An improved example for an autoconvolution inequality

We give a nonnegative step function with 575 equally spaced intervals such that $$\frac{\mathrm{\parallel }f\ast f{\mathrm{\parallel }}_{L^2(\mathbb{R})}^2}{\mathrm{\parallel }f\ast f{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathbb{R})}\mathrm{\parallel }f\ast f{\mathrm{\parallel }}_{L^1(\mathbb{R})}}\ge \mathrm{0.901564.}$$ 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 $\ge 0.8962$. Our function was found using simulated annealing and gradient based methods rather than using large language models.