{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:BULHYLEUPVQCKCFFMCCIJLO3LL","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":"6fdfaefdf97f9e9a27392d4485b15b18b9eaf35abad83c6d2f144de20b5a1ddb","cross_cats_sorted":["math.CA","math.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-02-21T01:11:05Z","title_canon_sha256":"e1a5b8b477ac058fd0438d441038e9717ed2558bd73451f1bd4a34acf22221c2"},"schema_version":"1.0","source":{"id":"2402.13451","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.13451","created_at":"2026-07-05T10:14:02Z"},{"alias_kind":"arxiv_version","alias_value":"2402.13451v3","created_at":"2026-07-05T10:14:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.13451","created_at":"2026-07-05T10:14:02Z"},{"alias_kind":"pith_short_12","alias_value":"BULHYLEUPVQC","created_at":"2026-07-05T10:14:02Z"},{"alias_kind":"pith_short_16","alias_value":"BULHYLEUPVQCKCFF","created_at":"2026-07-05T10:14:02Z"},{"alias_kind":"pith_short_8","alias_value":"BULHYLEU","created_at":"2026-07-05T10:14:02Z"}],"graph_snapshots":[{"event_id":"sha256:301ba57f34ca55b4c1d8f30d5f37732a1271a2edee17133f3a678d9248a0048b","target":"graph","created_at":"2026-07-05T10:14:02Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2402.13451/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the Folklore set of Dirichlet improvable matrices in $\\mathbb R^{m\\times n}$ which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs $m,n$ apart from $\\{m,n\\}=\\{ 1,1\\}$ and $\\{m,n\\}=\\{ 2,3\\}$ in a constructive manner. For a wide range of integer pairs $(m,n)$ we construct subsets of the Folklore set with an exact prescribed Dirichlet constant (in some right neighbourhood of $0$). This enables us to provide information on the Dirichlet Spectrum of matrices. The key technique of our construction is to build first vectors of a given Di","authors_text":"Benjamin Ward, Johannes Schleischitz, Mumtaz Hussain","cross_cats":["math.CA","math.DS"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-02-21T01:11:05Z","title":"On the Folklore set and Dirichlet spectrum for matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.13451","kind":"arxiv","version":3},"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":"sha256:6358b252b4a89ff3234d8e6fca958bf0beb4a073fa0cdf5c4519cfbe57e6a1bb","target":"record","created_at":"2026-07-05T10:14:02Z","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":"6fdfaefdf97f9e9a27392d4485b15b18b9eaf35abad83c6d2f144de20b5a1ddb","cross_cats_sorted":["math.CA","math.DS"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-02-21T01:11:05Z","title_canon_sha256":"e1a5b8b477ac058fd0438d441038e9717ed2558bd73451f1bd4a34acf22221c2"},"schema_version":"1.0","source":{"id":"2402.13451","kind":"arxiv","version":3}},"canonical_sha256":"0d167c2c947d602508a5608484addb5acb2f7eb767b9bff87fd037ae64a99bab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d167c2c947d602508a5608484addb5acb2f7eb767b9bff87fd037ae64a99bab","first_computed_at":"2026-07-05T10:14:02.503894Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:14:02.503894Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Hl/ZkSNv2OgZgXW/Sm+/ENX4pPRdjreTsOpLvfbh0Y2I2PwPQr101DeqU/3laajfbOPZ//0MZlb/tm+NGo2nCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:14:02.504398Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.13451","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6358b252b4a89ff3234d8e6fca958bf0beb4a073fa0cdf5c4519cfbe57e6a1bb","sha256:301ba57f34ca55b4c1d8f30d5f37732a1271a2edee17133f3a678d9248a0048b"],"state_sha256":"6fe89a6a566df8fe863f0ba23fe0977fb4ce7fb389743a1cadb908073cd18e80"}