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