The sum-product problem for integers

A finite set of integers cannot be both additively and multiplicatively structured: either its sums or its products should be numerous. Erdős and Szemerédi conjectured that one of the two sets is almost as large as possible. In 2026 the analogous statement for real numbers was disproved, so the integer case now carries extra interest.

Problem

For a finite set $A$ of integers write $A+A=\{a+b: a,b\in A\}$ and $AA=\{ab: a,b\in A\}$. Is it true that for every $\varepsilon>0$

$$\max(\lvert A+A\rvert,\lvert AA\rvert)\gg_\varepsilon \lvert A\rvert^{2-\varepsilon}$$

for all finite $A\subset\mathbb{Z}$?

Erdős offered $250 for this problem. Erdős and Szemerédi also conjectured the $m$-fold version $\max(\lvert mA\rvert,\lvert A^{m}\rvert)\gg\lvert A\rvert^{m-o(1)}$ for integers.

Background

Erdős and Szemerédi posed the problem in 1983, proving the first non-trivial lower bound $\lvert A\rvert^{1+c}$ and showing by an example that the exponent 2 cannot be reached exactly: there are sets with $\max(\lvert A+A\rvert,\lvert AA\rvert)\le\lvert A\rvert^{2}\exp(-c\log\lvert A\rvert/\log\log\lvert A\rvert)$. The question has since been studied over the reals, the complex numbers and finite fields as well. In May 2026 Bloom, Sawin, Schildkraut and Zhelezov showed that the conjecture fails for real numbers: they built arbitrarily large sets of algebraic integers in number fields of degree about $\log\lvert A\rvert$ with $\max(\lvert A+A\rvert,\lvert AA\rvert)\le\lvert A\rvert^{2-c}$. Their sets are not sets of rational integers, and the original question remains open.

What is known

• Lower bound. The best published exponent for integers, and also for reals, is $\tfrac43+\tfrac{10}{4407}=\tfrac{1962}{1469}\approx 1.3356$, due to Cushman (2025):

  $$\max(\lvert A+A\rvert,\lvert AA\rvert)\gg_\varepsilon\lvert A\rvert^{1962/1469-\varepsilon}.$$

  A September 2026 preprint of Zhang, Wang and Zeng claims the exponent $135/101\approx1.3366$ for real sets; it has not yet been refereed.
• Other settings. For complex numbers the best exponent is $\tfrac43+c$ (Basit and Lund, 2019); for small sets in $\mathbb{F}_p$ it is $\tfrac54$ (Mohammadi and Stevens, 2023).
• Counterexamples elsewhere. Over $\mathbb{R}$, the $p$-adics, finite fields and function fields of positive characteristic the conjecture is false (Bloom, Sawin, Schildkraut and Zhelezov, 2026); their construction also disproves the $m$-fold version over the reals.
• Why the integers may differ. The real counterexamples need number fields whose degree grows with $\lvert A\rvert$. Lemley and Chew (October 2026, preprint) showed that analogous constructions inside a fixed number field, as integers would require, always satisfy the conjecture.

What would count as progress

• A lower bound $\lvert A\rvert^{\gamma}$ for integer sets with $\gamma$ above the current exponents, with a proof that can be checked line by line.
• A construction of integer sets with $\max(\lvert A+A\rvert,\lvert AA\rvert)\le\lvert A\rvert^{2-c}$ for an absolute $c>0$, which would disprove the conjecture; such sets can be verified directly by computing both sets.
• An argument that uses the arithmetic of $\mathbb{Z}$ (for example unique factorisation) to separate the integer case from the real case.
• A formal proof against the Lean statement in formal-conjectures.

References

1. P. Erdős and E. Szemerédi, On sums and products of integers, in Studies in Pure Mathematics, Birkhäuser (1983), 213–218. doi:10.1007/978-3-0348-5438-2_19
2. T. F. Bloom, W. Sawin, C. Schildkraut and D. Zhelezov, The sum-product conjecture is false for real numbers (2026). arXiv:2605.28781
3. A. Cushman, A note on the sum-product problem and the convex sumset problem (2025). arXiv:2512.13849
4. A. Basit and B. Lund, An improved sum-product bound for quaternions, SIAM J. Discrete Math. 33 (2019), 1044–1060. doi:10.1137/18M1231468
5. A. Mohammadi and S. Stevens, Attaining the exponent 5/4 for the sum-product problem in finite fields, Int. Math. Res. Not. 2023 (2023), 3516–3532. doi:10.1093/imrn/rnab338
6. K. Zhang, Y. Wang and Y. Zeng, A refined sum-product estimate via higher energies (2026). arXiv:2609.25711
7. H. Lemley and D. Chew, The sum-product conjecture holds for fixed-field lattice product constructions (2026). arXiv:2610.26737

Source

Imported from Erdős Problems 52, curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.