Improved Lower Bound on the Second Autocorrelation Inequality

We let AI agents tackle an open problem in harmonic analysis — the second autocorrelation inequality — and obtained the best publicly available lower bound. Our approach uses projected gradient ascent to optimize a 100,000-point discretized function, starting from the best previously known solution.

Note: ImprovEvolve (Kravatskiy et al., Feb 2026) reports a higher bound of 0.96258, but the explicit solution is not publicly available. Our solution (0.961206) is the best with reproducible, publicly released data.

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Problem Statement

Find a non-negative function $f: \mathbb{R} \to \mathbb{R}$ that maximizes the constant $C_2$ in the second autocorrelation inequality

$$|f \star f|_2^2 \leq C_2 |f \star f|_1 |f \star f|_\infty$$

where $f \star f(t) = \int f(t-x)f(x)dx$ is the autoconvolution. By Hölder's inequality, $C_2 \leq 1$ trivially; the question is how close to $1$ the ratio can be when $F = f \star f$ is constrained to be an autoconvolution.

Discretized Formulation

Discretize $f$ as $n$ non-negative values. The autoconvolution $f \star f$ is computed via numpy.convolve. The score is

$$C_2 = \frac{|f \star f|_2^2}{|f \star f|_1 \cdot |f \star f|_\infty}.$$

Higher $C_2$ is better (tighter lower bound).

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Results Comparison

Method	Source	Date	Points	Lower Bound (higher is better)
AlphaEvolve	Novikov et al. (Colab)	June 2025	50	0.896280
AlphaEvolve V2	Georgiev et al. (Colab)	Nov 2025	50,000	0.961021
TTT-Discover	Yuksekgonul et al.	Jan 2026	50,000	0.959180
Ours (Together AI)	This repo	Mar 2026	100,000	0.961206
ImprovEvolve	arXiv:2602.10233	Feb 2026	—	0.962580*

*Solution not publicly available.

For full verification and additional analysis, see analysis.ipynb.

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References

• Novikov et al., "Alphaevolve: A coding agent for scientific and algorithmic discovery," arXiv:2506.13131, 2025.
• B. Georgiev, J. Gómez-Serrano, T. Tao, L. Wagner, "Mathematical exploration and discovery at scale," arXiv:2511.02864, 2025.
• M. Yuksekgonul et al., "Learning to Discover at Test Time," arXiv:2601.16175, 2026.
• A. Kravatskiy, V. Khrulkov, I. Oseledets, "ImprovEvolve: Ask AlphaEvolve to improve the input solution and then improvise," arXiv:2602.10233, 2026.