{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:HDWLUEUSONRTZ4JENTQAZ276VV","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":"026976a9153fbf19278e5d8e935cbb7e30a007bc8a3c2a46fd9462ff5e8f2c2d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-08-19T18:30:58Z","title_canon_sha256":"9ce1aa19617d8d489314645038863e76f1fdc1a80e5598a243dc8431ea46c100"},"schema_version":"1.0","source":{"id":"1908.07014","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.07014","created_at":"2026-07-05T04:02:50Z"},{"alias_kind":"arxiv_version","alias_value":"1908.07014v2","created_at":"2026-07-05T04:02:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.07014","created_at":"2026-07-05T04:02:50Z"},{"alias_kind":"pith_short_12","alias_value":"HDWLUEUSONRT","created_at":"2026-07-05T04:02:50Z"},{"alias_kind":"pith_short_16","alias_value":"HDWLUEUSONRTZ4JE","created_at":"2026-07-05T04:02:50Z"},{"alias_kind":"pith_short_8","alias_value":"HDWLUEUS","created_at":"2026-07-05T04:02:50Z"}],"graph_snapshots":[{"event_id":"sha256:b7e8c73e9324ff331e43e8f3fd2091037307b6903b36ecc2410cd654172adb53","target":"graph","created_at":"2026-07-05T04:02:50Z","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/1908.07014/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\\rm GL}_{n_0}(\\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions.\n  Here is a more precise formulation of our main result. For a positive integer $c_0$, we say a square-free polynomial is $c_0$-admissible if degree of irreducible factors of $f$ are distinct integers with prime factors at least $c_0$. Suppose $\\Omega$ is a finite symmetric subset of ${\\rm GL}_{n_0}(\\mathbb{F}_p(t))$, where $p$ is a prime more than $5$. Let $\\Gamma$","authors_text":"Alireza Salehi Golsefidy, Brian Longo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-08-19T18:30:58Z","title":"Towards super-approximation in positive characteristic"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.07014","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:ee9e09bbc553fea923cfc816b6b474b8f14704f2b5a70880c05250c9d4964f1b","target":"record","created_at":"2026-07-05T04:02:50Z","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":"026976a9153fbf19278e5d8e935cbb7e30a007bc8a3c2a46fd9462ff5e8f2c2d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-08-19T18:30:58Z","title_canon_sha256":"9ce1aa19617d8d489314645038863e76f1fdc1a80e5598a243dc8431ea46c100"},"schema_version":"1.0","source":{"id":"1908.07014","kind":"arxiv","version":2}},"canonical_sha256":"38ecba129273633cf1246ce00cebfead7e3567786db959b6cb7fe3acdbd7b611","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"38ecba129273633cf1246ce00cebfead7e3567786db959b6cb7fe3acdbd7b611","first_computed_at":"2026-07-05T04:02:50.679417Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:02:50.679417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Hf+8fvEqK1SQugo/NhR14ZhZSo/dwReqqnAgevFZOinUQEDqQVB9CY9G8seu9YHlvJiAns5ZFS1I1/s8P3cfBA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:02:50.679838Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.07014","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee9e09bbc553fea923cfc816b6b474b8f14704f2b5a70880c05250c9d4964f1b","sha256:b7e8c73e9324ff331e43e8f3fd2091037307b6903b36ecc2410cd654172adb53"],"state_sha256":"78f3f316cd6f6bdaeb3110b3532ed15328916a2070b9c32d6fbe2890329026c3"}