{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:YSIWLHNJ2FVDNGB5ON36TRPPX4","short_pith_number":"pith:YSIWLHNJ","canonical_record":{"source":{"id":"1702.02383","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-02-08T11:44:41Z","cross_cats_sorted":[],"title_canon_sha256":"c238aaa876aeb4198234f1485fbf7eb8140b466a1c0107d56fd7774a20cf3225","abstract_canon_sha256":"e138431ba7c9162efd4f9f03c9b44415dcd1603265c68c9b2fafb4dbd21e1f7f"},"schema_version":"1.0"},"canonical_sha256":"c491659da9d16a36983d7377e9c5efbf29da73325801cd5d86f955217169246b","source":{"kind":"arxiv","id":"1702.02383","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02383","created_at":"2026-05-17T23:46:11Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02383v2","created_at":"2026-05-17T23:46:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02383","created_at":"2026-05-17T23:46:11Z"},{"alias_kind":"pith_short_12","alias_value":"YSIWLHNJ2FVD","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YSIWLHNJ2FVDNGB5","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YSIWLHNJ","created_at":"2026-05-18T12:31:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:YSIWLHNJ2FVDNGB5ON36TRPPX4","target":"record","payload":{"canonical_record":{"source":{"id":"1702.02383","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-02-08T11:44:41Z","cross_cats_sorted":[],"title_canon_sha256":"c238aaa876aeb4198234f1485fbf7eb8140b466a1c0107d56fd7774a20cf3225","abstract_canon_sha256":"e138431ba7c9162efd4f9f03c9b44415dcd1603265c68c9b2fafb4dbd21e1f7f"},"schema_version":"1.0"},"canonical_sha256":"c491659da9d16a36983d7377e9c5efbf29da73325801cd5d86f955217169246b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:11.370136Z","signature_b64":"cqQlfQBlYkELtT49K4otzasceW+2q7kDd/iH1/hHYn0agdJCwWoUnKfKWngdXHuLBri1lfNXhLSwBqoZ/77gAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c491659da9d16a36983d7377e9c5efbf29da73325801cd5d86f955217169246b","last_reissued_at":"2026-05-17T23:46:11.369668Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:11.369668Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1702.02383","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:46:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RI70/Nn4HPWAlEqtTxrOYqP8POnn3IECTpvjyAttsOkGvYqcQGjL6aXy31rOU9FsMA3ioRnr9R6GRSuxyCxCDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:00:31.717287Z"},"content_sha256":"d510402c4a2ff9dff7778def4d1e4191074a23fc7fb0095164f4e261aebddc58","schema_version":"1.0","event_id":"sha256:d510402c4a2ff9dff7778def4d1e4191074a23fc7fb0095164f4e261aebddc58"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:YSIWLHNJ2FVDNGB5ON36TRPPX4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Periods and factors of weak model sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Christoph Richard, Gerhard Keller","submitted_at":"2017-02-08T11:44:41Z","abstract_excerpt":"There is a renewed interest in weak model sets due to their connection to $\\mathcal B$-free systems, which emerged from Sarnak's program on the M\\\"obius disjointness conjecture. Here we continue our recent investigation [arXiv:1511.06137] of the extended hull ${\\mathcal M}^{\\scriptscriptstyle G}_{\\scriptscriptstyle W}$, a dynamical system naturally associated to a weak model set in an abelian group $G$ with relatively compact window $W$. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of ${\\mathcal M}^{\\scriptscriptstyl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02383","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":""},"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":"2026-05-17T23:46:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9NCfgap99KXYpHGmsHhQsr0HhxhapkAhX3/VQGHFbcgCLs7WtRo2uBjUoMi4thadLuIew1UIVGwWN0K/42uhBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:00:31.717634Z"},"content_sha256":"77532ed477c81f0787775a434b1deb92057fb5661b26f612085d8b1e01bffdd3","schema_version":"1.0","event_id":"sha256:77532ed477c81f0787775a434b1deb92057fb5661b26f612085d8b1e01bffdd3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YSIWLHNJ2FVDNGB5ON36TRPPX4/bundle.json","state