{"paper":{"title":"Almost Affine Invariance Over Prime Fields: Green Problem 90","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A density-1/2 set in the prime field F_p is almost affine invariant under all maps ax+b with |a| and |b| at most o(log p).","cross_cats":["math.DS","math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Jie Ma, Max Wenqiang Xu, Quanyu Tang","submitted_at":"2026-05-13T12:48:56Z","abstract_excerpt":"Let $A\\subset \\mathbb{F}_p$ with density 1/2. We call a set $A$ almost affine invariant under an affine transformation $\\phi(x)=ax+b$ if \\[|A \\triangle \\phi(A)| =o(p).\\] We determine that, the threshold value of $K$ such that $A$ is almost affine invariant simultaneously under all $\\phi(x)$ with $|a|, |b|\\le K$ and $a\\neq 0$, is $K=o(\\log p)$. This solves Ben Green's Open Problem 90."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the threshold value of K such that A is almost affine invariant simultaneously under all ϕ(x) with |a|, |b|≤K and a≠0, is K=o(log p). This solves Ben Green's Open Problem 90.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The o(p) error is uniform over all maps with |a|,|b|≤K; if the error term grows with K in a non-uniform way or depends on the specific density-1/2 assumption, the claimed threshold may fail to hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The threshold K for almost affine invariance of density-1/2 sets in F_p is o(log p), solving Green's Problem 90.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A density-1/2 set in the prime field F_p is almost affine invariant under all maps ax+b with |a| and |b| at most o(log p).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"94e6497a14a55defda8d39a7c54ab5b9b278952f482b612b96ae5b6b5611c61e"},"source":{"id":"2605.13454","kind":"arxiv","version":1},"verdict":{"id":"15362345-01a7-48aa-8812-c707bc1eb908","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:01:22.393496Z","strongest_claim":"the threshold value of K such that A is almost affine invariant simultaneously under all ϕ(x) with |a|, |b|≤K and a≠0, is K=o(log p). This solves Ben Green's Open Problem 90.","one_line_summary":"The threshold K for almost affine invariance of density-1/2 sets in F_p is o(log p), solving Green's Problem 90.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The o(p) error is uniform over all maps with |a|,|b|≤K; if the error term grows with K in a non-uniform way or depends on the specific density-1/2 assumption, the claimed threshold may fail to hold.","pith_extraction_headline":"A density-1/2 set in the prime field F_p is almost affine invariant under all maps ax+b with |a| and |b| at most o(log p)."},"references":{"count":8,"sample":[{"doi":"","year":null,"title":"S. Eberhard, B. Green, R. Mrazović,Translation and dilation invariance inZ/qZ, Unpublished notes","work_id":"50eb5109-6b72-4d2f-a634-3ab0893dcae7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Green,100 Open Problems, available athttps://people.maths.ox.ac.uk/greenbj/papers/ open-problems.pdf","work_id":"5b661a3c-ad7f-4ddc-b1e4-f09fcfcb913c","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Some constructions in the inverse spectral theory of cyclic groups,","work_id":"5e5cb305-1445-4eaa-ba89-b20085d167b5","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1007/s00222-020-00967-6","year":2020,"title":"T. Hutchcroft and G. Pete,Kazhdan groups have cost 1, Invent. Math.221(2020), no. 3, 873–891.https: //doi.org/10.1007/s00222-020-00967-6","work_id":"9cb2739e-c031-4a4a-acd0-36fb5b1c2c07","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Freddie Manners,Is it known that(F × p ⋉F p, Fp)is not a relative expander family?, MathOverflow, URL (version: 2012-03-19):https://mathoverflow.net/q/91657","work_id":"9a6f2bc1-47c8-4c78-b9d9-d6472b66a28c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":8,"snapshot_sha256":"ef63467b29962366290887cc7b018e1bb6fd192c872bf3e2e80373d8cb47bfaa","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"773e173ee64d2bd54234eb08028b982c1702d0c5be8ca4f991bfa2a62896c31b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}