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An autocorrelation problem with functions that can take negative values (AlphaEvolve problem 4)

An autocorrelation problem with functions that can take negative values (AlphaEvolve problem 4). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 42A85 · Convolution, factorization for one variable harmonic analysis

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

Let CC be the best constant for which one has

max⁡−1/2≤t≤1/2∣∫Rf(t−x)f(x)dx∣≥C(∫−1/41/4f(x)dx)2\max_{-1/2 \leq t \leq 1/2}\left |\int_\mathbb{R} f(t-x) f(x) dx\right| \geq C \left(\int_{-1/4}^{1/4} f(x) dx\right)^2

for all f ⁣:[−1/4,1/4]→Rf \colon [-1/4,1/4] \to \mathbb{R} (note ff can take negative values). What is CC?

Lean 4

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Background

Repository result category: world record — AlphaEvolve improved on the best known result recorded at the time. See Section 6.2 in "Mathematical Exploration and Discovery at Scale " paper: arXiv:2511.02864.

Progress & source assessment

This is one of two very similar optimization problems, both of which have been studied in the literature, with some results for one problem incorrectly attributed to another. We will update the references on this problem (and add its sibling) soon.

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). — An autocorrelation problem with functions that can take negative values (AlphaEvolve problem 4) linkApache-2.0 (repository license); problem pages reproduced with attribution.

Recorded progress

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Discussion

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