Erdős Problem #1147

Let \(\alpha>0\) be an irrational number. Is the set \[A=\left\{ n\geq 1: \| \alpha n^2\| < \frac{1}{\log n}\right\},\] where \(\|\cdot\|\) denotes the distance to the nearest integer, an additive basis of order \(2\)? 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 \(A=\{ n\geq 1: \| \alpha n^2\| < \epsilon(n)\}\) is not an additive basis of order \(2\) for almost every \(\alpha>0\).