{"paper":{"title":"Bourgain-Chang's proof of the weak Erd\\H{o}s-Szemer\\'edi conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitrii Zhelezov","submitted_at":"2017-10-25T15:54:52Z","abstract_excerpt":"This is an exposition of the following `weak' Erd\\H{o}s-Szemer\\'edi conjecture for integer sets proved by Bourgain and Chang in 2004. For any $\\gamma > 0$ there exists $\\Lambda(\\gamma) > 0$ such that for an arbitrary $A \\subset \\mathbb{N}$, if $|AA| \\leq K|A|$ then $$E_{+}(A) \\leq K^{\\Lambda}|A|^{2+\\gamma}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09316","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"}