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arXiv · 2508.02803

Further Improvements to the Lower Bound for an Autoconvolution Inequality

Aaron Jaech·Alan Joseph
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Abstract

We construct a nonnegative step function comprising 2,399 equally spaced intervals such that \[ \frac{\|f * f\|_{L^{2}(\mathbb{R})}^{2}}{\|f * f\|_{L^{\infty}(\mathbb{R})}\,\|f * f\|_{L^{1}(\mathbb{R})}} \;\ge\; .926529. \] 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.

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Aaron Jaech, Alan Joseph. 2025-08-04. Further Improvements to the Lower Bound for an Autoconvolution Inequality. https://arxiv.org/abs/2508.02803

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