The sum-product problem. Erdős and Szemerédi [ErSz83] proved a lower bound of $\lvert A\rvert^{1+c}$ for some constant $c>0$, and an upper bound of\[\lvert A\rvert^2 \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right)\]for some constant $c>0$. The lower bound has been improved a number of times. The current record is\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{1962}{1469}-o(1)}\]due to Cushman [Cu25] (note $1962/1469=1.3356\cdots$). A complete history of sum-product bounds can be found at this webpage.

There is likely nothing special about the integers in this question, and indeed Erdős and Szemerédi also ask a similar question about finite sets of real or complex numbers. The current best lower bound for sets of reals is the same bound of Cushman above; the original conjecture is false for sets of reals, see below. The best bound for complex numbers is\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{4}{3}+c}\]for some absolute constant $c>0$, due to Basit and Lund [BaLu19].


One can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that there exists $c>0$ such that if $A\subseteq \mathbb{F}_p$ with $\lvert A\rvert <p^{c}$ then\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{5}{4}+o(1)},\]due to Mohammadi and Stevens [MoSt23].

There is also a natural generalisation to higher-fold sum and product sets. For example, in [ErSz83] (and in [Er91]) Erdős and Szemerédi also conjecture that for any $m\geq 2$ and finite set of integers $A$\[\max( \lvert mA\rvert,\lvert A^m\rvert)\gg \lvert A\rvert^{m-o(1)}.\]The conjecture is false for $A\subset \mathbb{R}$, as proved by Bloom, Sawin, Schildkraut, and Zhelezov [BSSZ26], who construct arbitrarily large $A$ such that\[\max(\lvert A+A\rvert, \lvert AA\rvert) \leq \lvert A\rvert^{2-c}\]for some absolute constant $c>0$. A similar construction also disproves this for small subsets of $\mathbb{F}_p$, and disproves the above higher-fold variant for sets of real numbers.

See [53] for more on this generalisation and [808] for a stronger form of the original conjecture. See also [818] for a special case.

View the LaTeX source

This page was last edited 28 May 2026. View history