Erd\H{o}s proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.\]This is discussed in problem C11 of Guy's collection \cite{Gu04}, in which Guy says Erd\H{o}s offered \$500 for the general problem of whether, for all $h\geq 2$,\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/h}}=0\]whenever the sum of $h$ terms in $A$ are distinct. This was proved for $h=4$ by Nash \cite{Na89} and for all even $h$ by Chen \cite{Ch96b}.


References


[Ch96b] Chen, Sheng, A note on {$B_{2k}$} sequences. J. Number Theory (1996), 1--3.

[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

[Na89] Nash, John C. M., On {$B_4$}-sequences. Canad. Math. Bull. (1989), 446--449.