An improved example for an autoconvolution inequality

TL;DR

This paper presents a new nonnegative step function with 575 intervals that improves the lower bound of a specific autoconvolution inequality, surpassing previous results obtained by AlphaEvolve, using optimization techniques.

Contribution

The authors introduce a significantly improved example for an autoconvolution inequality using optimization methods, surpassing prior bounds achieved by AlphaEvolve.

Findings

Achieved a lower bound of 0.901564 for the autoconvolution inequality.

Used simulated annealing and gradient methods instead of large language models.

Demonstrated the effectiveness of optimization techniques in mathematical inequality improvements.

Abstract

We give a nonnegative step function with 575 equally spaced intervals such that $$\frac{\|f \ast f\|_{L^{2}(\mathbb{R})}^{2}}{\|f \ast f\|_{L^{\infty}(\mathbb{R})}\|f \ast f\|_{L^{1}(\mathbb{R})}} \geq 0.901564.$$ This improves upon a recent result of Deepmind's AlphaEvolve, which found a nonnegative step function with 50 equally space intervals for which the left hand side is $\geq 0.8962$. Our function was found using simulated annealing and gradient based methods rather than using large language models.