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An autocorrelation problem related to Sidon sets (AlphaEvolve problem 2)

An autocorrelation problem related to Sidon sets (AlphaEvolve problem 2). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 42A85 · Convolution, factorization for one variable harmonic analysisMSC 11B83 · Special sequences and polynomials

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 denote the largest 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} \int_\mathbb{R} f(t-x) f(x) dx \geq C \left(\int_{-1/4}^{1/4} f(x) dx\right)^2

for all non-negative f ⁣:R→Rf \colon \mathbb{R} \to \mathbb{R}. 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. Experiment notebook.

Progress & source assessment

Upper bounds for CC

Bound Source Date
π/2=1.57059\pi/2 = 1.57059 Schinzel-Schmidt 2002
1.509921.50992 Matolcsi-Vinuesa 8 Jul 2009
1.50531.5053 Georgiev-Gómez-Serrano-Tao-Wagner [Colab 1] [Colab 2] 14 May 2025
1.50321.5032 Georgiev-Gómez-Serrano-Tao-Wagner [Colab 1] [Colab 2] 20 Dec 2025
1.50291.5029 Yuksekgonul et al. Jan 2026

Lower bounds for CC

Bound Source Date
11 Trivial
1.1827781.182778 Martin-O'Bryant 30 Sep 2004
1.2621.262 Martin-O'Bryant 1 Apr 2009
1.27481.2748 Matolcsi-Vinuesa 8 Jul 2009

See also Damek Davis's meta-analysis of 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). — An autocorrelation problem related to Sidon sets (AlphaEvolve problem 2) linkApache-2.0 (repository license); problem pages reproduced with attribution.

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