{"paper":{"title":"Sets of unit fractions without two members whose average is a unit fraction","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Will Sawin","submitted_at":"2026-07-16T19:44:56Z","abstract_excerpt":"We show that there is a constant $c>0$ such that, for all sufficiently large $N$, there is a subset $A \\subseteq \\{1,\\dots,N\\}$ of size $>cN$ such that for any two distinct elements $a,b$ in $A$, the average of $\\frac{1}{a}$ and $\\frac{1}{b}$ is not a unit fraction, negatively answering a question of Erd\\H{o}s and Graham. This also gives the best known lower bounds on the maximum size of a set of unit fractions without non-trivial three-term arithmetic progressions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15419","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/2607.15419/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"}