Let $\alpha$ be an irrational number. Is it true that if, for all large $n$,$$\#\{1\le m\le n:\{\alpha m\}\in [u,v)\}=n(v-u)+O(1)$$then $u=\{\alpha k\}$ and $v=\{\alpha \ell \}$ for some integers $k$ and $\ell$?