A long gap in a dilate of a square-root-sized set modulo a prime
Let $p$ be a prime and $A \subset \mathbb{Z}/p\mathbb{Z}$ a set of size $\sqrt{p}$. A gap of length $L$ in a set $B \subset \mathbb{Z}/p\mathbb{Z}$ is a set of $L$ consecutive residues disjoint from $B$, and a dilate of $A$ is a set $\lambda A$ with $\lambda \ne 0$. Shakan showed, by the polynomial method, that some dilate of $A$ has a gap of length $2\sqrt{p} - 2$, and this seems to be as far as his method goes.
Problem. Must some dilate of $A$ have a gap of length $100\sqrt{p}$?
Source: B. Green, as recorded by G. Shakan, A large gap in a dilate of a set, SIAM J. Discrete Math. 34 (2020). Listed as Problem 32 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- NT — Number Theory
- Secondary
- CO
- Keywords
- cyclic groups of prime order, dilates, gaps, polynomial method
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{shakan2020,
author = {Shakan, George},
title = {A large gap in a dilate of a set},
journal = {SIAM Journal on Discrete Mathematics},
volume = {34},
number = {4},
pages = {2553--2555},
year = {2020},
doi = {10.1137/20M1335030},
eprint = {2004.14828},
archivePrefix = {arXiv}
}
@article{alon1992peres,
author = {Alon, Noga and Peres, Yuval},
title = {Uniform dilations},
journal = {Geometric and Functional Analysis},
volume = {2},
number = {1},
pages = {1--28},
year = {1992},
doi = {10.1007/BF01895704}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 32; 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 It will not change if the problem is solved, reclassified or withdrawn.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-00937, "A long gap in a dilate of a square-root-sized set modulo a prime", clopen.solutions.