Additive bases of cyclic groups of nearly optimal size
If $A \subset \mathbb{Z}/q\mathbb{Z}$ satisfies $A + A = \mathbb{Z}/q\mathbb{Z}$, then $|A| \ge (\sqrt{2} + o(1)) q^{1/2}$, since $A + A$ has at most $|A|(|A|+1)/2$ elements. For prime $q$, Granville, and more recently Caprace and de la Harpe, asked whether this bound can be attained.
Problem. Do infinitely many moduli $q$ admit a set $A \subset \mathbb{Z}/q\mathbb{Z}$ with $A + A = \mathbb{Z}/q\mathbb{Z}$ and $|A| = (\sqrt{2} + o(1)) q^{1/2}$?
Source: B. Green, 100 open problems (version of January 2026), Problem 33; for prime moduli the question is Granville's, and Caprace and de la Harpe's (Question 6.1).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- NT — Number Theory
- Secondary
- CO
- Keywords
- additive bases, cyclic groups, Sidon sets, sumsets
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
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author = {Croot, Ernie and Lev, Vsevolod F.},
title = {Open problems in additive combinatorics},
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publisher = {American Mathematical Society},
year = {2007},
doi = {10.1090/crmp/043/10}
}
@article{caprace2020harpe,
author = {Caprace, Pierre-Emmanuel and de la Harpe, Pierre},
title = {Groups with irreducibly unfaithful subsets for unitary representations},
journal = {Confluentes Mathematici},
volume = {12},
number = {1},
pages = {31--68},
year = {2020},
doi = {10.5802/cml.61},
eprint = {1807.04992},
archivePrefix = {arXiv}
}
@article{jia2017shen,
author = {Jia, Xingde and Shen, Jian},
title = {Extremal bases for finite cyclic groups},
journal = {SIAM Journal on Discrete Mathematics},
volume = {31},
number = {2},
pages = {796--804},
year = {2017},
doi = {10.1137/15M1013237}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 33; 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 It will not change if the problem is solved, reclassified or withdrawn.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-00938, "Additive bases of cyclic groups of nearly optimal size", clopen.solutions.