{"paper":{"title":"The sum-product conjecture is false for real numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Carl Schildkraut, Dmitrii Zhelezov, Thomas F Bloom, Will Sawin","submitted_at":"2026-05-27T17:42:41Z","abstract_excerpt":"We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\\subset \\mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\\asymp \\log\\lvert A\\rvert$) such that \\[\\max(\\lvert A+A\\rvert ,\\lvert AA\\rvert)\\leq \\lvert A\\rvert^{2-c}\\] where $c>0$ is an absolute constant.\n  We also disprove the many sums and products conjecture by constructing, for any $k\\geq 3$, arbitrarily large $A\\subset \\mathbb{R}$ such that \\[\\max(\\lvert kA\\rvert,\\lvert A^{(k)}\\rvert)\\leq \\lvert A\\rvert^{C\\frac{\\log k}{\\log\\log k}}\\] for some constant $C>0$. We obtain sim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28781","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.28781/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"}