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For any $q \\ge 1$ and $\\alpha \\in [0,1]$, and any $S \\subseteq G$ with $|S| = \\frac{|G|}{q+\\alpha}$, we show $\\frac{T_3(S)}{|S|^2}$ and Prob[$S$] are bounded above by $\\max\\left(\\frac{q^2-\\alpha q+\\alpha^2}{q^2},\\frac{q^2+2\\alpha q+4\\alpha^2-6\\alpha+3}{(q+1)^2},\\gamma_0\\right)$, where $\\gamma_0 < 1$ is an absolute constant. As a consequence, we verify a graph theoretic conjecture of Gan, Loh, a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1809.03729","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-11T08:21:24Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"40958d0be2e6166c2126921abf05513693eb0ca3a3ca4e592fcc105b7f08dccc","abstract_canon_sha256":"6d35f5bc6ae3843d491ab40ed4a81e6d43953939d949a71a0d031d606b881825"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:59.983581Z","signature_b64":"kzJsh3e5DdVwvV7O9FZGt68I9tOfKdmK6eswKQ0PMRNp9s0j+3xg/x1IMep6p1ZrxqFQsUcUtA3amgxuvbN9AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8ff68bbf668e4949d8c7a71db0ef04273e713b1499a5fd179bd60858cb674b9","last_reissued_at":"2026-05-18T00:05:59.983163Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:59.983163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Maximum Number of Three Term Arithmetic Progressions, and Triangles in Cayley Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zachary Chase","submitted_at":"2018-09-11T08:21:24Z","abstract_excerpt":"Let $G$ be a finite Abelian group. For a subset $S \\subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $= \\frac{1}{|S|^2}\\sum_{x,y \\in S} 1_S(x+y)$. For any $q \\ge 1$ and $\\alpha \\in [0,1]$, and any $S \\subseteq G$ with $|S| = \\frac{|G|}{q+\\alpha}$, we show $\\frac{T_3(S)}{|S|^2}$ and Prob[$S$] are bounded above by $\\max\\left(\\frac{q^2-\\alpha q+\\alpha^2}{q^2},\\frac{q^2+2\\alpha q+4\\alpha^2-6\\alpha+3}{(q+1)^2},\\gamma_0\\right)$, where $\\gamma_0 < 1$ is an absolute constant. 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