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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2201.00104","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-01-01T03:26:00Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f81abb105adef2653f18681ba1df8913ef7836940a7224013a54b4f099bcbc2c","abstract_canon_sha256":"a6726da7d8eced0ae46e19dbf3a2b8729cfb9942be511eed32790c7aa5002605"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:34:16.721611Z","signature_b64":"8J3sIIPzBr7lCqnkGV7GR1RMy+O+iqGjzDTqf1mzPmbQD7nHD8dOEz/EldKBm4KtjTcdFqV7fO8jfHMwCNFbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98a891264f9a6b5071386c0e435810b294773a63b6fa6145cdaa2d736d968265","last_reissued_at":"2026-07-05T06:34:16.721154Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:34:16.721154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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