{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:FUKMODIHKIORQDYUDYAI6N23DC","short_pith_number":"pith:FUKMODIH","canonical_record":{"source":{"id":"1008.4289","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-08-25T14:19:18Z","cross_cats_sorted":[],"title_canon_sha256":"138140a8a27e4e74fff6bb1eb7d921235bd8347b53b5710672e3ec4d54f9887e","abstract_canon_sha256":"3f42adc7fa69a28026f0fdf58ab98816d0075d18e2f09c868ab2051c63f891a2"},"schema_version":"1.0"},"canonical_sha256":"2d14c70d07521d180f141e008f375b189c7d4bea6eb52f2647e59ed575604ec1","source":{"kind":"arxiv","id":"1008.4289","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.4289","created_at":"2026-05-18T04:41:53Z"},{"alias_kind":"arxiv_version","alias_value":"1008.4289v1","created_at":"2026-05-18T04:41:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4289","created_at":"2026-05-18T04:41:53Z"},{"alias_kind":"pith_short_12","alias_value":"FUKMODIHKIOR","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FUKMODIHKIORQDYU","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FUKMODIH","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:FUKMODIHKIORQDYUDYAI6N23DC","target":"record","payload":{"canonical_record":{"source":{"id":"1008.4289","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-08-25T14:19:18Z","cross_cats_sorted":[],"title_canon_sha256":"138140a8a27e4e74fff6bb1eb7d921235bd8347b53b5710672e3ec4d54f9887e","abstract_canon_sha256":"3f42adc7fa69a28026f0fdf58ab98816d0075d18e2f09c868ab2051c63f891a2"},"schema_version":"1.0"},"canonical_sha256":"2d14c70d07521d180f141e008f375b189c7d4bea6eb52f2647e59ed575604ec1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:53.247513Z","signature_b64":"MhhcfREo6CqqNSFG6NtxdERt+qI70KuXjmiCIrAX0yG7ii0eYS/yz2rzVo3/3eFsA9Vb4NOe/PmwgKIvLSDyBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d14c70d07521d180f141e008f375b189c7d4bea6eb52f2647e59ed575604ec1","last_reissued_at":"2026-05-18T04:41:53.247005Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:53.247005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.4289","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":"2026-05-18T04:41:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y9CGlyZDczt9sTK2fyWF7iiHjuO44D36us2bThV+2BEgpavtKEk026clLYxWfsPHJJoRbtBSeejPOll5xKTfCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:33:18.374173Z"},"content_sha256":"40f04550b735de7689e541cb9c26d53fb8f0db035da9b87d91146028607ae904","schema_version":"1.0","event_id":"sha256:40f04550b735de7689e541cb9c26d53fb8f0db035da9b87d91146028607ae904"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:FUKMODIHKIORQDYUDYAI6N23DC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Transformations generating negative $\\beta$-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Charlene Kalle, Karma Dajani","submitted_at":"2010-08-25T14:19:18Z","abstract_excerpt":"We introduce a family of dynamical systems that generate negative $\\beta$-expansions and study the support of the invariant measure which is absolutely continuous with respect to Lebesgue measure. We give a characterization of the set of digit sequences that is produced by a typical member of this family of transformations. We discuss the meaning of greedy expansions in the negative sense, and show that there is no transformation in the introduced family of dynamical systems that generates negative greedy. However, if one looks at random algorithms, then it is possible to define a greedy expan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4289","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":"2026-05-18T04:41:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uvGSlwWxq7Xr/S6YeIPKRXfjPtOZpx63jECyifeHixYKHIvon72mRUZQM8O9W8f8MVxlkpVDt1kPfhAlEF0FDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:33:18.374617Z"},"content_sha256":"99bb9d97e67a18d214a5b375976e05305f8ddb9d9057361d58d1219cb73525c1","schema_version":"1.0","event_id":"sha256:99bb9d97e67a18d214a5b375976e05305f8ddb9d9057361d58d1219cb73525c1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FUKMODIHKIORQDYUDYAI6N23DC/bundle.json","state_url":"https://pith.science/pith