Erdős–Szemerédi sum-product conjecture

Let $A$ be a finite subset of $\mathbb{C}$ . For $g\in\mathbb{N}$ , define

gA=\{a_1+\dots+a_g:a_1,\dots,a_g\in A\},\qquad A^{(g)}=\{a_1\dots a_g:a_1,\dots,a_g\in A\}.

Erdős–Szemerédi conjecture. For any $g\in\mathbb{N}$ , any $\varepsilon>0$ and any set $A\subseteq\mathbb{C}$ , one should have

\max\{\lvert gA\rvert,\lvert A^{(g)}\rvert\}\gg_{g,\varepsilon}\lvert A\rvert^{g-\varepsilon}.

The conjecture asserts that additive and multiplicative structure cannot coexist too strongly. It generalizes the assertion that either the sumset or product set of a finite complex set must be close in size to $\lvert A\rvert^2$ ; the supplied text gives no resolution status.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Erdős–Szemerédi sum-product conjecture
Let $A\subset \mathbb{N}$ be a finite set. For a finite set $S$ , write $S+S=\{s_1+s_2:s_1,s_2\in S\}$ and $S\cdot S=\{s_1s_2:s_1,s_2\in S\}$ . Write $a(x)\gg b(x)$ when $a(x)\geq c\,b(x)$ for some constant $c>0$ and all relevant $x$ . Erdős–Szemerédi conjecture. For every $\delta<1$ ,
$\max\{|A+A|,|A\cdot A|\}\gg |A|^{1+\delta}.$
This is a classical sum-product conjecture in additive combinatorics; the source uses it to motivate lower bounds for sets of spectral radii.
source: J. A. Dias da Silva and Pedro J. Freitas, “Counting Spectral Radii of Matrices with Positive Entries”, arXiv:1305.1139 (2013).

Sources & referencesView supporting material

Primary source

Joseph Harrison, Akshat Mudgal and Harry Schmidt, “Uniform sum-product phenomenon for algebraic groups and Bremner's conjecture”, arXiv:2603.06483 (2026).

Additional references

26 papers in this index state this conjecture (2006–2026). The statement above is taken from the most recent of them; the others are arXiv:2601.03797, arXiv:2512.13849, arXiv:2410.20618, arXiv:2310.07964, arXiv:2303.15910, arXiv:2207.08221, arXiv:2111.03586, arXiv:2109.04932, arXiv:2102.05446, arXiv:2010.01377, arXiv:2005.09893, arXiv:2005.11145, and 13 more.

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Erdős–Szemerédi sum-product conjecture

Equivalent formulations 1