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Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be a lacunary sequence (so there exists some $\lambda>1$ with $a_{k+1}\geq \lambda a_k$ for all $k$). Is there an irrational $\alpha$ such that \[\{ \{\alpha a_k\} : k\geq 1\}\] is not everywhere dense in $[0,1]$ (where $\{x\}=x-\lfloor x\rfloor$ is the fractional part).
Erdős and Graham write the existence of such an $\alpha$ has 'very recently been shown', but frustratingly give neither a name nor a reference. I will try to track down this solution soon.