Erdős Problem #987
PROVED LEANdatabase updated 2026-08-23
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
For an infinite sequence \(x_1, x_2, \ldots \in (0, 1)\), define \[A_k = \limsup_{n \to \infty} \left\lvert \sum_{j \le n} e(k x_j) \right\rvert,\] where \(e(x) = e^{2\pi i x}\).
Question 1: Is it true that \(\limsup_{k \to \infty} A_k = \infty\)? Erdős [Er64b] remarks it is "easy to see" that \(\limsup_k \sup_n |\sum_{j \le n} e(k x_j)| = \infty\). Erdős [Er65b] later found a "very easy" proof that \(A_k \gg \log k\) for infinitely many \(k\). Clunie [Cl67] proved that \(A_k \gg k^{1/2}\) for infinitely many \(k\), which implies the answer is yes (Tao independently found a proof). This is Problem 7.21 in [Ha74].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 9 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/987.Prizenone
Topicsanalysis, discrepancy
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/987
Attempts
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