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OP-12140Status unassessedNumber theory

Erdős discrepancy problem (AlphaEvolve problem 40)

Erdős discrepancy problem (AlphaEvolve problem 40). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 11K38 · Irregularities of distribution, discrepancy

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository at commit 8f44745 (2026-07-11); the repository records AlphaEvolve results, not current research statusSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

The discrepancy of a sign pattern a1,…,aN∈{−1,+1}a_1,\dots,a_N \in \{-1,+1\} is the maximum value of ∣ad+a2d+⋯+akd∣|a_d + a_{2d} + \dots + a_{kd}| for homogeneous progressions d,…,kdd,\dots,kd in {1,…,N}\{1,\dots,N\}. For any D≥1D \geq 1, let C(D)C(D) denote the largest NN for which there exists a sign pattern a1,…,aNa_1,\dots,a_N of discrepancy at most CC. Establish upper and lower bounds on C(D)C(D) that are as strong as possible.

Lean 4

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Background

Repository result category: worse than record — AlphaEvolve did not reach the best known result. See Section 6.23 in "Mathematical Exploration and Discovery at Scale " paper: arXiv:2511.02864. Experiment notebook.

Progress & source assessment

The repository page records no bounds for this problem.

References
  1. [1]Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Repository of problems accompanying “Mathematical exploration and discovery at scale” (arXiv:2511.02864), github.com/google-deepmind/alphaevolve_repository_of_problems (commit 8f44745, 11 July 2026). — Erdős discrepancy problem (AlphaEvolve problem 40) linkApache-2.0 (repository license); problem pages reproduced with attribution.

Recorded progress

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Machine attempts

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Discussion

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