Construct a sequence.

Question

We know that, if a sequence $\{\xi_n\}_{n=1}^{\infty}\subset [0,1)$ is equidistributed, then it must be dense in $[0,1)$.

My problem is, how to construct a sequence $\{\xi_n\}$ that is dense but not equidistributed?

Answer 1

Let $\{\alpha_n\}_1^\infty \subset [0, \frac12)$ and $\{\beta_n\}_1^\infty \subset [\frac12, 1)$ be equidistributed on their respective sets. Then let $$ \xi_1=\alpha_1,\ \xi_2=\beta_1,\ \xi_3=\beta_2,\\ \xi_4=\alpha_2,\ \xi_5=\beta_3,\ \xi_6=\beta_4,\\ \xi_7=\alpha_3,\ \xi_8=\beta_5,\ \xi_9=\beta_6,\\ \ldots$$ This isn't equidistributed on $[0, 1),$ is it?