{"paper":{"title":"Counting sets with small sumset and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Ben Green, Robert Morris","submitted_at":"2013-05-14T09:42:47Z","abstract_excerpt":"We study the number of $k$-element sets $A \\subset \\{1,\\ldots,N\\}$ with $|A + A| \\leq K|A|$ for some (fixed) $K > 0$. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of $2^{o(k)} N^{o(1)}$ for most $N$ and $k$. As a consequence of this and a further new result concerning the number of sets $A \\subset \\mathbf{Z}/N\\mathbf{Z}$ with $|A +A| \\leq c |A|^2$, we deduce that the random Cayley graph on $\\mathbf{Z}/N\\mathbf{Z}$ with edge density~$\\frac{1}{2}$ has no clique or independent set of size greater than $\\big( 2 + "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.3079","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"}