{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:MXDFDZDLQ6XMU4QQPN6PVBFAMK","short_pith_number":"pith:MXDFDZDL","canonical_record":{"source":{"id":"1606.02868","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2016-06-09T08:33:25Z","cross_cats_sorted":["cs.FL","math.CO"],"title_canon_sha256":"cd3c3c514474996b3334c6d80109802b54067b2fd6773edd2af9166827d70ca8","abstract_canon_sha256":"bd95148b191c60d2c87035308660374e8be4feb71f1bbf84a32046d50f8f659a"},"schema_version":"1.0"},"canonical_sha256":"65c651e46b87aeca72107b7cfa84a062a882ec1a425b08dcd9cb5be1db7bce6b","source":{"kind":"arxiv","id":"1606.02868","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02868","created_at":"2026-05-18T00:15:00Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02868v3","created_at":"2026-05-18T00:15:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02868","created_at":"2026-05-18T00:15:00Z"},{"alias_kind":"pith_short_12","alias_value":"MXDFDZDLQ6XM","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MXDFDZDLQ6XMU4QQ","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MXDFDZDL","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:MXDFDZDLQ6XMU4QQPN6PVBFAMK","target":"record","payload":{"canonical_record":{"source":{"id":"1606.02868","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2016-06-09T08:33:25Z","cross_cats_sorted":["cs.FL","math.CO"],"title_canon_sha256":"cd3c3c514474996b3334c6d80109802b54067b2fd6773edd2af9166827d70ca8","abstract_canon_sha256":"bd95148b191c60d2c87035308660374e8be4feb71f1bbf84a32046d50f8f659a"},"schema_version":"1.0"},"canonical_sha256":"65c651e46b87aeca72107b7cfa84a062a882ec1a425b08dcd9cb5be1db7bce6b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:00.275506Z","signature_b64":"n01EQTUQftd0flx0d5FNVG6CgqTgUwXzc9EJJ+UvQ/vLNSP5ZiRrym8Hcd7xwgvPfAjdwHFNbJXOMkL7JqOrAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65c651e46b87aeca72107b7cfa84a062a882ec1a425b08dcd9cb5be1db7bce6b","last_reissued_at":"2026-05-18T00:15:00.274696Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:00.274696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.02868","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:15:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8oyrNGRhzD+LYCvFZUgZFxsSRG78rrJMEqLGjgN9LVsjzvJXi2PZ9cnruffxrcXZX7vNlu3LWJanvNwJ34U9Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T15:42:10.916663Z"},"content_sha256":"2b48fefc81f0e11551d2357df9383cca82a1b7a28efd11b2d972b3b9d6bb56a3","schema_version":"1.0","event_id":"sha256:2b48fefc81f0e11551d2357df9383cca82a1b7a28efd11b2d972b3b9d6bb56a3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:MXDFDZDLQ6XMU4QQPN6PVBFAMK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Anti-Powers in Infinite Words","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.FL","math.CO"],"primary_cat":"cs.DM","authors_text":"Antonio Restivo, Gabriele Fici, Luca Q. Zamboni, Manuel Silva","submitted_at":"2016-06-09T08:33:25Z","abstract_excerpt":"In combinatorics of words, a concatenation of $k$ consecutive equal blocks is called a power of order $k$. In this paper we take a different point of view and define an anti-power of order $k$ as a concatenation of $k$ consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary expo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02868","kind":"arxiv","version":3},"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":"2026-05-18T00:15:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/hAeinMSOrkLveah654k82oJ2V77xFBHNuHtj1+6DMLQGqnLGgXodDpvyUyE0BqRKlGNKqOP2jOqqMOt/qnCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T15:42:10.917337Z"},"content_sha256":"e5f7564bdc6151304c1b6c56d2d2069e7899312ebf103f365370724dccc88bda","schema_version":"1.0","event_id":"sha256:e5f7564bdc6151304c1b6c56d2d2069e7899312ebf103f365370724dccc88bda"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MXDFDZDLQ6XMU4QQPN6PVBFAMK/bundle.json","state_url":"https://pith.science/pith/MXDFDZDLQ6XMU4QQPN