SOLVED

Let $z_1,z_2,\ldots \in [0,1]$ be an infinite sequence, and define the discrepancy
\[D_N(I) = \#\{ n\leq N : z_n\in I\} - N\lvert I\rvert.\]
Must there exist some interval $I\subseteq [0,1]$ such that
\[\limsup_{N\to \infty}\lvert D_N(I)\rvert =\infty?\]

The answer is yes, as proved by Schmidt [Sc68], who later showed [Sc72] that in fact this is true for all but countably many intervals of the shape $[0,x]$.

Essentially the best possible result was proved by Tijdeman and Wagner [TiWa80], who proved that, for almost all intervals of the shape $[0,x)$, we have \[\limsup_{N\to \infty}\frac{\lvert D_N([0,x))\rvert}{\log N}\gg 1.\]