{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:HJB34BCDHOELYGPRY3ZIF26NOJ","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":"8c929f4d449eb1ff761d2e073192e848d7bb48d3d00a6fc0534b93be39842aca","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-03-28T02:44:14Z","title_canon_sha256":"8958b9282026446ed08af8f5314f15d542c39ed904e81961d9d67759dd92777c"},"schema_version":"1.0","source":{"id":"1003.5335","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.5335","created_at":"2026-05-18T04:33:15Z"},{"alias_kind":"arxiv_version","alias_value":"1003.5335v2","created_at":"2026-05-18T04:33:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.5335","created_at":"2026-05-18T04:33:15Z"},{"alias_kind":"pith_short_12","alias_value":"HJB34BCDHOEL","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"HJB34BCDHOELYGPR","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"HJB34BCD","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:fd33df950e13306549c9dc0e35b60a5f166e32482cc459162652f999c865ae8b","target":"graph","created_at":"2026-05-18T04:33:15Z","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 $\\mathbf{J} \\subset \\mathbb{R}^2$ be the set of couples $(x,q)$ with $q>1$ such that $x$ has at least one representation of the form $x=\\sum_{i=1}^{\\infty} c_i q^{-i}$ with integer coefficients $c_i$ satisfying $0 \\leq c_i < q$, $i \\ge 1$. In this case we say that $(c_i)=c_1c_2...$ is an expansion of $x$ in base $q$. Let $\\mathbf{U}$ be the set of couples $(x,q) \\in \\mathbf{J}$ such that $x$ has exactly one expansion in base $q$. In this paper we deduce some topological and combinatorial properties of the set $\\mathbf{U}$. We characterize the closure of $\\mathbf{U}$, and we determine its H","authors_text":"Martijn de Vries, Vilmos Komornik","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-03-28T02:44:14Z","title":"A two-dimensional univoque set"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.5335","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":"sha256:123352485d666e5b55f1cc889a6aa36535ac8f5dec9eda0178bed7569e7dc4de","target":"record","created_at":"2026-05-18T04:33:15Z","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":"8c929f4d449eb1ff761d2e073192e848d7bb48d3d00a6fc0534b93be39842aca","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-03-28T02:44:14Z","title_canon_sha256":"8958b9282026446ed08af8f5314f15d542c39ed904e81961d9d67759dd92777c"},"schema_version":"1.0","source":{"id":"1003.5335","kind":"arxiv","version":2}},"canonical_sha256":"3a43be04433b88bc19f1c6f282ebcd724599588e933121b2b99c83da121e9df7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a43be04433b88bc19f1c6f282ebcd724599588e933121b2b99c83da121e9df7","first_computed_at":"2026-05-18T04:33:15.812286Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:33:15.812286Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eqcm2AEfgkHhVsi4Ginih2injPrmJJR27BiHNyV8JriZTO86FVV4ugQHGzlvyAYYAMdfSupJQoiELHfdGp4RDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:33:15.812869Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.5335","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:123352485d666e5b55f1cc889a6aa36535ac8f5dec9eda0178bed7569e7dc4de","sha256:fd33df950e13306549c9dc0e35b60a5f166e32482cc459162652f999c865ae8b"],"state_sha256":"23b5906e565e8c52f8091cb0c514df46fb22aa5dbcfd43e9b370a756b820479c"}