How much can an autocorrelation of a non-negative function resemble an indicator function?
Let C be the best constant for which one has ‖f f‖_L^2(ℝ)^2 ≤ C ‖ff‖_L^1(ℝ) ‖f * f‖_L^∞(ℝ) for non-negative f : ℝ → ℝ. What is C?
From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.
Cite
@misc{cairn-how-much-can-an-autocorrelation-of-a-non-negative-function-resemble-an,
title = {How much can an autocorrelation of a non-negative function resemble an indicator function?},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/how-much-can-an-autocorrelation-of-a-non-negative-function-resemble-an}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
} Also: CITATION.cff · Atom feed of results
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The problem
The problem
Let be the best constant for which one has for non-negative . What is ?
Known results
Upper bounds for
| Bound | Source | Date |
|---|---|---|
| Trivial |
Lower bounds for
| Bound | Source | Date |
|---|---|---|
| Martin-O'Bryant | 1 Apr 2009 | |
| Matolcsi-Vinuesa | 8 Jul 2009 | |
| Georgiev-Gómez-Serrano-Tao-Wagner [Colab 1] [Colab 2] | 14 May 2025 | |
| Boyer-Li | Nov 2024 / 10 Aug 2025 |
AlphaEvolve set a record that has since been improved.
What counts as progress
- A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
- A proof that a known construction is optimal, or a better bound on the other side.
- Literature: earlier or newer records (literature claims).
Source and licence
Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.