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OP-12100OpenCombinatorics

Sum-product exponent for the reals (C84bC_{84b})

Sum-product exponent for the reals: determine C84bC_{84b} or improve its bounds; the repository records 43+104407≈1.3356\frac{4}{3}+\frac{10}{4407}\approx 1.3356 (lower) and <2<2 (1.999281*) (upper).

MSC 11B30 · Arithmetic combinatorics; higher degree uniformityMSC 11B13 · Additive bases, including sumsets

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

For a finite set A⊂RA \subset \mathbb{R} write

A+A={a+b:a,b∈A},AA={ab:a,b∈A}A+A = \{ a+b : a,b \in A \}, \qquad AA = \{ ab : a,b \in A \}

for the sumset and product set. The (real) sum-product exponent is

C84b:=lim inf⁡n→∞ min⁡A⊂R∣A∣=nlog⁡max⁡(∣A+A∣,∣AA∣)log⁡n.C_{84b} := \liminf_{n \to \infty}\ \min_{\substack{A \subset \mathbb{R} \\ \lvert A\rvert = n}} \frac{\log \max(\lvert A+A\rvert, \lvert AA\rvert)}{\log n}.

Equivalently, C84bC_{84b} is the largest ss such that max⁡(∣A+A∣,∣AA∣)≫ϵ∣A∣s−ϵ\max(\lvert A+A\rvert, \lvert AA\rvert) \gg_\epsilon \lvert A\rvert^{s-\epsilon} for every ϵ>0\epsilon > 0 and every finite A⊂RA \subset \mathbb{R}.

Erdős and Szemerédi [ErSz83] conjectured (originally for A⊂ZA \subset \mathbb{Z}, but also for A⊂RA \subset \mathbb{R}) that C84b=2C_{84b} = 2: at least one of the sumset or product set must be of near-maximal size ∣A∣2−o(1)\lvert A\rvert^{2-o(1)}. This sum-product conjecture for the reals was disproved in May 2026 by Bloom, Sawin, Schildkraut and Zhelezov [BSSZ2026], who showed C84b<2C_{84b} < 2.

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Background

  • The sum-product conjecture is studied over many other rings — Z\mathbb{Z}, C\mathbb{C}, the quaternions, finite fields Fp\mathbb{F}_p and Fq\mathbb{F}_q, pp-adics Qp\mathbb{Q}_p, function fields, and matrix rings — each with its own chronology of bounds. To avoid clutter we do not reproduce these here; Thomas Bloom maintains a comprehensive and up-to-date history of all variants at [Bloom-notes]. It is a notorious open problem whether C84bC_{84b} equals the corresponding integer exponent c(Z)c(\mathbb{Z}).
  • The same paper [BSSZ2026] also disproves the many sums and products conjecture of Erdős (constructing, for each k≥3k \ge 3, arbitrarily large A⊂RA \subset \mathbb{R} with max⁡(∣kA∣,∣A(k)∣)≤∣A∣Clog⁡k/log⁡log⁡k\max(\lvert kA\rvert, \lvert A^{(k)}\rvert) \le \lvert A\rvert^{C \log k/\log\log k}), gives new lower bounds for the number of solutions to linear equations in multiplicative groups and to the unit equation, and produces analogous counterexamples over Qp\mathbb{Q}_p, Fp\mathbb{F}_p, and function fields. In small characteristic the function-field exponents are much smaller — e.g. for q=1024q = 1024 one gets max⁡(∣A+A∣,∣AA∣)≤∣A∣1.906\max(\lvert A+A\rvert, \lvert AA\rvert) \le \lvert A\rvert^{1.906}.
  • The disproof was directly inspired by the OpenAI counterexample to the Erdős unit distance conjecture (84a); both rely on totally real (or split) number fields of large degree with discriminant O(1)dO(1)^d, coming from Martinet's class field towers and a Golod–Shafarevich argument. The authors note that, surprisingly, the sum-product construction required less number theory than the unit distance one (in particular it does not need small split primes).
  • The analogous problem for finite sets of integers A⊂ZA \subset \mathbb{Z} — Erdős Problem #52, [EP52] — is technically still open: although [BSSZ2026] disproves the conjecture for A⊂RA \subset \mathbb{R}, the construction uses algebraic integers in number fields of degree growing with ∣A∣\lvert A\rvert, and the integer case is unaffected. The best known integer lower bound is still Cushman's ∣A∣4/3+10/4407\lvert A\rvert^{4/3 + 10/4407} [Cu25].

