{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:I3XQQAVNIGNOWGQ6VUVCPDHNAB","short_pith_number":"pith:I3XQQAVN","canonical_record":{"source":{"id":"1110.3192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-14T12:53:22Z","cross_cats_sorted":[],"title_canon_sha256":"04c4344e3140a531f08adbf1be28553f2434ac5f441d97f009d947b3c2b1de39","abstract_canon_sha256":"94aeb26f651864adaab95ea959183762ea173fc47344ba623e3e719a1daf5cb9"},"schema_version":"1.0"},"canonical_sha256":"46ef0802ad419aeb1a1ead2a278ced0062379906c93cfe0bf78c3ca077d5b37d","source":{"kind":"arxiv","id":"1110.3192","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.3192","created_at":"2026-05-18T04:10:57Z"},{"alias_kind":"arxiv_version","alias_value":"1110.3192v1","created_at":"2026-05-18T04:10:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.3192","created_at":"2026-05-18T04:10:57Z"},{"alias_kind":"pith_short_12","alias_value":"I3XQQAVNIGNO","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I3XQQAVNIGNOWGQ6","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I3XQQAVN","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:I3XQQAVNIGNOWGQ6VUVCPDHNAB","target":"record","payload":{"canonical_record":{"source":{"id":"1110.3192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-14T12:53:22Z","cross_cats_sorted":[],"title_canon_sha256":"04c4344e3140a531f08adbf1be28553f2434ac5f441d97f009d947b3c2b1de39","abstract_canon_sha256":"94aeb26f651864adaab95ea959183762ea173fc47344ba623e3e719a1daf5cb9"},"schema_version":"1.0"},"canonical_sha256":"46ef0802ad419aeb1a1ead2a278ced0062379906c93cfe0bf78c3ca077d5b37d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:57.816909Z","signature_b64":"kQ0JabeRtYZ+WVRu5J1MLnci36j1ZLUhbnIbNyotjXvd4cp/RhepMoA5VsAXRuDueHNDGAGtaxLZe9klINS+CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"46ef0802ad419aeb1a1ead2a278ced0062379906c93cfe0bf78c3ca077d5b37d","last_reissued_at":"2026-05-18T04:10:57.816363Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:57.816363Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.3192","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":"2026-05-18T04:10:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WeXZn7/HY8IuJaIwp3Ur3qm8kyFCBg2xsj3ZuuZ1BrHHvfZmzNielhVRwXB/+XRhGLz5i1VxzspNMLq7PVAUDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T01:42:31.052004Z"},"content_sha256":"ee65322a5aca4be12bebbaef1bae920db6b38899d85c2f77f899ef4ca9e8f102","schema_version":"1.0","event_id":"sha256:ee65322a5aca4be12bebbaef1bae920db6b38899d85c2f77f899ef4ca9e8f102"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:I3XQQAVNIGNOWGQ6VUVCPDHNAB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Intersections of homogeneous Cantor sets and beta-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Derong Kong, Michel Dekking, Wenxia Li","submitted_at":"2011-10-14T12:53:22Z","abstract_excerpt":"Let $\\Gamma_{\\beta,N}$ be the $N$-part homogeneous Cantor set with $\\beta\\in(1/(2N-1),1/N)$. Any string $(j_\\ell)_{\\ell=1}^\\N$ with $j_\\ell\\in\\{0,\\pm 1,...,\\pm(N-1)\\}$ such that $t=\\sum_{\\ell=1}^\\N j_\\ell\\beta^{\\ell-1}(1-\\beta)/(N-1)$ is called a code of $t$. Let $\\mathcal{U}_{\\beta,\\pm N}$ be the set of $t\\in[-1,1]$ having a unique code, and let $\\mathcal{S}_{\\beta,\\pm N}$ be the set of $t\\in\\mathcal{U}_{\\beta,\\pm N}$ which make the intersection $\\Gamma_{\\beta,N}\\cap(\\Gamma_{\\beta,N}+t)$ a self-similar set. We characterize the set $\\mathcal{U}_{\\beta,\\pm N}$ in a geometrical and algebraical w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3192","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":"2026-05-18T04:10:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OAZXaC1sgETdVzNsX2UeH6/Qkgv7tToS4Ajo0QgqlQHfEVzc2daL5/F6G93y+j+DVai8BPeuABKm/eJFTm7eCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T01:42:31.052353Z"},"content_sha256":"55f65a80dd23e59f62b85e729e33064e47ada3bcec73e80f2b08c8059606d068","schema_version":"1.0","event_id":"sha256:55f65a80dd23e59f62b85e729e33064e47ada3bcec73e80f2b08c8059606d068"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I3XQQAVNIGNOWGQ6VUVCPDHNAB/bundle.json","state_url":"https://pith.science