Erdős Problem #1147

The problem

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 $2$?

Statement quoted from erdosproblems.com/1147, maintained by Thomas Bloom. Original source: [Va99,1.21]. Current status there: disproved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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