Further Improvements to the Lower Bound for an Autoconvolution Inequality

We construct a nonnegative step function comprising 2,399 equally spaced intervals such that $$\frac{\mathrm{\parallel }f\ast f{\mathrm{\parallel }}_{L^2(\mathbb{R})}^2}{\mathrm{\parallel }f\ast f{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathbb{R})}\mathrm{\parallel }f\ast f{\mathrm{\parallel }}_{L^1(\mathbb{R})}}\ge \mathrm{.926529.}$$ Using a 4x upsampling procedure on this 559-interval optimizer, we further increase the bound to $.94136$, closing roughly 40% of the gap between the previous best bound (.901562 on 575 intervals) and the trivial upper limit of 1.