OP-00984 Open

Polynomial bounds for a large set S with S^8 inside A^4 in approximate groups

Let $A$ be a $K$-approximate group in a group that need not be abelian, and write $S^8$ and $A^4$ for iterated product sets. By work of Croot and Sisask and of Sanders there is a set $S$ with $|S| \gg_K |A|$ and $S^8 \subset A^4$, but not with a polynomial dependence on $K$. The version that also requires $S \subset A$ is false, even with $|S| \gg_K |A|$ only (Kitamura); the question of Breuillard, Green and Tao puts no such condition on $S$.

Problem. Is there a set $S$ with $|S| \gg K^{-O(1)} |A|$ and $S^8 \subset A^4$?

Source: E. Breuillard, B. Green and T. Tao, Small doubling in groups, Erdős Centennial (2013), Problem 6.6. Listed as Problem 29 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: GR — Group Theory
Secondary: CO
Keywords: approximate groups, Croot–Sisask lemma, nonabelian additive combinatorics, product sets
Literature checked: Oct. 26, 2026 by clemens-admin

References

@incollection{breuillard2013small,
  author = {Breuillard, Emmanuel and Green, Ben and Tao, Terence},
  title = {Small doubling in groups},
  booktitle = {Erd{\H o}s Centennial},
  series = {Bolyai Society Mathematical Studies},
  volume = {25},
  pages = {129--151},
  publisher = {Springer},
  year = {2013},
  doi = {10.1007/978-3-642-39286-3_5},
  eprint = {1301.7718},
  archivePrefix = {arXiv},
  note = {Theorem 6.5 and Problem 6.6}
}

@article{croot2010sisask,
  author = {Croot, Ernie and Sisask, Olof},
  title = {A probabilistic technique for finding almost-periods of convolutions},
  journal = {Geometric and Functional Analysis},
  volume = {20},
  number = {6},
  pages = {1367--1396},
  year = {2010},
  doi = {10.1007/s00039-010-0101-8}
}

@article{sanders2010nonabelian,
  author = {Sanders, Tom},
  title = {On a nonabelian {B}alog--{S}zemer{\'e}di-type lemma},
  journal = {Journal of the Australian Mathematical Society},
  volume = {89},
  number = {1},
  pages = {127--132},
  year = {2010},
  doi = {10.1017/S1446788710000236}
}

@misc{kitaken2026green29,
  author = {Kitamura, Kenta},
  title = {green-29-counterexample: a {L}ean counterexample to the version with $S \subset A$},
  howpublished = {\url{https://github.com/KitaKen1/green-29-counterexample}},
  year = {2026},
  note = {commit 1cbf80c; concerns the literal wording with $S \subset A$, not the question stated here}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 29; version of January 2026, most recent update December 2025},
  year = {2026}
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00984, "Polynomial bounds for a large set S with S^8 inside A^4 in approximate groups", clopen.solutions.