A problem of Erdős and Newman [ErNe77], who proved that there exist $A$ with $\lvert A\rvert\asymp n^{1/2}$ such that if $A\subseteq B+B$ then\[\lvert B\rvert \gg \frac{\log\log n}{\log n}n^{1/2}.\]Resolved by Alon, Bukh, and Sudakov [ABS09], who proved that for any $A\subseteq \{1,\ldots,n\}$ with $\lvert A\rvert \leq n^{1/2}$ there exists some $B$ such that $A\subseteq B+B$ and\[\lvert B\rvert \ll \frac{\log\log n}{\log n}n^{1/2}.\]See also [333].

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