Erdős minimum overlap problem

The problem

Let $C$ be the largest constant for which

$$\sup_{x \in [-2,2]} \int_{-1}^1 f(t) g(x+t) dt\geq C$$

for all non-negative $f,g: [-1,1] \to [0,1]$ with $f+g=1$ on $[-1,1]$ and $\int_\mathbb{R} f = 1$, where we extend $f,g$ by zero outside of $[-1,1]$. What is $C$?

Statement quoted from problem 5 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 $C$

Lower bounds for $C$

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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