Let $A=\{n_1<n_2<\cdots \}\subset \mathbb{N}$ be a lacunary sequence (so there exists some $ϵ>0$ with $n_{k+1}\ge (1+ϵ)n_k$ for all $k$). Must there exist an irrational $\theta$ such that$$\{\Vert \theta n_k\Vert :k\ge 1\}$$is not dense in $[0,1]$ (where $\Vert x\Vert$ is the distance to the nearest integer)?