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Updated Oct 8

Problem 52

Statement. Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$

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

Status. Open.

Source. erdosproblems.com/52, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #52, https://www.erdosproblems.com/52.

References.

[BSSZ26] T. F. Bloom, W. Sawin, C. Schildkraut, D. Zhelezov, The sum-product conjecture is false for real numbers. arXiv:2605.28781 (2026).
[BaLu19] Abdul Basit, Ben Lund, An improved sum-product bound for quaternions. SIAM J. Discrete Math. 33 (2019), no. 2, 1044-1060.
[Cu25] A. Cushman, A Note on the Sum-Product Problem and the Convex Sumset Problem. arXiv:2512.13849 (2025).
[Er91] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.
[ErSz83] Erdős, P. and Szemerédi, E., On sums and products of integers. Studies in pure mathematics (1983), 213-218.
[MoSt23] Mohammadi, Ali and Stevens, Sophie, Attaining the exponent 5/4 for the sum-product problem in finite fields. Int. Math. Res. Not. IMRN (2023), 3516-3532.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (44)

Sumsets and Arithmetic Progressions Sumsets and Arithmetic Progressions More on the sum-product problem for integers with few prime factors additive_combinatorics/alon_2020_sums_products_ratios_along_edges_graph Theorem 5 (p. 3): small sumset and product set give few sums and ratios along a graph Theorem 7 and Corollary 8 (p. 5): sums and ratios along a graph from a small sumset and product set A low-energy decomposition theorem Theorem 1.1: every finite real set splits into low additive and low multiplicative energy parts Theorem 1.2: the best low-energy decomposition exponent lies between 1/3 and 31/33 An improved sum-product bound for quaternions Theorem 1.2 (p. 2): |A + A| + |AA| >> |A|^(4/3 + c) for finite sets of quaternions additive_combinatorics/bloom_2026_sum_product_conjecture_is_false_real Sum-product theorems in algebraic number fields Corollary 12 (p. 20): for bounded-degree algebraic numbers some l-fold product set or l-fold sumset exceeds N^m Corollary 14 (p. 23): l-fold product sets of bounded-degree, bounded-height algebraic integers are nearly full Proposition 10 (p. 17): small multiplicative doubling bounds every 2q-fold additive energy of a bounded-degree set Proposition 13 (p. 22): multiplicative energy bound for algebraic integers of bounded degree and bounded height Proposition 6 (p. 10): small multiplicative doubling puts a large part of a bounded-degree set in one bounded-degree field Theorem 11 (p. 19): few products force many sums for algebraic numbers of bounded degree additive_combinatorics/chang_2003_erdos_szemeredi_problem_sum_set_product_set Theorem 1 (p. 4): a finite set of positive integers with |AA| < α|A| has |2A| > 36^(-α)|A|^2 and |hA| > (2h^2-h)^(-hα)|A|^h additive_combinatorics/cushman_2025_note_sum_product_problem_convex_sumset additive_combinatorics/erdos_1983_sums_products_integers Lemma (p. 217): t integers in (m, 2m] give more than eps t^{1+alpha} distinct sums and products of pairs Theorem 1 (p. 213): n^{1+c_1} < f(n) < n^2 exp(-c_2 log n / log log n) for the least number of sums and products of n positive integers The sum-product problem for integers with few prime factors Autonomous disproofs of the sum-product conjecture over the real numbers with GPT-5.5 Pro Theorem 1 (p. 2): finite real sets with both sumset and product set at most |A|^(2-c)additive_combinatorics/mohammadi_2023_attaining_exponent_5_4_sum_product More sum-product type counterexamples: products with shifts and $AA+A$Bounding multiplicative energy by the sumset Corollary 2.2 (p. 2): max(|A+A|, |AA|) >= |A|^(4/3) / (2 ceil(log |A|)^(1/3)) for finite sets of positive reals Mapping Incidences Theorem 1.1: a finite part of a characteristic-zero integral domain maps to Z/pZ for a positive-density set of primes, keeping prescribed elements nonzero Theorem 3.2: max(|A+A|, |AA|) is at least C|A|^(14/13)(log|A|)^alpha in any characteristic-zero integral domain Query complexity and the polynomial Freiman-Ruzsa conjecture Lemma 4.1 (p. 8): every rooted tree has a large low-branching subtree or a large binary subtree Proposition 1.5 (p. 6): integer sets with |kA| + |A^(k)| <= |A|^(C log k / log log k)Theorem 1.1 (p. 3): a set of doubling K has a K^(-2/eps)-dense subset of coordinate query complexity at most eps log_2 |A|Theorem 1.2 (p. 4): few products, many sums, |kA| >>_k |A|^(k - 2 eps k log_2 k) K_*^(-2k/eps) for integer sets Theorem 1.3 (p. 5): lambda_k(A) is at most 10 beta_*(A)^(1/eps) |A|^(2 eps log_2 k) for integer sets Theorem 1.4 (p. 5): iterated sum-product, max(|kA|, |A^(k)|) >= |A|^(c log_2 k / log_2 log_2 k) for integer sets discrete_geometry/erdos_1981_applications_graph_theory_combinatorial_methods_number Sums and products, p. 146: the Erdős–Szemerédi bounds (1) and the bound (2) on all subset sums and products

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