Humans have disproved the sum-product conjectures for real numbers
Researchers have disproved the sum-product conjecture for real numbers, presenting new mathematical constructions. They demonstrated that for certain sets of algebraic integers, the conjecture does not hold true. Additionally, they addressed related conjectures and provided new insights into solutions for linear equations in multiplicative groups.
- ▪The sum-product conjecture is shown to be false for real numbers by constructing specific sets of algebraic integers.
- ▪For any k greater than or equal to 3, the researchers disproved the many sums and products conjecture.
- ▪The study also offers new lower bounds for the number of solutions to linear equations in multiplicative groups.
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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 View a PDF of the paper titled The sum-product conjecture is false for real numbers, by Thomas F Bloom and 3 other authors View PDF HTML (experimental) 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.
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