{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:RWDPBMSSEK2N6VDFPJRQP47X5W","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":"d507ab9e94c0f3858974f19278dc53f917ba9bd03f64db7341bb2fbbc48fe087","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-09-08T17:47:38Z","title_canon_sha256":"0173ff3fc009b0473761f824780d869b7df78d94ec2dc1373c88b6dd68a3046d"},"schema_version":"1.0","source":{"id":"2509.06935","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.06935","created_at":"2026-07-05T12:06:55Z"},{"alias_kind":"arxiv_version","alias_value":"2509.06935v1","created_at":"2026-07-05T12:06:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.06935","created_at":"2026-07-05T12:06:55Z"},{"alias_kind":"pith_short_12","alias_value":"RWDPBMSSEK2N","created_at":"2026-07-05T12:06:55Z"},{"alias_kind":"pith_short_16","alias_value":"RWDPBMSSEK2N6VDF","created_at":"2026-07-05T12:06:55Z"},{"alias_kind":"pith_short_8","alias_value":"RWDPBMSS","created_at":"2026-07-05T12:06:55Z"}],"graph_snapshots":[{"event_id":"sha256:de7bd4eb5879be8a0ee8f255a0107198953f9c14fe943f794a86ba253cd8ee97","target":"graph","created_at":"2026-07-05T12:06:55Z","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/2509.06935/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For integers $1 < k < d-1$ and $r \\ge k+2$, we establish new lower bounds on the maximum number of points in $[n]^d$ such that no $r$ lie in a $k$-dimensional affine (or linear) subspace. These bounds improve on earlier results of Sudakov-Tomon and Lefmann. Further, we provide a randomised construction for the no-four-on-a-circle problem posed by Erd\\H{o}s and Purdy, improving Thiele's bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for $d \\geq 4$. In each case, we apply the deletion method, using results from number theory and incid","authors_text":"Anubhab Ghosal, Peter Keevash, Ritesh Goenka","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-09-08T17:47:38Z","title":"On subsets of lattice cubes avoiding affine and spherical degeneracies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.06935","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":"sha256:0e602e386909b711e1d7b995f3499ada1d690178ec6751540587d551cdfc4b15","target":"record","created_at":"2026-07-05T12:06:55Z","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":"d507ab9e94c0f3858974f19278dc53f917ba9bd03f64db7341bb2fbbc48fe087","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-09-08T17:47:38Z","title_canon_sha256":"0173ff3fc009b0473761f824780d869b7df78d94ec2dc1373c88b6dd68a3046d"},"schema_version":"1.0","source":{"id":"2509.06935","kind":"arxiv","version":1}},"canonical_sha256":"8d86f0b25222b4df54657a6307f3f7eda743b7e68f54beaeaad7d9430bea2858","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8d86f0b25222b4df54657a6307f3f7eda743b7e68f54beaeaad7d9430bea2858","first_computed_at":"2026-07-05T12:06:55.391480Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:06:55.391480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lteqFU5iN7P2Ca5FJQX/qqxPeHfmyqjwgHcpwwmx2AjuXHnI7qLnNC5ASgEeGvkIUlVSMyRKfWurRpUv0W1sAg==","signature_status":"signed_v1","signed_at":"2026-07-05T12:06:55.391949Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.06935","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e602e386909b711e1d7b995f3499ada1d690178ec6751540587d551cdfc4b15","sha256:de7bd4eb5879be8a0ee8f255a0107198953f9c14fe943f794a86ba253cd8ee97"],"state_sha256":"4c9ffb7c97ab5e93ebeb98c80cb661653426b29da97352079fc9871b2b560360"}