A generalization of Boppana's entropy inequality

Technical Summary: A Generalization of Boppala's Entropy Inequality

Problem Statement
The paper addresses a key lemma in the study of the union-closed sets conjecture, specifically focusing on Boppala's entropy inequality. The original conjecture posits that for any non-empty family of sets closed under unions, there exists an element contained in at least half of the sets. Recent progress by Gilmer (2022) utilized information-theoretic methods, relying heavily on Boppala's inequality: $h(x^2) \geq \phi x h(x)$, where $\phi = (1+\sqrt{5})/2$ and $h(x)$ is the binary entropy function.

While Yuster previously conjectured a generalization of this inequality for integer $k \geq 2$ (relating to "approximate $k$-union closed set systems"), and subsequent work verified it for specific integers ($k=3,4$ and $5 \leq k \leq 20$), a proof for real $k > 1$ remained open. This paper aims to prove this generalized inequality for all real $k > 1$.

Methodology
The author employs standard calculus to analyze the behavior of a specific ratio function derived from the entropy function. The core of the proof involves:

1. Defining a Ratio Function: The paper defines $q(x) = \frac{x^{k-1}h(x)}{h(x^k)}$ for $x \in (0, 1)$, extended continuously to the boundaries $[0, 1]$.
2. Target Inequality: The goal is to show that $q(x) \leq \alpha_k$, where $\alpha_k$ is the unique positive solution to $x(1+x)^{k-1} = 1$. This is equivalent to proving the generalized inequality $\alpha_k h(x^k) \geq x^{k-1}h(x)$.
3. Critical Point Analysis: The proof demonstrates that the derivative $q'(x)$ vanishes if and only if $x = 1/(1+\alpha_k)$.
4. Auxiliary Function: To solve $q'(x)=0$, the author introduces the function $U(x) = \frac{\log(x)\log(1-x)}{h(x)}$. The condition $q'(x)=0$ reduces to the equation $U(x) = U(x^k)$.
5. Monotonicity and Symmetry: By establishing that $U(x)$ is strictly decreasing on $(0, 1/2]$ and symmetric ($U(x) = U(1-x)$), the author deduces that $U(x) = U(x^k)$ implies $x^k = 1-x$ (given $k>1$ and $0<x<1$).
6. Boundary Verification: The proof confirms that the maximum value of $q(x)$ occurs at the unique critical point $x = 1/(1+\alpha_k)$, where $q(x) = \alpha_k$, and that the boundary values $q(0)$ and $q(1)$ are strictly less than $\alpha_k$.

Key Contributions and Results

• Theorem 1: The paper proves that for any real $k > 1$, the inequality $\alpha_k h(x^k) \geq x^{k-1}h(x)$ holds for all $0 \leq x \leq 1$. Equality holds if and only if $x = 0$, $x = 1/(1+\alpha_k)$, or $x = 1$.
• Generalization: This result generalizes Boppala's inequality (the case $k=2$) and confirms Yuster's conjecture for all real $k > 1$, extending beyond previous integer-specific proofs.
• Corollary 1: The paper derives a direct application to set systems: If a finite set system $\mathcal{F}$ is $(1-\epsilon)$-approximate $k$-union closed, there exists an element contained in an $\frac{\alpha_k}{1+\alpha_k} - \delta$ fraction of the sets, where $\delta$ depends on $\epsilon, k$, and the size of the system.
• Formalization: The author notes that the proof has been formally verified in Lean 4.

Significance and Claims
The paper claims to resolve the generalization of Boppala's entropy inequality for real exponents, a step previously conjectured by Yuster. By establishing this inequality, the work provides the theoretical foundation for an analogue of the union-closed sets conjecture applicable to "approximate $k$-union closed set systems." The author emphasizes that the proof relies on standard calculus techniques, though key steps were assisted by AI tools (GPT-5.2 pro) and the formalization utilized AI assistance (Harmonic Aristotle and Gemini 3 Pro Preview). The work does not propose new experimental applications but rather solidifies the mathematical bounds used in recent information-theoretic approaches to the union-closed sets problem.