{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:3PBQZLT2EW7V37Q6ZCRC3NAHA6","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":"339172b970755d9af355d218fb63c51217ec747bb200ae776c922a9016198e90","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-16T04:34:41Z","title_canon_sha256":"b0167535810e1df8ed3bcf3f2ca393ff80fbbbdd66044faff52a5af97085e766"},"schema_version":"1.0","source":{"id":"1409.4501","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.4501","created_at":"2026-05-18T02:39:47Z"},{"alias_kind":"arxiv_version","alias_value":"1409.4501v3","created_at":"2026-05-18T02:39:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.4501","created_at":"2026-05-18T02:39:47Z"},{"alias_kind":"pith_short_12","alias_value":"3PBQZLT2EW7V","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"3PBQZLT2EW7V37Q6","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"3PBQZLT2","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:65ab1f7d002db023551894e5d412bb7d65410d1f686377ca9a212343df29d8a4","target":"graph","created_at":"2026-05-18T02:39:47Z","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":"We show that the equation $\\lambda_1 n_1^2 + ... + \\lambda_s n_s^2 = 0$ admits non-trivial solutions in any subset of $[N]$ of density $(\\log N)^{-c_s}$, provided that $s \\geq 7$ and the coefficients $\\lambda_i$ sum to zero and satisfy certain sign conditions. This improves upon previous known density bounds of the form $(\\log\\log N)^{-c}$.","authors_text":"Kevin Henriot","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-16T04:34:41Z","title":"Logarithmic bounds for translation-invariant equations in squares"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.4501","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:e9e8174df75f35d146cf6ff1c8ccdd18bd6dd933069ce27a77ad12e5ba37c1a8","target":"record","created_at":"2026-05-18T02:39:47Z","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":"339172b970755d9af355d218fb63c51217ec747bb200ae776c922a9016198e90","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-16T04:34:41Z","title_canon_sha256":"b0167535810e1df8ed3bcf3f2ca393ff80fbbbdd66044faff52a5af97085e766"},"schema_version":"1.0","source":{"id":"1409.4501","kind":"arxiv","version":3}},"canonical_sha256":"dbc30cae7a25bf5dfe1ec8a22db40707bb3b1237a0f242dbcd0ffa4b773fe705","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dbc30cae7a25bf5dfe1ec8a22db40707bb3b1237a0f242dbcd0ffa4b773fe705","first_computed_at":"2026-05-18T02:39:47.825192Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:39:47.825192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i0zRbI6LuVNRCrpTnUGwMVJ1okPFwP8Mpi1ON7tVzC26mxZqTdJypfpUd/T95eYXfeawcdLRC8brhr1FVvrtBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:39:47.825713Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.4501","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9e8174df75f35d146cf6ff1c8ccdd18bd6dd933069ce27a77ad12e5ba37c1a8","sha256:65ab1f7d002db023551894e5d412bb7d65410d1f686377ca9a212343df29d8a4"],"state_sha256":"c3b5d71442e27a5b083d707672230b2bef7b60da7d75a5e5cb4e55ab392a9961"}