Erdős Problem #333

Let $A\subseteq \mathbb{N}$ be a set of density zero. Does there exist a $B$ such that $A\subseteq B+B$ and\[\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2})\]for all large $N$?

#333: [ErGr80]

number theory | additive basis

Erdős and Newman [ErNe77] have proved this is true when $A$ is the set of squares. In fact, Theorem 2 of [ErNe77] already implies a negative answer to this problem, but this seems to have been overlooked by Erdős and Graham.

See also [806].

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This page was last edited 27 December 2025.

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