Erdős Problem #1109

Let $f(N)$ be the size of the largest subset $A\subseteq \{1,\ldots,N\}$ such that every $n\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\leq N^{o(1)}$, or even $f(N) \leq (\log N)^{O(1)}$?

#1109: [ErSa87]

number theory

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First studied by Erdős and Sárközy [ErSa87], who proved\[\log N \ll f(N) \ll N^{3/4}\log N,\]and guessed the lower bound is nearer the truth. Sárközy [Sa92c] extended this to consider the case of $A+B$ and also looking for sumsets which are $k$-power-free.

Gyarmati [Gy01] gave an alternative proof of $f(N)\gg \log N$, and also gave new bounds for the case of $A+B$. Konyagin [Ko04] improved this to\[ \log\log N(\log N)^2\ll f(N) \ll N^{11/15+o(1)}.\]The infinite analogue of this problem is [1103]. (In particular upper bounds for this $f(N)$ directly imply lower bounds for the size of the $a_j$ considered there.)

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This page was last edited 03 December 2025.

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Related OEIS sequences: A392164 A392165

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