Erdős Problem #987 — 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 $e(x)=e^{2\pi ix}$.

The problem is attributed to Erdős and was recorded as Problem 7.21 in 1974. It asks whether the exponential-sum amplitudes must be unbounded and whether some sequence can make them grow slower than linearly.

Known results

• Erdős, 1964: established divergence and later proved $\widetilde A_k\gg\log k$ for infinitely many $k$.
• Clunie, 1967: proved $\widetilde A_k\gg k^{1/2}$ for infinitely many $k$, and constructed a sequence with $A_k\le k$ for every $k$.
• Liu, 1969: under finitely many distinct points, proved $A_k\gg k^{1-\epsilon}$ infinitely often; Clunie observed that in fact $A_k=\infty$ infinitely often.

April 2026 construction

A recent paper gives a sequence satisfying $A_k\ll\sqrt{k\log(2k)}$ for every $k$, hence $A_k=o(k)$. A separate proof establishes the first question directly; the historical lower bound also implies unboundedness. The problem page additionally credits an internal OpenAI model with the construction, but this attribution is not independently corroborated there.

Current status (as of April 2026): both questions are settled affirmatively, with the stronger uniform bound $A_k\ll\sqrt{k\log(2k)}$ and universal unboundedness established.