Uniform exponential-sum bounds in Erdős Problem 987

Can one real sequence (xₙ) have partial exponential sums uniformly bounded in the cutoff N for every positive frequency k, with the bound growing sublinearly in k? The target compares A_k=sup_N|Σ_{n<N}exp(2πikxₙ)| with known lower bounds.

Why it matters and what counts as progress

Why it matters: Uniform bounds over all cutoffs test cancellation in one sequence simultaneously at every integer frequency.

What counts as progress: Construct a sequence with A_k=o(k), improve it toward √k or prove a stronger universal lower bound.

How the problem developed

1965 · Lower bound [1]
Paul Erdős Showed that the limsup exponential sum Aₖ is at least a constant times log k for infinitely many k.
1967 · Partial result [1]
J. Clunie Improved the infinitely-often lower bound to order √k and gave a sequence with Aₖ≤k for all k.
8 Apr 2026 · Reported claim [2]
Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke and Gregory Valiant Reported a sequence with every prefix exponential sum bounded by O(√(k log(2k))) for all k, improving the earlier all-k upper bound; this implies the weaker o(k) question.

AI contribution

Short proofs in combinatorics, probability and number theory II

Boris Alexeev, Moe Putterman, Mehtaab Sawhney, Mark Sellke, Gregory Valiant

A single real sequence has uniformly bounded partial exponential sums A_k= sup_N |Σ_{n<N} exp(2πikx_n)|=O(√(k log(2k))) for every positive integer k. This implies the weaker limsup Ã_k=o(k) sought in Erdős Problem 987.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull