{"paper":{"title":"On product sets of arithmetic progressions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Max Wenqiang Xu, Yunkun Zhou","submitted_at":"2022-01-01T03:26:00Z","abstract_excerpt":"We prove that the size of the product set of any finite arithmetic progression $\\mathcal{A}\\subset \\mathbb{Z}$ satisfies\n  \\[|\\mathcal A \\cdot \\mathcal A| \\ge \\frac{|\\mathcal A|^2}{(\\log |\\mathcal A|)^{2\\theta +o(1)} } ,\\] where $2\\theta=1-(1+\\log\\log 2)/(\\log 2)$ is the constant appearing in the celebrated Erd\\H{o}s multiplication table problem. This confirms a conjecture of Elekes and Ruzsa from about two decades ago.\n  If instead $\\mathcal{A}$ is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that \\[|\\mathcal A \\cdot \\mathcal A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.00104","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.00104/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}