The sum-product conjecture is false for real numbers

@misc{bloom2026sumproduct,
  title = {{The sum-product conjecture is false for real numbers}},
  author = {Thomas F. Bloom and Will Sawin and Carl Schildkraut and Dmitrii Zhelezov},
  howpublished = {arXiv (Cornell University)},
  year = {2026},
  month = may,
  doi = {10.48550/arxiv.2605.28781},
  url = {https://arxiv.org/abs/2605.28781},
  note = {CC BY},
}

TY  - UNPB
TI  - The sum-product conjecture is false for real numbers
AU  - Thomas F. Bloom
AU  - Will Sawin
AU  - Carl Schildkraut
AU  - Dmitrii Zhelezov
PY  - 2026
DA  - 2026/05/27
PB  - arXiv (Cornell University)
DO  - 10.48550/arxiv.2605.28781
UR  - https://arxiv.org/abs/2605.28781
AB  - We disprove the sum-product conjecture for real numbers by constructing arbitrarily large A⊂ ℝ (whose elements are algebraic integers in a number field of degree asymp loglvert Arvert) such that max(lvert A+Arvert ,lvert AArvert)≤ lvert Arvert^2−c where c>0 is an absolute constant. We also disprove the many sums and products conjecture by constructing, for any k≥ 3, arbitrarily large A⊂ ℝ such that max(lvert kArvert,lvert A^(k)rvert)≤ lvert Arvert^C(log k)/(loglog 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.
SN  - 2331-8422
ER  -