{"paper":{"title":"Large Sets of Integers with No Harmonic Triples","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Samuel Korsky","submitted_at":"2026-07-07T04:40:06Z","abstract_excerpt":"Let $f(N)$ denote the largest size of a set $A\\subseteq [N]=\\{1,\\ldots,N\\}$ containing no distinct $a,b,c$ such that \\[\n  \\frac2a=\\frac1b+\\frac1c . \\] We prove \\[\n  f(N)\\gg N\\exp\\!\\left(-(2\\sqrt{\\log(24/7)}+o(1))\\sqrt{\\log\\log N}\\right). \\] The construction filters the odd integers up to $N$ by a random affine image of a dense three-term-progression-free set in a prime field $\\mathbb{F}_q$ with $q\\asymp\\log N$, and then deletes a controlled family of collapsed triples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05823","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.05823/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}