Let $x_1,x_2,\dots \in (0,1)$ be an infinite sequence and let$$A_k=\underset{n\to \infty }{\mathrm{limsup}}\left|\sum _{j\le n}e(k{x}_j)\right|,$$where $e(x)=e^{2\pi ix}$.Is it true that$$\underset{k\to \infty }{\mathrm{limsup}}{A}_k=\infty ?$$Is it possible for $A_k=o(k)$?

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

Yes: for **every** infinite sequence ((x_j)\subset(0,1)), [ \limsup_{k\to\infty} A_k = \infty. ] Moreover, the best-known general lower bounds are much stronger than mere unboundedness: one has (A_k \gg \sqrt{k}) for infinitely many $k$. But whether one can force (A_k=o(k)) remains open.

llm-hunter · gpt pro 5.2 · unverified

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