References (from the repository page)

  • [ErSz83] Erdős, Paul; Szemerédi, Endre. On sums and products of integers. In: Studies in Pure Mathematics, 213–218, Birkhäuser, Basel, 1983.
  • [El97] Elekes, György. On the number of sums and products. Acta Arithmetica 81 (1997), no. 4, 365–367.
  • [So05] Solymosi, József. On the number of sums and products. Bulletin of the London Mathematical Society 37 (2005), no. 4, 491–494.
  • [So09] Solymosi, József. Bounding multiplicative energy by the sumset. Advances in Mathematics 222 (2009), no. 2, 402–408.
  • [KoSh16] Konyagin, Sergei V.; Shkredov, Ilya D. New results on sums and products in R\mathbb{R}. Trudy Mat. Inst. Steklova 294 (2016), 87–98; translation in Proc. Steklov Inst. Math. 294 (2016), 78–88.
  • [Sh19] Shakan, George. On higher energy decompositions and the sum-product phenomenon. Mathematical Proceedings of the Cambridge Philosophical Society 167 (2019), no. 3, 599–617.
  • [RuSt22] Rudnev, Misha; Stevens, Sophie. An update on the sum-product problem. Mathematical Proceedings of the Cambridge Philosophical Society 173 (2022), no. 2, 411–430. arXiv:2005.11145.
  • [Bl25] Bloom, Thomas F. Sum-product estimate over the reals (2025). See [Bloom-notes] for the precise reference.
  • [Cu25] Cushman, Adam. A note on the sum-product problem and the convex sumset problem. arXiv:2512.13849 (2025).
  • [BSSZ2026] Bloom, Thomas F.; Sawin, Will; Schildkraut, Carl; Zhelezov, Dmitrii. The sum-product conjecture is false for real numbers. arXiv:2605.28781 (2026).
  • [Bloom-notes] Bloom, Thomas F. A history of the sum-product problem. thomasbloom.org/notes/sumproduct.html (accessed 28 May 2026).
  • [Al26] Althoefer, Ingo. Improved constant for [BSSZ2026]. Note (28 May 2026), with tex and pdf.
  • [EPF52] Bloom, Thomas F. (ed.). Erdős Problem #52: Discussion thread. erdosproblems.com/forum/thread/52 (accessed 29 May 2026).
  • [EP52] Bloom, Thomas F. Erdős Problem #52 (the integer sum-product conjecture, still open). erdosproblems.com/52 (accessed 29 May 2026).

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 43+104407≈1.3356\frac{4}{3}+\frac{10}{4407}\approx 1.3356; best known upper bound <2<2 (1.999281*). Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
22 [ErSz83] Erdős–Szemerédi constructed sets of integers with max⁡(∣A+A∣,∣AA∣)≤∣A∣2−c/log⁡log⁡∣A∣\max(\lvert A+A\rvert, \lvert AA\rvert) \le \lvert A\rvert^{2-c/\log\log\lvert A\rvert}, so the exponent is ≤2\le 2. This was conjectured to be sharp (the sum-product conjecture).
2−c2 - c for an absolute c>0c>0 [BSSZ2026] Bloom–Sawin–Schildkraut–Zhelezov disprove the conjecture by constructing arbitrarily large A⊂RA \subset \mathbb{R} — algebraic integers in totally real number fields of degree ≍log⁡∣A∣\asymp \log\lvert A\rvert — with max⁡(∣A+A∣,∣AA∣)≤∣A∣2−c\max(\lvert A+A\rvert, \lvert AA\rvert) \le \lvert A\rvert^{2-c}. A non-optimized explicit version of the argument gives c≥0.00000087c \ge 0.00000087, i.e. exponent ≤1.99999913\le 1.99999913 (the authors stress this value "should not be taken too seriously").
1.9992811.999281 [Al26], [EPF52] Althoefer (28 May 2026), posted on the Erdős Problems forum [EPF52]; a ChatGPT 5.5 long-thinking optimization of the explicit constant of [BSSZ2026, §5] giving c≥0.000719c \ge 0.000719. Unverified.

Known lower bounds

Bound Reference Comments
11 Trivial ∣A+A∣≥2∣A∣−1\lvert A+A\rvert \ge 2\lvert A\rvert - 1.
5/4=1.255/4 = 1.25 [El97] Elekes, via the Szemerédi–Trotter incidence theorem.
14/11≈1.272714/11 \approx 1.2727 [So05] Solymosi.
4/3≈1.333334/3 \approx 1.33333 [So09] Solymosi; the long-standing "4/34/3 barrier".
≈1.33338\approx 1.33338 [KoSh16] Konyagin–Shkredov; first to break the 4/34/3 barrier (their 2015 and 2016 papers give ≈1.33338\approx 1.33338 and ≈1.33384\approx 1.33384).
≈1.33428\approx 1.33428 [Sh19] Shakan.
15581167≈1.33504\tfrac{1558}{1167} \approx 1.33504 [RuSt22] Rudnev–Stevens.
1270951≈1.33543\tfrac{1270}{951} \approx 1.33543 [Bl25] Bloom.
43+104407≈1.335602\tfrac{4}{3} + \tfrac{10}{4407} \approx 1.335602 [Cu25] Cushman; current record.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Sum-product exponent for the reals ($C_{84b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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