Grokability in five inequalities

2026-05-06

AI Summary

This work addresses several classical problems in convex geometry, harmonic analysis, and extremal combinatorics—including lower bounds for Gaussian perimeter of high-dimensional convex sets, moment inequalities on the Hamming cube, self-convolution inequalities, asymptotic size estimates for g-Sidon sets, and Szarek’s inequality—by introducing novel methods that refine both precision and boundary estimates. Innovatively integrating the large language model Grok with rigorous mathematical verification, this study presents the first systematic AI-driven discovery and proof of strengthened forms of multiple key inequalities. The results include an improved lower bound for Gaussian perimeter, a sharp refinement of L₂–L₁ moment comparison on the Hamming cube, an enhanced self-convolution inequality, a larger asymptotic upper bound for g-Sidon sets, and an optimally balanced version of Szarek’s inequality, collectively yielding substantial advances across these mathematical domains.

Abstract

In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in $\mathbb{R}^n$, sharper $L_2$-$L_1$ moment comparison inequalities on the Hamming cube $\{-1,1\}^n$, a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest $g$-Sidon sets in $\{1,\dots,n\}$, and an optimal balanced Szarek's inequality.

Problem

• Gaussian perimeter
• moment comparison inequalities
• autoconvolution inequality
• Sidon sets
• Szarek's inequality

Innovation

• Gaussian perimeter
• moment comparison inequalities
• autoconvolution inequality
• g-Sidon sets
• Szarek's inequality