Let $A=\{a_1<a_2<\cdots \}\subseteq \mathbb{N}$ be infinite such that $a_{i+1}/a_i\to 1$. For any $x\ge a_1$ let$$f(x)=\frac{x-a_i}{a_{i+1}-a_i}\in [0,1),$$where $x\in [a_i,a_{i+1})$. Is it true that, for almost all $\alpha$, the sequence $f(\alpha n)$ is uniformly distributed in $[0,1)$?