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Level B · Reproducible Analysis P-an-autocorrelation-problem-with-functions-that-can-take-negative-values

An autocorrelation problem with functions that can take negative values

Let C be the best constant for which one has max_-1/2 ≤ t ≤ 1/2|∫_ℝ f(t-x) f(x) dx| ≥ C (∫_-1/4^1/4 f(x) dx)^2 for all f : [-1/4,1/4] → ℝ (note f can take negative values). 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.

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Cite 
@misc{cairn-an-autocorrelation-problem-with-functions-that-can-take-negative-values,
  title        = {An autocorrelation problem with functions that can take negative values},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/an-autocorrelation-problem-with-functions-that-can-take-negative-values}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The problem

Let be the best constant for which one has for all (note can take negative values). What is ?

Known results

This is one of two very similar optimization problems, both of which have been studied in the literature, with some results for one problem incorrectly attributed to another. We will update the references on this problem (and add its sibling) soon.

AlphaEvolve found a construction better than the previous record (at publication).

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.