PROVED
This has been solved in the affirmative.
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\[\limsup_{N\to \infty}\frac{\lvert D_N([0,x))\rvert}{\log N}\gg 1.\]
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Additional thanks to: Cedric Pilatte and Stefan Steinerberger
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #255, https://www.erdosproblems.com/255, accessed 2026-02-13
