Open subsets of [0, 1] of measure above 1/3 contain a solution of xy = z
For every $\theta$ the set $A_\theta = \{x \in (0, 1) : \theta \log x \in (\frac13, \frac23) \pmod 1\}$ is open and product-free, and its measure is $t/(t^2 + t + 1)$ with $t = e^{1/(3\theta)}$, which tends to $\frac13$ as $\theta \to \infty$; so $\frac13$ cannot be lowered.
Problem. If $A \subset [0, 1]$ is open and has measure greater than $\frac13$, must there be $x, y, z \in A$ with $xy = z$?
Source: B. Green, 100 open problems (version of January 2026), Problem 3; it came up during the work of S. Eberhard, B. Green and F. Manners on sum-free sets.
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Secondary
- CO
- Keywords
- additive combinatorics, measure theory, product-free sets, sum-free sets
References
@article{eberhard2014sumfree,
author = {Eberhard, Sean and Green, Ben and Manners, Freddie},
title = {Sets of integers with no large sum-free subset},
journal = {Annals of Mathematics (2)},
volume = {180},
number = {2},
pages = {621--652},
year = {2014},
doi = {10.4007/annals.2014.180.2.5}
}
@misc{franchi2026productfree,
author = {Franchi, Leonardo and Gowers, W. T. and Yip, Fredy},
title = {Product-free subsets of $(0,1)$},
year = {2026},
eprint = {2607.06073},
archivePrefix = {arXiv}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 3; version of January 2026, most recent update December 2025},
year = {2026}
}
Resolution claims
The third problem in Ben Green's collection of 100 open problems asks whether an open subset of $(0,1)$ that does not contain $x,y,z$ with $xy=z$ must have measure at most 1/3. We give an affirmative answer to this question. As part of the proof we obtain a result of independent interest that gives a lower bound for the size of …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: arXiv:2607.06073v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01029, "Open subsets of [0, 1] of measure above 1/3 contain a solution of xy = z", clopen.solutions.