What was already known
Clunie proved Ã_k≫√k infinitely often for every sequence and constructed a dyadic sequence with A_k≤k.
AI involvement
An internal OpenAI model found the prefix-randomized binary van der Corput construction and its dyadic-block estimate. Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke and Gregory Valiant report checking the proof and explain its mechanism.
The construction and proof are explicitly attributed to the model; detailed prompt independence is undisclosed.
Importance
The bound answers the Erdős question and approaches Clunie’s lower bound up to a logarithmic factor.
Limitations
The result is an existence theorem for one sequence, with a uniform bound over all partial sums and positive k.
Technical verification
The coauthors explicitly report verifying this model solution. The probabilistic existence proof is not a machine certificate, but completed human checking qualifies under the policy.
Independent checking
human verification of the model proof
- Checker
- Alexeev, Putterman, Sawhney, Sellke and Valiant
- Scope
- Problem 987 simultaneous uniform exponential-sum bound
- Basis
- Reported by the linked checker
Human understanding
The paper's human-authored exposition explains the randomized dyadic mechanism and explicitly connects the uniform bound to Clunie's baseline.
Correction and publication history
The bound answers the Erdős question and approaches Clunie’s lower bound up to a logarithmic factor.
Read this published revision