/pith/I3XQQAVNIGNOWGQ6VUVCPDHNAB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I3XQQAVNIGNOWGQ6VUVCPDHNAB/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":"2026-06-22T01:42:31Z","links":{"resolver":"https://pith.science/pith/I3XQQAVNIGNOWGQ6VUVCPDHNAB","bundle":"https://pith.science/pith/I3XQQAVNIGNOWGQ6VUVCPDHNAB/bundle.json","state":"https://pith.science/pith/I3XQQAVNIGNOWGQ6VUVCPDHNAB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I3XQQAVNIGNOWGQ6VUVCPDHNAB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:I3XQQAVNIGNOWGQ6VUVCPDHNAB","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":"94aeb26f651864adaab95ea959183762ea173fc47344ba623e3e719a1daf5cb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-14T12:53:22Z","title_canon_sha256":"04c4344e3140a531f08adbf1be28553f2434ac5f441d97f009d947b3c2b1de39"},"schema_version":"1.0","source":{"id":"1110.3192","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.3192","created_at":"2026-05-18T04:10:57Z"},{"alias_kind":"arxiv_version","alias_value":"1110.3192v1","created_at":"2026-05-18T04:10:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.3192","created_at":"2026-05-18T04:10:57Z"},{"alias_kind":"pith_short_12","alias_value":"I3XQQAVNIGNO","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"I3XQQAVNIGNOWGQ6","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"I3XQQAVN","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:55f65a80dd23e59f62b85e729e33064e47ada3bcec73e80f2b08c8059606d068","target":"graph","created_at":"2026-05-18T04:10:57Z","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":"Let $\\Gamma_{\\beta,N}$ be the $N$-part homogeneous Cantor set with $\\beta\\in(1/(2N-1),1/N)$. Any string $(j_\\ell)_{\\ell=1}^\\N$ with $j_\\ell\\in\\{0,\\pm 1,...,\\pm(N-1)\\}$ such that $t=\\sum_{\\ell=1}^\\N j_\\ell\\beta^{\\ell-1}(1-\\beta)/(N-1)$ is called a code of $t$. Let $\\mathcal{U}_{\\beta,\\pm N}$ be the set of $t\\in[-1,1]$ having a unique code, and let $\\mathcal{S}_{\\beta,\\pm N}$ be the set of $t\\in\\mathcal{U}_{\\beta,\\pm N}$ which make the intersection $\\Gamma_{\\beta,N}\\cap(\\Gamma_{\\beta,N}+t)$ a self-similar set. We characterize the set $\\mathcal{U}_{\\beta,\\pm N}$ in a geometrical and algebraical w","authors_text":"Derong Kong, Michel Dekking, Wenxia Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-14T12:53:22Z","title":"Intersections of homogeneous Cantor sets and beta-expansions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3192","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:ee65322a5aca4be12bebbaef1bae920db6b38899d85c2f77f899ef4ca9e8f102","target":"record","created_at":"2026-05-18T04:10:57Z","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":"94aeb26f651864adaab95ea959183762ea173fc47344ba623e3e719a1daf5cb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-10-14T12:53:22Z","title_canon_sha256":"04c4344e3140a531f08adbf1be28553f2434ac5f441d97f009d947b3c2b1de39"},"schema_version":"1.0","source":{"id":"1110.3192","kind":"arxiv","version":1}},"canonical_sha256":"46ef0802ad419aeb1a1ead2a278ced0062379906c93cfe0bf78c3ca077d5b37d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"46ef0802ad419aeb1a1ead2a278ced0062379906c93cfe0bf78c3ca077d5b37d","first_computed_at":"2026-05-18T04:10:57.816363Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:57.816363Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kQ0JabeRtYZ+WVRu5J1MLnci36j1ZLUhbnIbNyotjXvd4cp/RhepMoA5VsAXRuDueHNDGAGtaxLZe9klINS+CA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:57.816909Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.3192","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee65322a5aca4be12bebbaef1bae920db6b38899d85c2f77f899ef4ca9e8f102","sha256:55f65a80dd23e59f62b85e729e33064e47ada3bcec73e80f2b08c8059606d068"],"state_sha256":"0734f68da6e29a6c87dd115950defaef76844a89a245807a02e011559980965d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yQKviaNLZ2/I6yfsW/xZ9DyDIl1RmWKzAcQbt7m5du4vbZ39VfiHdjfnxXWwSt2CwGHYVjdWkL1cjUXh0g9WCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T01:42:31.054233Z","bundle_sha256":"0485dcc6f10d7532024ace828d4da0bf7d55a8ad51dc2bc1ee15c3e7b783b235"}}