{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:BUU3BFPGYTXOM7OC3F22YHIEEE","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":"fd89253f7df6642c7e44dba1933648e95f3df67b502b957ef8ee61bac9d75e99","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T14:55:15Z","title_canon_sha256":"5d55817c2d909f340d60333c7112be10af9d7ba0e09b2f25121503f04d33d7ba"},"schema_version":"1.0","source":{"id":"2605.13628","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13628","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13628v1","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13628","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"pith_short_12","alias_value":"BUU3BFPGYTXO","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"BUU3BFPGYTXOM7OC","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"BUU3BFPG","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:44db37a9fe2888193147518d14a44290e753b1041684f68ee2abfee380d22f96","target":"graph","created_at":"2026-05-18T02:44:17Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"if q is an odd prime power, there is ε_q>0 such that if S ⊆ F_q with 0 ∈ S and |S|>(q+1)/2 and A ⊆ F_q^n contains no three-term arithmetic progression whose common difference is in S^n, then |A| ≤ q^{(1-ε_q)n}."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That Tao's slice rank method can be directly adapted to the setting of restricted differences without losing the polynomial rank bounds or requiring additional unstated conditions on S or q."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Adapting the slice rank method yields that sets in F_q^n without 3-APs with differences in S^n have size at most q^{(1-ε_q)n} when |S|>(q+1)/2 and q is an odd prime power."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"When S is a large subset of the finite field containing zero, any subset of the n-dimensional vector space over the field that avoids three-term arithmetic progressions with differences in S to the n has size at most q to the power (1 minus"}],"snapshot_sha256":"6012f011326620c9658b27defdd6e5bc194d043ad0a2a755bb8b855c80cf5fbb"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricted differences. In particular, we show that if $q$ is an odd prime power, there is $\\varepsilon_q>0$ such that if $S \\subseteq \\mathbb{F}_q$ with $0 \\in S$ and $|S|>(q+1)/2$ and $A \\subseteq \\mathbb{F}_q^n$ contains no three-term arithmetic progression whose common difference is in $S^n$, then $|A| \\leq q^{(1-\\varepsilon_q)n}$.","authors_text":"David Conlon, Huy Tuan Pham, Jacob Fox","cross_cats":["math.NT"],"headline":"When S is a large subset of the finite field containing zero, any subset of the n-dimensional vector space over the field that avoids three-term arithmetic progressions with differences in S to the n has size at most q to the power (1 minus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T14:55:15Z","title":"A note on arithmetic progressions with restricted differences"},"references":{"count":16,"internal_anchors":0,"resolved_work":16,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"A. Bhangale, S. Khot, Y. P. Liu, and D. Minzer, On inverse theorems and combinatorial lines,FOCS 2025, 1672–1684","work_id":"21a60064-74aa-47f2-879a-c1d24d1a596c","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"A. Bhangale, S. Khot, and D. Minzer, Effective bounds for restricted 3-arithmetic progressions inFn p, Discrete Anal.2024, Paper No. 16, 22 pp","work_id":"fd36ba77-6168-4997-9123-69a3e0276210","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"J. Blasiak, T. Church, H. Cohn, J. A. Grochow, E. Naslund, W. F. Sawin, and C. Umans, On cap sets and the group-theoretic approach to matrix multiplication,Discrete Anal.2017, Paper No. 3, 27 pp","work_id":"dbcfabb5-d858-45fb-965f-b8dd562d96cc","year":2017},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"E. Croot, V. F. Lev, and P. P. Pach, Progression-free sets inZn 4 are exponentially small,Ann. of Math.185(2017), 331–337","work_id":"c3b6dcf2-6037-45ef-a893-81bef04eb154","year":2017},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"J. S. Ellenberg and D. Gijswijt, On large subsets ofFn q with no three-term arithmetic progression, Ann. of Math.185(2017), 339–343","work_id":"bff001af-e58f-4af9-baa0-659028ade260","year":2017}],"snapshot_sha256":"2254809aa286acca99c60e62e0dfea86cbb56eab4d0fb86d23ba33fcfa520ff5"},"source":{"id":"2605.13628","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T17:47:29.202143Z","id":"47331c4b-97c8-4031-90ba-d1b3bd75f613","model_set":{"reader":"grok-4.3"},"one_line_summary":"Adapting the slice rank method yields that sets in F_q^n without 3-APs with differences in S^n have size at most q^{(1-ε_q)n} when |S|>(q+1)/2 and q is an odd prime power.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"When S is a large subset of the finite field containing zero, any subset of the n-dimensional vector space over the field that avoids three-term arithmetic progressions with differences in S to the n has size at most q to the power (1 minus","strongest_claim":"if q is an odd prime power, there is ε_q>0 such that if S ⊆ F_q with 0 ∈ S and |S|>(q+1)/2 and A ⊆ F_q^n contains no three-term arithmetic progression whose common difference is in S^n, then |A| ≤ q^{(1-ε_q)n}.","weakest_assumption":"That Tao's slice rank method can be directly adapted to the setting of restricted differences without losing the polynomial rank bounds or requiring additional unstated conditions on S or q."}},"verdict_id":"47331c4b-97c8-4031-90ba-d1b3bd75f613"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:62ea52181f3166df9fdcabf4733129694946f526345c700dc09442c3004d3b60","target":"record","created_at":"2026-05-18T02:44:17Z","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":"fd89253f7df6642c7e44dba1933648e95f3df67b502b957ef8ee61bac9d75e99","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T14:55:15Z","title_canon_sha256":"5d55817c2d909f340d60333c7112be10af9d7ba0e09b2f25121503f04d33d7ba"},"schema_version":"1.0","source":{"id":"2605.13628","kind":"arxiv","version":1}},"canonical_sha256":"0d29b095e6c4eee67dc2d975ac1d04212158527880204188eb74c98ab7610905","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d29b095e6c4eee67dc2d975ac1d04212158527880204188eb74c98ab7610905","first_computed_at":"2026-05-18T02:44:17.781766Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:17.781766Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qGxGsmJjqT3jHYB/qcVub0Inxt/Kt215oF1a6rpXVPbQGGscLGqiHNRRB1hvx+I8aKIrz20kBb9xrVnb9YESCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:17.782195Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13628","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62ea52181f3166df9fdcabf4733129694946f526345c700dc09442c3004d3b60","sha256:44db37a9fe2888193147518d14a44290e753b1041684f68ee2abfee380d22f96"],"state_sha256":"c11d212470ece26a5fc0d321520dd04b811a9abc9e13e9eceeb43426c67cabbc"}