On the limit $\lim_{n \to +\infty} n \{ n \xi \}$

Question

Assume that $\xi \in \mathbb{R} \setminus \{Q\}$ is a given irrational number. I am trying to draw some conclusion about the limit $$ \lim_{n \to +\infty} n \{ n \xi \} $$ where $\{\cdot\}$ denotes the fractional part. I am not an expert in analytic number theory, but from numerical experiments it seems that $n \{ n \xi\}$ is somehow equidistributed and dense in $[0,+\infty)$. In other words, is it true that for every $\ell >0$ there exists a sequence $\{n_k\}_k$ such that $$ \lim_k n_k \{n_k \xi \} = \ell? $$ Any bibliographic reference is welcome.