{"paper":{"title":"Dense sets without large sumsets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gabriel Dahia, Jo\\~ao Pedro Marciano, Victor Souza","submitted_at":"2026-07-16T17:57:34Z","abstract_excerpt":"We prove, for all fixed $0 < \\delta < 1$, and all sufficiently large $n$, that there exists $S \\subset [n]$ with $|S| \\ge \\delta n$ such that $A + B \\not \\subset S$ for all ${A, B \\subset \\mathbb{N}}$ satisfying $$\\min\\big\\{|A|, |B|\\big\\} \\ge \\big(3 + o(1)\\big) \\frac{\\log n }{ \\log (1 / \\delta)}.$$ A very recent result of Hern\\'andez and Hetzel shows that our bound is sharp up to a factor of 3, and together our results settle a conjecture of Kra, Moreira, Richter, and Robertson. In fact, we prove that a $\\delta$-dense random subset of $[n]$ is a valid choice for $S$ with high probability, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15269","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/2607.15269/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"}