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The square-root scale $k=\\sqrt N$ is a natural transition point. Truss proved \\[\n  f(n^2,n)>n+\\frac12 n^{1/2}-2. \\] We improve the leading constant in the second-order term, proving \\[\n  f(n^2,n)\\ge n+\\left(\\frac1{\\sqrt2}+o(1)\\right)n^{1/2}. \\] On the upper-bound side, Brown and Freedman pr"},"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":"2606.01780","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-01T07:02:08Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"4426db3621107f13b108ae5a5fb90e6d7be04a6677fd4e53f0868a69cc99d7da","abstract_canon_sha256":"62ec1dbc31d49380f3062591ae56bdf09484f90f3d32ff62616a9796bdc8e607"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:56.522027Z","signature_b64":"iXp0HQzZZaeja6SbYWRrXZU6Rc2Iu1pzAXATv158kE0RUL/gJG9qj8nuPVhS31D19j7yLkApv5VbZiSHNSHEDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b47bbd90f43ad96ea71507735bef4103a68e8029c97dd9e96f468848522030d8","last_reissued_at":"2026-06-02T02:04:56.521676Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:56.521676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hitting Arithmetic Progressions at the Square-Root Scale","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Samuel Korsky","submitted_at":"2026-06-01T07:02:08Z","abstract_excerpt":"For positive integers $N$ and $k$, let $f(N,k)$ be the minimum size of a set $A\\subseteq\\{0,1,\\ldots,N-1\\}$ which intersects every $k$-term arithmetic progression contained in $\\{0,1,\\ldots,N-1\\}$. 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