{"paper":{"title":"On a sumset conjecture of Erd\\H{o}s","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR","math.LO"],"primary_cat":"math.NT","authors_text":"Isaac Goldbring, Karl Mahlburg, Martino Lupini, Mauro Di Nasso, Renling Jin, Steven Leth","submitted_at":"2013-07-02T17:14:07Z","abstract_excerpt":"Erd\\H{o}s conjectured that for any set $A\\subseteq \\mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\\subseteq \\mathbb{N}$ such that $B+C\\subseteq A$. We verify Erd\\H{o}s' conjecture in the case that $A$ has Banach density exceeding $\\frac{1}{2}$. As a consequence, we prove that, for $A\\subseteq \\mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\\subseteq \\mathbb{N}$ such that $B+C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0767","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}