This was disproved by Konieczny [Ko16b], and is false both for almost every $\alpha>0$, and also is false specifically for $\alpha=\sqrt{2}$.

More generally, given any $\epsilon(n)\to 0$, the set\[A=\left\{ n\geq 1: \| \alpha n^2\| < \epsilon(n)\right\}\]is not an additive basis of order $2$ for almost every $\alpha>0$.

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This page was last edited 27 January 2026.