{"paper":{"title":"A note on arithmetic progressions with restricted differences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","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","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"David Conlon, Huy Tuan Pham, Jacob Fox","submitted_at":"2026-05-13T14:55:15Z","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 $|A| \\leq q^{(1-\\varepsilon_q)n}$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","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}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","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","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6012f011326620c9658b27defdd6e5bc194d043ad0a2a755bb8b855c80cf5fbb"},"source":{"id":"2605.13628","kind":"arxiv","version":1},"verdict":{"id":"47331c4b-97c8-4031-90ba-d1b3bd75f613","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:47:29.202143Z","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}.","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","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.","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"},"references":{"count":16,"sample":[{"doi":"","year":2025,"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","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"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","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"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","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"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","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"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","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":16,"snapshot_sha256":"2254809aa286acca99c60e62e0dfea86cbb56eab4d0fb86d23ba33fcfa520ff5","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}