Erdős–Szemerédi sum–product problem: is $\max(|A+A|,|AA|)\gg |A|^{2-\epsilon}$ for integer sets? (Erdős #52)
Statement
Let $A$ be a finite set of integers, and write $A+A=\{a+b : a,b\in A\}$ (the sumset) and $AA=\{ab : a,b\in A\}$ (the product set). Is it true that for every $\epsilon>0$ $$\max(\lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$$
Acceptance. FULLY RESOLVES: a proof of the conjecture for finite sets of integers, for every $\epsilon>0$ — machine-checkable (Lean/Coq) preferred, otherwise a complete written proof; OR a disproof: an explicitly described infinite family of finite integer (or rational) sets $A$ of unbounded size with a proof that $\max(\lvert A+A\rvert,\lvert AA\rvert)\leq\lvert A\rvert^{2-c}$ for some fixed $c>0$ (finitely many examples cannot suffice, since the statement is asymptotic). ADVANCES: a proven lower-bound exponent for integer or real sets strictly exceeding the best exponent stated in the background; a strict improvement of the complex-set or finite-field bounds stated in the background; a quantitative barrier theorem showing BSSZ26-type constructions cannot exist over the rationals (or constructing rational ones with $\max\leq\lvert A\rvert^{2-c}$, which would resolve the problem); or exhaustively computed extremal values of $\min_{\lvert A\rvert=n}\max(\lvert A+A\rvert,\lvert AA\rvert)$ for integer sets for new $n$, each with an optimal example and a reproducible exhaustiveness certificate. Deliver the proof file, or the construction/values plus verification code.
Background
The sum–product problem of Erdős and Szemerédi [ErSz83], reiterated by Erdős in many papers ([Er77c] through [Va99, 1.26]); Erdős offered \$250 for a solution. Listed as open on erdosproblems.com/52 (fetched 2026-07-13, status 'open', tagged 'number theory | additive combinatorics'). Erdős and Szemerédi proved a lower bound of $\lvert A\rvert^{1+c}$ for some $c>0$ and an upper bound of $\lvert A\rvert^2\exp(-c\log\lvert A\rvert/\log\log\lvert A\rvert)$ (so the $\epsilon$ in the conjecture cannot be removed). After many improvements, the record lower bound for integer (and real) sets is $\max(\lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{1962/1469-o(1)}$, due to Cushman [Cu25] (note $1962/1469\approx 1.3356$); a complete history is maintained at thomasbloom.org/notes/sumproduct.html. A dramatic recent development: Bloom, Sawin, Schildkraut, and Zhelezov [BSSZ26] DISPROVED the analogous conjecture for finite sets of reals, constructing arbitrarily large $A\subset\mathbb{R}$ with $\max(\lvert A+A\rvert,\lvert AA\rvert)\leq\lvert A\rvert^{2-c}$ for an absolute $c>0$; variants of the construction also disprove it for small subsets of $\mathbb{F}_p$ and kill the higher-fold real conjecture $\max(\lvert mA\rvert,\lvert A^m\rvert)\gg\lvert A\rvert^{m-o(1)}$ of [ErSz83, Er91]. Since both $\lvert A+A\rvert$ and $\lvert AA\rvert$ are invariant under dilation, the integer case is equivalent to the rational case — so the BSSZ26 counterexamples are essentially irrational, and any proof of the integer conjecture must genuinely exploit integrality/rationality. In other settings: $\max\gg\lvert A\rvert^{4/3+c}$ for complex sets (Basit–Lund [BaLu19]) and $\max\gg\lvert A\rvert^{5/4+o(1)}$ for $A\subseteq\mathbb{F}_p$ with $\lvert A\rvert<p^{c}$ (Mohammadi–Stevens [MoSt23]). Related: Erdős #53, #808, #818 (erdosproblems.com/53, /808, /818); OEIS A263996 is listed as related. The attacker's tools: incidence-geometry proof machinery for exponent improvements; adapting the BSSZ26 construction toward or away from rational sets; and exhaustive small-$n$ extremal computations of $\min_{|A|=n}\max(\lvert A+A\rvert,\lvert AA\rvert)$ with certificates.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #52 (T. F. Bloom) | website |
| REF-02 | T. F. Bloom — complete history of sum-product bounds | website |
| REF-03 | OEIS A263996 — sequence related to the sum-product problem (Erdős #52) | website |
| REF-04 | Lean formalisation of Erdős Problem #52 (formal-conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.