{"paper":{"title":"A structure theorem for sets of small popular doubling","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Przemys{\\l}aw Mazur","submitted_at":"2015-06-01T11:16:08Z","abstract_excerpt":"In this paper we prove that every set $A\\subset\\mathbb{Z}$ satisfying the inequality $\\sum_{x}\\min(1_A*1_A(x),t)\\le(2+\\delta)t|A|$ for $t$ and $\\delta$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset $A\\subset\\mathbb{N}$ satisfies $|\\mathbb{N}\\setminus(A+A)|\\ge k$; specifically we show that $\\mathbb{P}(|\\mathbb{N}\\setminus(A+A)|\\ge k)=\\Theta(2^{-k/2})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00445","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":""},"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"}