Erdős minimum overlap problem

AlphaEvolve found a construction that improved on the best bound known at the time; the problem itself remains open.

Problem
Problem 5 in the AlphaEvolve repository: Erdős minimum overlap problem
Credit
AlphaEvolve (Google DeepMind) · LLM-guided evolutionary code search, with Georgiev, Gómez-Serrano, Tao and Wagner
Checked
Construction can be checked directlyPreprint (arXiv, 2025)
Timeline
Paper submitted 2025-11-03
Sources
Problem page (AlphaEvolve repository)Paper (arXiv:2511.02864)
PARTIALImproved bound · new construction

The problem

Let CC be the largest constant for which

sup⁡x∈[−2,2]∫−11f(t)g(x+t)dt≥C\sup_{x \in [-2,2]} \int_{-1}^1 f(t) g(x+t) dt\geq C

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

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 CC

Bound Source Date
0.3809270.380927 Haugland 2016
0.3809240.380924 Georgiev-Gómez-Serrano-Tao-Wagner [Colab 1] [Colab 2] 14 May 2025
0.3808760.380876 Yuksekgonul et al. Jan 2026

Lower bounds for CC

Bound Source Date
11 Trivial
0.3790050.379005 White 2022

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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