It is clear that $\{\log n\pmod 1: n\in\mathbb{N} \}$ is dense in $[0,1]$ but not uniformly distributed.
How about $\{n\log n \pmod 1: n\in\mathbb{N} \} ?$ Is it dense in $[0,1]?$ If so, is it uniformly distributed?
Definition of uniformly distributed sequence, i.e., equidistributed sequence: https://en.wikipedia.org/wiki/Equidistributed_sequence#Definition
In Uniform Distribution of Sequences by Kuipers and Neidrreiter, after several lemmas, it is proved (page 18, Example 2.8) that for any integer $h\neq 0$, $$ \left|\frac1N \sum_{n=1}^N e^{2\pi i h n \log n}\right|=O\left( \frac{\sqrt{|h|}\log N}{\sqrt{N}}\right). $$ Together with Weyl's criterion, the sequence $n\log n$ is uniformly distributed modulo $1$.
