Such a set $A$ is called an additive complement of the set of squares. Erdős observed that there exist $A$ for which the $\limsup$ is finite and $>1$. Moser [Mo65] proved that, for any such $A$,\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>1.06.\]The best-known lower bound is\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\geq\frac{4}{\pi}\approx 1.273\]proved by Cilleruelo [Ci93], Habsieger [Ha95], and Balasubramanian and Ramana [BaRa01].

The problem of minimising the $\limsup$ appears to have been much less studied. van Doorn has a construction of such an $A$ in which, for all $N$,\[\frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}< 2\phi^{5/2}\approx 6.66,\]where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio.

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This page was last edited 27 December 2025.