An autocorrelation problem related to Sidon sets

The problem

Let $C$ denote the largest constant for which one has
$\max_{-1/2 \leq t \leq 1/2} \int_\mathbb{R} f(t-x) f(x) dx \geq C \left(\int_{-1/4}^{1/4} f(x) dx\right)^2$
for all non-negative $f \colon \mathbb{R} \to \mathbb{R}$ . What is $C$ ?

Statement quoted from problem 2 of the AlphaEvolve repository of problems (Google DeepMind, CC BY 4.0).

What AI did

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open. Status in the repository: New result. See Section 6.2 in "Mathematical Exploration and Discovery at Scale" paper. Records may have moved since; the repository's notes below list the latest bounds it knows of.

Notes from the repository

Upper bounds for $C$

Bound	Source	Date
$\pi/2 = 1.57059$	Schinzel-Schmidt	2002
$1.50992$	Matolcsi-Vinuesa	8 Jul 2009
$1.5053$	Georgiev-Gómez-Serrano-Tao-Wagner [Colab 1] [Colab 2]	14 May 2025
$1.5032$	Georgiev-Gómez-Serrano-Tao-Wagner [Colab 1] [Colab 2]	20 Dec 2025
$1.5029$	Yuksekgonul et al.	Jan 2026

Lower bounds for $C$

Bound	Source	Date
$1$	Trivial
$1.182778$	Martin-O'Bryant	30 Sep 2004
$1.262$	Martin-O'Bryant	1 Apr 2009
$1.2748$	Matolcsi-Vinuesa	8 Jul 2009

How it was checked

The results are explicit constructions or programs; the repository's notebooks include the verification code used to score them.

Limitations

An improved bound is progress, not a solution: the exact answer to the problem is still unknown. The authors report that results were better when experts set up and guided the search.

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