6cdbaf6e706a72cfb8e48a4881f535b"},"schema_version":"1.0"},"canonical_sha256":"b05e27de75826a89bef03653305a1f6019cbfa1fe66caf9afbba885befc42835","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:39.211566Z","signature_b64":"jzMHZ9vQl7rCRf2M4J08/Lm2wjrlwfThvE2U2nUFgEd2Sk4yL9HZLD1jifDUq7C/1gXSq+DRPWBYqNtyneQ9Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b05e27de75826a89bef03653305a1f6019cbfa1fe66caf9afbba885befc42835","last_reissued_at":"2026-05-18T01:15:39.210805Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:39.210805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.7947","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:15:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DghwhNHaunb5YVG34zPKE0RoVzs1KzGAyiPO6isgrrzjOGV75AlMEXYIU81o8zOi0pUXp83+gdnwXTtn3HrMCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T14:24:55.072728Z"},"content_sha256":"79b50fb17830d075c39abab6c596a81f38158926b0b2e0da273b96ff4d24b482","schema_version":"1.0","event_id":"sha256:79b50fb17830d075c39abab6c596a81f38158926b0b2e0da273b96ff4d24b482"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:WBPCPXTVQJVITPXQGZJTAWQ7MA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Weighted Dirac Combs Supported Inside Model Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Nicolae Strungaru","submitted_at":"2013-09-30T18:11:35Z","abstract_excerpt":"In this paper we prove that given a weakly almost periodic measure $\\mu$ supported inside some model set $\\Lambda(W)$ with closed window $W$, then the strongly almost periodic component $\\mu_S$ and the null weakly almost periodic component $\\mu_0$ are both supported inside $\\Lambda(W)$. As a consequence we prove that given any translation bounded measure $\\omega$, supported inside some model set, then each of the pure point diffraction spectrum $\\hat{\\gamma}_{pp}$ and the continuous diffraction spectrum $\\hat{\\gamma}_c$ is either trivial or have a relatively dense support."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7947","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:15:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d7H2g4/qm9it42xv4Sm1JQals6KM+fzFZHkbeTZoT3oYbBbYQ4JpuZzSNlSDw1Io2BvKo/RoHU/0xcjNsmGXBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T14:24:55.073550Z"},"content_sha256":"3138950201a8ad6822aaff296264793a15ec0a74be76b9cc70afad2cd38041e9","schema_version":"1.0","event_id":"sha256:3138950201a8ad6822aaff296264793a15ec0a74be76b9cc70afad2cd38041e9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WBPCPXTVQJVITPXQGZJTAWQ7MA/bundle.json","state_url":"https://pith.science/pith/WBPCPXTVQJVITPXQGZJTAWQ7MA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WBPCPXTVQJVITPXQGZJTAWQ7MA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-16T14:24:55Z","links":{"resolver":"https://pith.science/pith/WBPCPXTVQJVITPXQGZJTAWQ7MA","bundle":"https://pith.science/pith/WBPCPXTVQJVITPXQGZJTAWQ7MA/bundle.json","state":"https://pith.science/pith/WBPCPXTVQJVITPXQGZJTAWQ7MA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WBPCPXTVQJVITPXQGZJTAWQ7MA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:WBPCPXTVQJVITPXQGZJTAWQ7MA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e11f4aafd92e0025f071f3159b607242e6cdbaf6e706a72cfb8e48a4881f535b","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-30T18:11:35Z","title_canon_sha256":"6b6e21737d60fd79d67a263c91caa15382691b13a5d898260db501e02aa7672e"},"schema_version":"1.0","source":{"id":"1309.7947","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.7947","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"arxiv_version","alias_value":"1309.7947v1","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.7947","created_at":"2026-05-18T01:15:39Z"},{"alias_kind":"pith_short_12","alias_value":"WBPCPXTVQJVI","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"WBPCPXTVQJVITPXQ","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"WBPCPXTV","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:3138950201a8ad6822aaff296264793a15ec0a74be76b9cc70afad2cd38041e9","target":"graph","created_at":"2026-05-18T01:15:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we prove that given a weakly almost periodic measure $\\mu$ supported inside some model set $\\Lambda(W)$ with closed window $W$, then the strongly almost periodic component $\\mu_S$ and the null weakly almost periodic component $\\mu_0$ are both supported inside $\\Lambda(W)$. As a consequence we prove that given any translation bounded measure $\\omega$, supported inside some model set, then each of the pure point diffraction spectrum $\\hat{\\gamma}_{pp}$ and the continuous diffraction spectrum $\\hat{\\gamma}_c$ is either trivial or have a relatively dense support.","authors_text":"Nicolae Strungaru","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-30T18:11:35Z","title":"On Weighted Dirac Combs Supported Inside Model Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7947","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:79b50fb17830d075c39abab6c596a81f38158926b0b2e0da273b96ff4d24b482","target":"record","created_at":"2026-05-18T01:15:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e11f4aafd92e0025f071f3159b607242e6cdbaf6e706a72cfb8e48a4881f535b","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-09-30T18:11:35Z","title_canon_sha256":"6b6e21737d60fd79d67a263c91caa15382691b13a5d898260db501e02aa7672e"},"schema_version":"1.0","source":{"id":"1309.7947","kind":"arxiv","version":1}},"canonical_sha256":"b05e27de75826a89bef03653305a1f6019cbfa1fe66caf9afbba885befc42835","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b05e27de75826a89bef03653305a1f6019cbfa1fe66caf9afbba885befc42835","first_computed_at":"2026-05-18T01:15:39.210805Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:39.210805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jzMHZ9vQl7rCRf2M4J08/Lm2wjrlwfThvE2U2nUFgEd2Sk4yL9HZLD1jifDUq7C/1gXSq+DRPWBYqNtyneQ9Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:39.211566Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.7947","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:79b50fb17830d075c39abab6c596a81f38158926b0b2e0da273b96ff4d24b482","sha256:3138950201a8ad6822aaff296264793a15ec0a74be76b9cc70afad2cd38041e9"],"state_sha256":"a54188a330a88958f83bd401537f63beb69a9131017f4d59aa8a04bb69a96bee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NJdED1krFV9fRG6HJR95B2TS0QITQR6c/5CDM0Ba+PX0ctPMq+BX1/iOmKvLPq1f13FHLOOlWa0DfuR+bj6QDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T14:24:55.080467Z","bundle_sha256":"60105527a67a97b1b3104c209eb750d8a6cc04c5360c3f9423c28addc9b54f3f"}}