Let $x_1<x_2<\cdots$ be an infinite sequence of integers. Is it true that, for almost all $\alpha \in [0,1]$, the discrepancy$$D(N)=\underset{I\subseteq [0,1]}{\max }|\#\{n\le N:\{\alpha x_n\}\in I\}-|I|N|$$satisfies$$D(N)\ll N^{1/2}(\log N{)}^{o(1)}?$$Or even$$D(N)\ll N^{1/2}(\log \log N{)}^{O(1)}?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

No—neither bound can hold **uniformly over all increasing integer sequences** ((x_n)).

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.