The sum-product conjecture is false for real numbers
- Published
- Source
- arXiv
- Paper number
- 258
- Field
- Research
- arXiv ID
- 2605.28781
Key points
- A counterexample to the sum-product conjecture over the reals (1.1), constructing sets A with both |A+A| and |AA| at most |A|^{2-c}.
- A higher-dimensional Balog-Wooley-type structure using algebraic integers from totally real number fields.
- A counterexample to the many sums and products conjecture as well: max(|kA|, |A^(k)|) ≤ |A|^{C log k / log log k}.
- Variants over various rings, including p-adics, the finite field Fp, and the positive-characteristic function field Fq((t)).
- New lower bounds for the number of solutions to the linear equation x1+...+xk=1 in multiplicative groups, along with applications to unit equations.
- GPT-5.5 Pro was used for initial exploration, but the final proof was generated mostly by humans, inspired by the OpenAI counterexample to the unit-distance conjecture.
Paper links
External research summaries. These are not HDATF publications or measured product results.