{"paper":{"title":"Small sumsets in real line : a continuous 3k-4 theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anne de Roton","submitted_at":"2016-05-15T19:25:27Z","abstract_excerpt":"We prove a continuous Freiman's $3k-4$ theorem for small sumsets in $\\mathbb{R}$ by using some ideas from Ruzsa's work on measure of sumsets in $\\mathbb{R}$ as well as some graphic representation of density functions of sets. We thereby get some structural properties of $A$, $B$ and $A+B$ when $\\lambda(A+B)<\\lambda(A)+\\lambda(B)+\\min(\\lambda(A),\\lambda(B))$. We also give some structural information for sets of large density with small sumset and characterize the extremal sets for which equality holds in the lower bounds for $\\lambda(A+B)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04597","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"}