{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:Z2DZU62M3UZL5WEZVXGWELPPIA","short_pith_number":"pith:Z2DZU62M","canonical_record":{"source":{"id":"1507.02581","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-09T16:26:51Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3a1c9be19adde7336a6afc7b8cbfb98260fa44ad18b068b444f42ed1635fd04d","abstract_canon_sha256":"12bce3280996fcb0690590552410580f7cb2c796dfdfe068f1cfc9bdb8ecb447"},"schema_version":"1.0"},"canonical_sha256":"ce879a7b4cdd32bed899adcd622def4007aca379926d7b5b47270ec89765f910","source":{"kind":"arxiv","id":"1507.02581","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.02581","created_at":"2026-05-18T01:37:06Z"},{"alias_kind":"arxiv_version","alias_value":"1507.02581v1","created_at":"2026-05-18T01:37:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.02581","created_at":"2026-05-18T01:37:06Z"},{"alias_kind":"pith_short_12","alias_value":"Z2DZU62M3UZL","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"Z2DZU62M3UZL5WEZ","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"Z2DZU62M","created_at":"2026-05-18T12:29:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:Z2DZU62M3UZL5WEZVXGWELPPIA","target":"record","payload":{"canonical_record":{"source":{"id":"1507.02581","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-09T16:26:51Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3a1c9be19adde7336a6afc7b8cbfb98260fa44ad18b068b444f42ed1635fd04d","abstract_canon_sha256":"12bce3280996fcb0690590552410580f7cb2c796dfdfe068f1cfc9bdb8ecb447"},"schema_version":"1.0"},"canonical_sha256":"ce879a7b4cdd32bed899adcd622def4007aca379926d7b5b47270ec89765f910","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:06.310510Z","signature_b64":"/Uni6BuwjnKgLNCOjNPkqoa+LTqOwmbffWFxro99jCECX6xzmUoaeRL53+ZUe59yu9Os+xnFO0td1y6xcjUjAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce879a7b4cdd32bed899adcd622def4007aca379926d7b5b47270ec89765f910","last_reissued_at":"2026-05-18T01:37:06.309892Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:06.309892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.02581","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":"2026-05-18T01:37:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L1M6/j999YowRW1+0lJgjKpsbbOyr8ORSp0gT9hbqHcO9q+E4WKV+H+A52MrmKXFqgI3mcQt05kke0BR32uhBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:03:30.144976Z"},"content_sha256":"73140aec6679c1552631d02ee703dade96a5a4988219c5b0884cff1a9e2a7d89","schema_version":"1.0","event_id":"sha256:73140aec6679c1552631d02ee703dade96a5a4988219c5b0884cff1a9e2a7d89"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:Z2DZU62M3UZL5WEZVXGWELPPIA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Avoidability of long $k$-abelian repetitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Matthieu Rosenfeld, Micha\\\"el Rao","submitted_at":"2015-07-09T16:26:51Z","abstract_excerpt":"We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are M\\\"akel\\\"a's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite binary words, and therefore answering negatively one question from M\\\"akel\\\"a. Then we show that one can avoid $3$-abelian-squares of period at least $3$ in infinite binary words and $2$-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct $k$-abelian-squares that must appear in an infinite binary w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02581","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":"2026-05-18T01:37:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wZGFDI2EBLBnNCYTep0/w3YGt+J80xSuOjJ+S+zNtPiSAT22Nq8ENfHbg4LTAqMwr9DydCgMHqcB06sh5E6hAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:03:30.145716Z"},"content_sha256":"e7879b47769aeb544d5dbe43cb2d80dae487a3f807568d06b33380c5505357fd","schema_version":"1.0","event_id":"sha256:e7879b47769aeb544d5dbe43cb2d80dae487a3f807568d06b33380c5505357fd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z2DZU62M3UZL5WEZVXGWELPPIA/bundle.json","state_url":"https://pith.science/pith/Z2DZU62M3UZL5WEZVXGWELPPIA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z2DZU62M3UZL5WEZVXGWELPPIA/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":"2026-05-28T17:03:30Z","links":{"resolver":"https://pith.science/pith/Z2DZU62M3UZL5WEZVXGWELPPIA","bundle":"https://pith.science/pith/Z2DZU62M3UZL5WEZVXGWELPPIA/bundle.json","state":"https://pith.science/pith/Z2DZU62M3UZL5WEZVXGWELPPIA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z2DZU62M3UZL5WEZVXGWELPPIA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Z2DZU62M3UZL5WEZVXGWELPPIA","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":"12bce3280996fcb0690590552410580f7cb2c796dfdfe068f1cfc9bdb8ecb447","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-09T16:26:51Z","title_canon_sha256":"3a1c9be19adde7336a6afc7b8cbfb98260fa44ad18b068b444f42ed1635fd04d"},"schema_version":"1.0","source":{"id":"1507.02581","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.02581","created_at":"2026-05-18T01:37:06Z"},{"alias_kind":"arxiv_version","alias_value":"1507.02581v1","created_at":"2026-05-18T01:37:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.02581","created_at":"2026-05-18T01:37:06Z"},{"alias_kind":"pith_short_12","alias_value":"Z2DZU62M3UZL","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"Z2DZU62M3UZL5WEZ","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"Z2DZU62M","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:e7879b47769aeb544d5dbe43cb2d80dae487a3f807568d06b33380c5505357fd","target":"graph","created_at":"2026-05-18T01:37:06Z","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":"We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are M\\\"akel\\\"a's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite binary words, and therefore answering negatively one question from M\\\"akel\\\"a. Then we show that one can avoid $3$-abelian-squares of period at least $3$ in infinite binary words and $2$-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct $k$-abelian-squares that must appear in an infinite binary w","authors_text":"Matthieu Rosenfeld, Micha\\\"el Rao","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-09T16:26:51Z","title":"Avoidability of long $k$-abelian repetitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.02581","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":"sha256:73140aec6679c1552631d02ee703dade96a5a4988219c5b0884cff1a9e2a7d89","target":"record","created_at":"2026-05-18T01:37:06Z","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":"12bce3280996fcb0690590552410580f7cb2c796dfdfe068f1cfc9bdb8ecb447","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-09T16:26:51Z","title_canon_sha256":"3a1c9be19adde7336a6afc7b8cbfb98260fa44ad18b068b444f42ed1635fd04d"},"schema_version":"1.0","source":{"id":"1507.02581","kind":"arxiv","version":1}},"canonical_sha256":"ce879a7b4cdd32bed899adcd622def4007aca379926d7b5b47270ec89765f910","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce879a7b4cdd32bed899adcd622def4007aca379926d7b5b47270ec89765f910","first_computed_at":"2026-05-18T01:37:06.309892Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:06.309892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/Uni6BuwjnKgLNCOjNPkqoa+LTqOwmbffWFxro99jCECX6xzmUoaeRL53+ZUe59yu9Os+xnFO0td1y6xcjUjAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:06.310510Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.02581","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73140aec6679c1552631d02ee703dade96a5a4988219c5b0884cff1a9e2a7d89","sha256:e7879b47769aeb544d5dbe43cb2d80dae487a3f807568d06b33380c5505357fd"],"state_sha256":"dca6ad8283202fb628d437d295f07104e01abd6c3b8caa15326d115548bd703e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F8YV7AgBT5dFwIcf0ZDWeYS+RnGMZe8nfoT4G3H9vOL6mRbFMI8hbPjsJFlP/iiSocJt/mX6OyTgCyYEOuo9Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T17:03:30.149509Z","bundle_sha256":"554484d78f89ab548abeb2b8d65f7e7f40b4720f3069784bbd835603f17ca330"}}