[2605.28781] The sum-product conjecture is false for real numbers
0$ is an absolute constant.<br>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 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."/>
0$ 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 $C>0$. 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." />
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Mathematics > Number Theory
arXiv:2605.28781 (math)
[Submitted on 27 May 2026]
Title:The sum-product conjecture is false for real numbers
Authors:Thomas F Bloom, Will Sawin, Carl Schildkraut, Dmitrii Zhelezov<br>View a PDF of the paper titled The sum-product conjecture is false for real numbers, by Thomas F Bloom and 3 other authors
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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 $c>0$ 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 $C>0$. 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.
Comments:<br>25 pages
Subjects:
Number Theory (math.NT); Combinatorics (math.CO)
Cite as:<br>arXiv:2605.28781 [math.NT]
(or<br>arXiv:2605.28781v1 [math.NT] for this version)
https://doi.org/10.48550/arXiv.2605.28781
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arXiv-issued DOI via DataCite (pending registration)
Submission history<br>From: Thomas Bloom [view email]<br>[v1]<br>Wed, 27 May 2026 17:42:41 UTC (29 KB)
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