Erdős Problem #1147 — Quadratic recurrence sets as additive bases

Let $\alpha>0$ be irrational and set

$$A_\alpha=\left\{n\geq1:\|\alpha n^2\|<1/\log n\right\},$$

where $\|x\|$ is the distance from $x$ to the nearest integer. Must $A_\alpha$ be an additive basis of order $2$?