A Lower Bound on the Constant in the Fourier Min-Entropy/Influence Conjecture

TL;DR

This work targets the Fourier Min-entropy/Influence (FMEI) conjecture by constructing new Boolean functions that push the lower bound on the universal constant $D$. It introduces a palindromic, disjoint-composition framework that preserves $H_{\infty}$ while increasing ${\rm Inf}$, enabling amplification of the min-entropy/influence ratio. Applying this method to a 5-variable seed from an exhaustive search yields a 30-variable function family with $\frac{H_{\infty}}{\mathrm{Inf}} = \frac{128}{45} \approx 2.8444$, the best known to date, and 384 such functions arise from the construction. The results advance understanding of FMEI, provide a concrete amplification technique, and highlight remaining open questions about achieving larger ratios and the broader FEI/FMEI landscape, including rotation-symmetric cases.

Abstract

We describe a new construction of Boolean functions. A specific instance of our construction provides a 30-variable Boolean function having min-entropy/influence ratio to be $128/45 \approx 2.8444$ which is presently the highest known value of this ratio that is achieved by any Boolean function. Correspondingly, $128/45$ is also presently the best known lower bound on the universal constant of the Fourier min-entropy/influence conjecture.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2: simplified form of Proposition $2$ in o2013composition
• Remark 1
• Theorem 3
• Theorem 4
• Example 1
• