Sumsets of sets of squares grow by a power
The question below is strictly easier than Hardy and Littlewood's question whether the squares form a $\Lambda(p)$-set for some $2 < p < 4$. A positive answer implies that an arithmetic progression of length $n$ contains $O(n^{\kappa})$ squares for some $\kappa < 1$, which is known, with $\kappa = \frac35 + o(1)$, by arguments involving rational points on curves; Rudin conjectured that $\kappa = \frac12$ is possible. Solymosi observed that a positive answer would follow if the squares contained no affine cube $\{x + \omega_1 h_1 + \dots + \omega_d h_d : \omega_i \in \{0, 1\}\}$ for some $d$.
Problem. Is there an absolute constant $c > 0$ such that every set $A$ of squares of positive integers with $|A| \ge 2$ satisfies $|A + A| \ge |A|^{1+c}$?
Source: A well-known problem, a weak form of a conjecture of Ruzsa; see J. Cilleruelo and A. Granville, Lattice points on circles, squares in arithmetic progressions and sumsets of squares (2007). Listed as Problem 60 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
- Rudin's conjecture, squares, sumsets, Λ(p) sets
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@incollection{cilleruelo2007granville,
author = {Cilleruelo, Javier and Granville, Andrew},
title = {Lattice points on circles, squares in arithmetic progressions and sumsets of squares},
booktitle = {Additive Combinatorics},
series = {CRM Proceedings and Lecture Notes},
volume = {43},
pages = {241--262},
publisher = {American Mathematical Society},
year = {2007},
doi = {10.1090/crmp/043/12},
eprint = {math/0608109},
archivePrefix = {arXiv}
}
@article{rudin1960,
author = {Rudin, Walter},
title = {Trigonometric series with gaps},
journal = {Journal of Mathematics and Mechanics},
volume = {9},
number = {2},
pages = {203--227},
year = {1960},
doi = {10.1512/iumj.1960.9.59013}
}
@article{chang2004,
author = {Chang, Mei-Chu},
title = {On problems of {E}rd{\H o}s and {R}udin},
journal = {Journal of Functional Analysis},
volume = {207},
number = {2},
pages = {444--460},
year = {2004},
doi = {10.1016/S0022-1236(03)00073-9}
}
@incollection{solymosi2007elementary,
author = {Solymosi, J{\'o}zsef},
title = {Elementary additive combinatorics},
booktitle = {Additive Combinatorics},
series = {CRM Proceedings and Lecture Notes},
volume = {43},
pages = {29--38},
publisher = {American Mathematical Society},
year = {2007},
doi = {10.1090/crmp/043/02}
}
@article{bombieri1992squares,
author = {Bombieri, Enrico and Granville, Andrew and Pintz, J{\'a}nos},
title = {Squares in arithmetic progressions},
journal = {Duke Mathematical Journal},
volume = {66},
number = {3},
pages = {369--385},
year = {1992},
doi = {10.1215/S0012-7094-92-06612-9}
}
@article{bombieri2002zannier,
author = {Bombieri, Enrico and Zannier, Umberto},
title = {A note on squares in arithmetic progressions, {II}},
journal = {Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni},
volume = {13},
number = {2},
pages = {69--75},
year = {2002}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 60; 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-00946, "Sumsets of sets of squares grow by a power", clopen.solutions.