6PVBFAMK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MXDFDZDLQ6XMU4QQPN6PVBFAMK/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":"2026-05-26T15:42:10Z","links":{"resolver":"https://pith.science/pith/MXDFDZDLQ6XMU4QQPN6PVBFAMK","bundle":"https://pith.science/pith/MXDFDZDLQ6XMU4QQPN6PVBFAMK/bundle.json","state":"https://pith.science/pith/MXDFDZDLQ6XMU4QQPN6PVBFAMK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MXDFDZDLQ6XMU4QQPN6PVBFAMK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MXDFDZDLQ6XMU4QQPN6PVBFAMK","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":"bd95148b191c60d2c87035308660374e8be4feb71f1bbf84a32046d50f8f659a","cross_cats_sorted":["cs.FL","math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2016-06-09T08:33:25Z","title_canon_sha256":"cd3c3c514474996b3334c6d80109802b54067b2fd6773edd2af9166827d70ca8"},"schema_version":"1.0","source":{"id":"1606.02868","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02868","created_at":"2026-05-18T00:15:00Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02868v3","created_at":"2026-05-18T00:15:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02868","created_at":"2026-05-18T00:15:00Z"},{"alias_kind":"pith_short_12","alias_value":"MXDFDZDLQ6XM","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MXDFDZDLQ6XMU4QQ","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MXDFDZDL","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:e5f7564bdc6151304c1b6c56d2d2069e7899312ebf103f365370724dccc88bda","target":"graph","created_at":"2026-05-18T00:15:00Z","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 combinatorics of words, a concatenation of $k$ consecutive equal blocks is called a power of order $k$. In this paper we take a different point of view and define an anti-power of order $k$ as a concatenation of $k$ consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary expo","authors_text":"Antonio Restivo, Gabriele Fici, Luca Q. Zamboni, Manuel Silva","cross_cats":["cs.FL","math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2016-06-09T08:33:25Z","title":"Anti-Powers in Infinite Words"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02868","kind":"arxiv","version":3},"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":"sha256:2b48fefc81f0e11551d2357df9383cca82a1b7a28efd11b2d972b3b9d6bb56a3","target":"record","created_at":"2026-05-18T00:15:00Z","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":"bd95148b191c60d2c87035308660374e8be4feb71f1bbf84a32046d50f8f659a","cross_cats_sorted":["cs.FL","math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DM","submitted_at":"2016-06-09T08:33:25Z","title_canon_sha256":"cd3c3c514474996b3334c6d80109802b54067b2fd6773edd2af9166827d70ca8"},"schema_version":"1.0","source":{"id":"1606.02868","kind":"arxiv","version":3}},"canonical_sha256":"65c651e46b87aeca72107b7cfa84a062a882ec1a425b08dcd9cb5be1db7bce6b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"65c651e46b87aeca72107b7cfa84a062a882ec1a425b08dcd9cb5be1db7bce6b","first_computed_at":"2026-05-18T00:15:00.274696Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:15:00.274696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n01EQTUQftd0flx0d5FNVG6CgqTgUwXzc9EJJ+UvQ/vLNSP5ZiRrym8Hcd7xwgvPfAjdwHFNbJXOMkL7JqOrAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:15:00.275506Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.02868","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b48fefc81f0e11551d2357df9383cca82a1b7a28efd11b2d972b3b9d6bb56a3","sha256:e5f7564bdc6151304c1b6c56d2d2069e7899312ebf103f365370724dccc88bda"],"state_sha256":"0d3e0f592d99a2cb0c62629d9751801d1bab9ed0e62ed3d75ab388d45bad4fd0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VuwQrmOmIPPpLf4/f3am8ERqRUeG0bj6jun1dA03TDhuK2W6fI4idRjvj56QbV2GDO8/l1KZomNdKSSyRCyuAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T15:42:10.920613Z","bundle_sha256":"1c688f7efc2e4348b06ec8f7b1bd06e38ccc7e1089076d76f805fb14f2a4c667"}}