Grokability in five inequalities

Technical Summary of "Grokability in five inequalities"

Overview
This note reports on five mathematical discoveries in analysis, probability theory, convex geometry, and additive combinatorics achieved through collaborative conversations with the Large Language Model (LLM) Grok. The authors, Paata Ivanisvili and Xinyuan Xie, confirm that these results are new in the public literature. The article serves as a case study for "conversational" mathematical discovery, demonstrating that logically reasoning LLMs can refine existing arguments, correct minor inaccuracies, and explore parameter spaces to achieve improved constants in established inequalities.

Problem Context and Methodology
The article addresses five distinct problems where the goal is to determine optimal or improved constants in various inequalities. The methodology relies on a "conversational approach," in which the authors employ Grok to:

1. Verify existing sophisticated proofs (e.g., Nazarov's construction).
2. Identify and correct small numerical inaccuracies or missing proofs in the literature.
3. Refine parameters in constructions to optimize bounds.
4. Apply known frameworks (e.g., semigroup methods, orthogonal bases) to new settings (e.g., conditional random variables).

In contrast to automated reasoning systems that parameterize problems in code and iteratively develop solutions through exhaustive search (a method used in previous work for the constant $C_{1a}$), this approach utilizes natural language dialogues to steer mathematical reasoning.

Main Contributions and Results

1. Improved Lower Bound for the Maximum Gaussian Surface Area

• Problem: Determining the growth of $\Gamma(n)$, the maximum Gaussian surface area of convex sets in $\mathbb{R}^n$. The order of growth is known to be $n^{1/4}$, yet the constant factor has been the subject of investigation since Nazarov's 2003 construction.
• Methodology: The authors followed Nazarov's strategy of constructing a random convex polytope. Grok assisted in providing detailed proofs for intermediate claims that remained unproven in the original article and corrected a minor inaccuracy in the exponent of an integrand. Subsequently, the model helped refine the parameters ($a, b, W$) in Nazarov's construction.
• Result: The authors prove $\liminf_{n\to\infty} \Gamma(n)/n^{1/4} \ge 0.31258$. This represents an improvement of approximately 9% over the previous best lower bound of $e^{-5/4} \approx 0.2865$.

2. Sharper $L^2-L^1$ Moment Comparison on the Hamming Cube

• Problem: Determining the optimal exponential base $C^*$ such that $\|f\|_2 \le C^d \|f\|_1$ holds for functions $f$ on $\{-1, 1\}^n$ with Fourier-Walsh degree $d$. A 2014 question on MathOverflow asked whether $C^* = \sqrt{2}$.
• Methodology: The authors utilized the observation that strong $L^4/L^2$ growth in a family of functions provides lower bounds for $C^*$. Grok suggested using Hermite polynomials, evaluated at long linear forms (a Gaussian model), to construct a counterexample to the $\sqrt{2}$ conjecture.
• Result: The article establishes $2.408\dots \ge C^* \ge \sqrt{3}$. The lower bound $\sqrt{3}$ answers the MathOverflow question in the negative. The improvement of the upper bound from $2.69\dots$ to $2.408\dots$ was noted by the authors of a related article but had not been explicitly published; Grok recovered this refinement.

3. Optimal Balanced Szarek Inequality

• Problem: Extending the Khintchine inequality to the "middle slice" of the Hamming cube $\Omega_n = \{x \in \{-1, 1\}^n : \sum x_i = 0\}$, where coordinates are dependent. Specifically, the goal is to find the optimal constant for the $L^1$ endpoint ($p=1$).
• Methodology: The authors adapted the semigroup framework of Kwapie'n, Lata\l a, and Oleszkiewicz, typically used for independent variables. To address the lack of independence, they relied on Filmus's recent results regarding orthogonal bases for the slice of the Hamming cube.
• Result: They prove that the inequality $c_n \sqrt{E|\sum a_i x_i|^2} \le E|\sum a_i x_i|$ holds with the optimal constant $c_n = \sqrt{\frac{n-2}{2(n-1)}}$. This is the first documentation of the optimal constant for this endpoint in the literature.

4. Improved Lower Bound for Auto-Convolution and $g$-Sidon Sets

• Problem: Finding the largest universal constant $c$ such that $\sup_x (f*f)(x) \ge c (\int f)^2$ holds for non-negative $f$ with support in $[-1/4, 1/4]$. This constant is related to the asymptotic size of $g$-Sidon sets.
• Methodology: The authors refined an estimate from a previous study [7], which had used extensive CPU time (20,000 hours) for exhaustive search in discrete spaces. Grok was prompted to tighten the inequality in the error term of the existing proof.
• Result: The lower bound is improved from $c \ge 1.28$ to $c \ge 1.2802$. The improvement results from a tighter bound for the $L^\infty$ norm of the auto-convolution of a specific error term, derived via a simple natural language prompt.

5. Significance and Claims
The article primarily claims significance as a demonstration of the "Grokability" of mathematical problems by current advanced reasoning models.

• Novelty of Approach: The authors highlight that automated reasoning systems often rely on explicit parameterization and code development, whereas this "conversational approach" successfully uncovered improvements in problems previously intensively studied via structured, exhaustive searches (particularly the problem of the constant $C_{1a}$).
• Capability of LLMs: The results suggest that LLMs can perform sophisticated analytical arguments, explore the limits of existing methods, and implement new ideas with significant precision.
• Modesty: The authors are modest regarding the magnitude of the mathematical contributions. They note that the improvement of the auto-convolution constant is mathematically "not particularly substantial" and represents a slight refinement. However, they emphasize the methodological implication: that conversational LLMs can extract further improvements from problems where one might have thought explicit structural search had exhausted the potential.

The article concludes by providing links to the specific chat sessions with Grok that led to each result, aiming to keep the mathematical community informed about the growing utility of these tools in mathematical exploration.