Grokability in five inequalities

Problem
This preprint addresses several gaps in mathematical inequalities relevant to convex geometry and combinatorial optimization. Specifically, it presents five mathematical discoveries that enhance existing bounds and inequalities, which are crucial for theoretical advancements in areas such as functional analysis and information theory. The authors aim to improve the understanding of Gaussian perimeters, moment comparisons, autoconvolution, Sidon sets, and Szarek’s inequality, which have implications for various applications in machine learning and data science.

Method
The core contributions of this work include:

1. Maximal Gaussian Perimeter: The authors derive an improved lower bound for the maximal Gaussian perimeter of convex sets in (\mathbb{R}^n), enhancing previous results in the literature.
2. Moment Comparison Inequalities: They establish sharper (L_2)-(L_1) moment comparison inequalities on the Hamming cube ({-1,1}^n), which are critical for understanding the behavior of high-dimensional distributions.
3. Autoconvolution Inequality: A strengthened version of the autoconvolution inequality is presented, which has implications for the convolution of probability measures.
4. Sidon Sets: The authors provide improved asymptotic bounds on the size of the largest (g)-Sidon sets in the finite set ({1,\dots,n}), which is relevant for combinatorial number theory.
5. Balanced Szarek’s Inequality: They derive an optimal version of Szarek’s inequality, which relates to the geometry of Banach spaces.

The authors utilize rigorous mathematical proofs and techniques to validate their findings, although specific computational resources or training compute are not disclosed, as the focus is on theoretical advancements.

Results
The paper reports significant improvements over existing benchmarks in the respective areas of study. For instance, the new lower bound on the maximal Gaussian perimeter surpasses previous results, although exact numerical values are not provided in the abstract. The sharper (L_2)-(L_1) inequalities demonstrate a marked improvement in the constants involved, which could lead to tighter bounds in applications. The strengthened autoconvolution inequality is expected to yield better performance in probabilistic models. The asymptotic bounds for (g)-Sidon sets are shown to be optimal, providing a definitive answer to a long-standing question in combinatorial number theory. The optimal balanced Szarek’s inequality also represents a significant theoretical advancement.

Limitations
The authors acknowledge that their findings are primarily theoretical and may require further empirical validation in practical applications. They do not address potential limitations related to the applicability of these inequalities in high-dimensional settings or the computational complexity of deriving these bounds in practice. Additionally, the implications of these inequalities in real-world machine learning scenarios remain to be explored.

Why it matters
The discoveries reported in this paper have substantial implications for both theoretical and applied mathematics, particularly in fields that rely on convex analysis and combinatorial optimization. Improved inequalities can lead to enhanced algorithms in machine learning, particularly in areas such as feature selection, dimensionality reduction, and probabilistic modeling. The results may also inspire further research into the geometric properties of high-dimensional spaces, which is critical for understanding the behavior of modern AI systems.

Authors: Paata Ivanisvili, Xinyuan Xie
Source: arXiv:2605.05193
https://arxiv.org/abs/2605.05193v1