{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:5RHPLWKZP6PBA5FVBSCWH2FAUN","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":"98e9f5a1a33bd1624343aff3bfa85af90ba1c65220c9eb25b81de1ebcd8f57c6","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-31T03:10:32Z","title_canon_sha256":"709e132c48b3c69c94cab41082f385e2028efa4b5a6e29f9a754f6e6a5ef7c2c"},"schema_version":"1.0","source":{"id":"1512.09205","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.09205","created_at":"2026-05-18T01:23:32Z"},{"alias_kind":"arxiv_version","alias_value":"1512.09205v1","created_at":"2026-05-18T01:23:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.09205","created_at":"2026-05-18T01:23:32Z"},{"alias_kind":"pith_short_12","alias_value":"5RHPLWKZP6PB","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"5RHPLWKZP6PBA5FV","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"5RHPLWKZ","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:415e7a6bfc7679222a7d0cac2519cc4d0b5bbe61b0168b076ef423ef815d4072","target":"graph","created_at":"2026-05-18T01:23:32Z","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":"This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $\\beta$-expansions. More precisely, let $([0,1),T_{\\beta})$ be the $\\beta$-dynamical system for a general $\\beta>1$ and $\\psi:[0,1]\\mapsto\\mathbb{R}$ be a continuous function. Denote by $\\textsf{A}(\\psi,x)$ all the accumulation points of $\\Big\\{\\frac{1}{n}\\sum_{j=0}^{n-1}\\psi(T^jx): n\\ge 1\\Big\\}$. The Hausdorff dimensions of the sets $$\\Big\\{x:\\textsf{A}(\\psi,x)\\supset[a,b]\\Big\\},\\ \\ \\Big\\{x:\\textsf{A}(\\psi,x)=[a,b]\\Big\\}, \\ \\Big\\{x:\\textsf{A}(\\psi,x)\\subset[a,b]\\Big\\}$$ i.e","authors_text":"Xiaojun Zhao, Yuanhong Chen, Zhenliang Zhang","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-31T03:10:32Z","title":"Multifractal analysis of the divergence points of Birkhoff averages in $beta$-dynamical systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.09205","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:ecb4ea4398c857231cf5e96f8b5caf693fc85caae7d21a02638550ac95b96323","target":"record","created_at":"2026-05-18T01:23:32Z","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":"98e9f5a1a33bd1624343aff3bfa85af90ba1c65220c9eb25b81de1ebcd8f57c6","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-31T03:10:32Z","title_canon_sha256":"709e132c48b3c69c94cab41082f385e2028efa4b5a6e29f9a754f6e6a5ef7c2c"},"schema_version":"1.0","source":{"id":"1512.09205","kind":"arxiv","version":1}},"canonical_sha256":"ec4ef5d9597f9e1074b50c8563e8a0a36e2ebb871f5633d6384b9950e4d91478","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ec4ef5d9597f9e1074b50c8563e8a0a36e2ebb871f5633d6384b9950e4d91478","first_computed_at":"2026-05-18T01:23:32.996630Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:23:32.996630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u6DvQj/wqRzzw+gdhOX9hHuB6MzTcwNYJg0liO3P0upwt8Fkd6rMz3HfpMfmkjhCcBY81KeXWQsn11lDGM/CCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:23:32.997281Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.09205","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ecb4ea4398c857231cf5e96f8b5caf693fc85caae7d21a02638550ac95b96323","sha256:415e7a6bfc7679222a7d0cac2519cc4d0b5bbe61b0168b076ef423ef815d4072"],"state_sha256":"a30e86026ee274129115d0042407eeb4b879a7bf0329d0835addda8d014cc7c0"}