Problem 987

Erdős Problem #987: declared status 'proved'. Formalized: no. Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let$$A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert,$$where $e(x)=e^{2\pi ix}$. Is it true that$$\limsup_{k\to \infty} A_k=\infty?$$Is it possible for $A_k=o(k)$? Current best: Erd\H{o}s [Er64b] remarks it is 'easy to see' that$$\limsup_{k\to \infty}\left(\sup_n\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert\right)=\infty.$$Erd\H{o}s [Er65b] later found a 'very easy' proof that $A_k\gg \log k$ for infinitely many $k$. Clunie [Cl67] proved that $A_k\gg k^{1/2}$ infinitely often, and that there exist sequences with $A_k\leq k$ for all $k$. Tao has independently found a proof that $A_k\gg k^{1/2}$ infinitely often (see the comment section). Liu [Li69] showed that, for any $\epsilon>0$, $A_k\gg k^{1-\epsilon}$ infinitely often, under the additional assumption that there are only a finite number of distinct points. Prize: no. Tags: analysis, discrepancy.