/FUKMODIHKIORQDYUDYAI6N23DC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FUKMODIHKIORQDYUDYAI6N23DC/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":"2026-05-28T22:33:18Z","links":{"resolver":"https://pith.science/pith/FUKMODIHKIORQDYUDYAI6N23DC","bundle":"https://pith.science/pith/FUKMODIHKIORQDYUDYAI6N23DC/bundle.json","state":"https://pith.science/pith/FUKMODIHKIORQDYUDYAI6N23DC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FUKMODIHKIORQDYUDYAI6N23DC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:FUKMODIHKIORQDYUDYAI6N23DC","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":"3f42adc7fa69a28026f0fdf58ab98816d0075d18e2f09c868ab2051c63f891a2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-08-25T14:19:18Z","title_canon_sha256":"138140a8a27e4e74fff6bb1eb7d921235bd8347b53b5710672e3ec4d54f9887e"},"schema_version":"1.0","source":{"id":"1008.4289","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.4289","created_at":"2026-05-18T04:41:53Z"},{"alias_kind":"arxiv_version","alias_value":"1008.4289v1","created_at":"2026-05-18T04:41:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4289","created_at":"2026-05-18T04:41:53Z"},{"alias_kind":"pith_short_12","alias_value":"FUKMODIHKIOR","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FUKMODIHKIORQDYU","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FUKMODIH","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:99bb9d97e67a18d214a5b375976e05305f8ddb9d9057361d58d1219cb73525c1","target":"graph","created_at":"2026-05-18T04:41:53Z","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 introduce a family of dynamical systems that generate negative $\\beta$-expansions and study the support of the invariant measure which is absolutely continuous with respect to Lebesgue measure. We give a characterization of the set of digit sequences that is produced by a typical member of this family of transformations. We discuss the meaning of greedy expansions in the negative sense, and show that there is no transformation in the introduced family of dynamical systems that generates negative greedy. However, if one looks at random algorithms, then it is possible to define a greedy expan","authors_text":"Charlene Kalle, Karma Dajani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-08-25T14:19:18Z","title":"Transformations generating negative $\\beta$-expansions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4289","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":"sha256:40f04550b735de7689e541cb9c26d53fb8f0db035da9b87d91146028607ae904","target":"record","created_at":"2026-05-18T04:41:53Z","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":"3f42adc7fa69a28026f0fdf58ab98816d0075d18e2f09c868ab2051c63f891a2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-08-25T14:19:18Z","title_canon_sha256":"138140a8a27e4e74fff6bb1eb7d921235bd8347b53b5710672e3ec4d54f9887e"},"schema_version":"1.0","source":{"id":"1008.4289","kind":"arxiv","version":1}},"canonical_sha256":"2d14c70d07521d180f141e008f375b189c7d4bea6eb52f2647e59ed575604ec1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2d14c70d07521d180f141e008f375b189c7d4bea6eb52f2647e59ed575604ec1","first_computed_at":"2026-05-18T04:41:53.247005Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:53.247005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MhhcfREo6CqqNSFG6NtxdERt+qI70KuXjmiCIrAX0yG7ii0eYS/yz2rzVo3/3eFsA9Vb4NOe/PmwgKIvLSDyBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:53.247513Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.4289","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40f04550b735de7689e541cb9c26d53fb8f0db035da9b87d91146028607ae904","sha256:99bb9d97e67a18d214a5b375976e05305f8ddb9d9057361d58d1219cb73525c1"],"state_sha256":"093dc27f9e4131581bba8c7f57bead0c86e0f9e36b6935d131d3fba3b37019ad"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SbmyvtdYioMPCbK6b9z0ddMhONYVGIU61qtAZrUNiO5stv+7Z/dCeRFPTCS1fgPFufqytt7c+OXvNVm0di/RCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T22:33:18.377390Z","bundle_sha256":"dd51d7b4f25ca4a8fab58dea5079ab3b85fd448f66bf9d4dea20ad71fb398b84"}}