{"ID":2941027,"CreatedAt":"2026-06-02T07:21:46.982601397Z","UpdatedAt":"2026-06-02T12:00:17.69116927Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2605.28781","arxiv_id":"2605.28781","title":"The sum-product conjecture is false for real numbers","abstract":"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 $c\u003e0$ is an absolute constant. 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 $C\u003e0$. We obtain similar constructions for $p$-adics, finite fields, and function fields in positive characteristic, and also obtain new lower bounds for the number of solutions to linear equations in a multiplicative group and the number of solutions to the unit equation in sufficiently many variables.","short_abstract":"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 $c\u003e0$ is an absolute consta...","url_abs":"https://arxiv.org/abs/2605.28781","url_pdf":"https://arxiv.org/pdf/2605.28781v1","authors":"[\"Thomas F Bloom\",\"Will Sawin\",\"Carl Schildkraut\",\"Dmitrii Zhelezov\"]","published":"2026-05-27T17:42:41Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"math.CO\"]","methods":"[]","has_code":false}