_url":"https://pith.science/pith/YSIWLHNJ2FVDNGB5ON36TRPPX4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YSIWLHNJ2FVDNGB5ON36TRPPX4/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":"2026-06-03T03:00:31Z","links":{"resolver":"https://pith.science/pith/YSIWLHNJ2FVDNGB5ON36TRPPX4","bundle":"https://pith.science/pith/YSIWLHNJ2FVDNGB5ON36TRPPX4/bundle.json","state":"https://pith.science/pith/YSIWLHNJ2FVDNGB5ON36TRPPX4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YSIWLHNJ2FVDNGB5ON36TRPPX4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:YSIWLHNJ2FVDNGB5ON36TRPPX4","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":"e138431ba7c9162efd4f9f03c9b44415dcd1603265c68c9b2fafb4dbd21e1f7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-02-08T11:44:41Z","title_canon_sha256":"c238aaa876aeb4198234f1485fbf7eb8140b466a1c0107d56fd7774a20cf3225"},"schema_version":"1.0","source":{"id":"1702.02383","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02383","created_at":"2026-05-17T23:46:11Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02383v2","created_at":"2026-05-17T23:46:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02383","created_at":"2026-05-17T23:46:11Z"},{"alias_kind":"pith_short_12","alias_value":"YSIWLHNJ2FVD","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YSIWLHNJ2FVDNGB5","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YSIWLHNJ","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:77532ed477c81f0787775a434b1deb92057fb5661b26f612085d8b1e01bffdd3","target":"graph","created_at":"2026-05-17T23:46:11Z","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":"There is a renewed interest in weak model sets due to their connection to $\\mathcal B$-free systems, which emerged from Sarnak's program on the M\\\"obius disjointness conjecture. Here we continue our recent investigation [arXiv:1511.06137] of the extended hull ${\\mathcal M}^{\\scriptscriptstyle G}_{\\scriptscriptstyle W}$, a dynamical system naturally associated to a weak model set in an abelian group $G$ with relatively compact window $W$. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of ${\\mathcal M}^{\\scriptscriptstyl","authors_text":"Christoph Richard, Gerhard Keller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-02-08T11:44:41Z","title":"Periods and factors of weak model sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02383","kind":"arxiv","version":2},"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:d510402c4a2ff9dff7778def4d1e4191074a23fc7fb0095164f4e261aebddc58","target":"record","created_at":"2026-05-17T23:46:11Z","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":"e138431ba7c9162efd4f9f03c9b44415dcd1603265c68c9b2fafb4dbd21e1f7f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-02-08T11:44:41Z","title_canon_sha256":"c238aaa876aeb4198234f1485fbf7eb8140b466a1c0107d56fd7774a20cf3225"},"schema_version":"1.0","source":{"id":"1702.02383","kind":"arxiv","version":2}},"canonical_sha256":"c491659da9d16a36983d7377e9c5efbf29da73325801cd5d86f955217169246b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c491659da9d16a36983d7377e9c5efbf29da73325801cd5d86f955217169246b","first_computed_at":"2026-05-17T23:46:11.369668Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:11.369668Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cqQlfQBlYkELtT49K4otzasceW+2q7kDd/iH1/hHYn0agdJCwWoUnKfKWngdXHuLBri1lfNXhLSwBqoZ/77gAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:11.370136Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.02383","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d510402c4a2ff9dff7778def4d1e4191074a23fc7fb0095164f4e261aebddc58","sha256:77532ed477c81f0787775a434b1deb92057fb5661b26f612085d8b1e01bffdd3"],"state_sha256":"9267e95192c8ed94913d756c448d70b6339dd14295991a59d3fd2bba6be8ab54"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P+FDOx9WmSnWeZ4N4zVQEgLiRHE6oiD6nohfccgt2b7XFvPLmsfFW3HJv5AZIq54P8SROSBC90YYnqn4CZlSDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T03:00:31.719482Z","bundle_sha256":"9297c733e1222d2a1350c03b5eecfbbcf042b65dc2627ebcfbe024e6abd5c594"}}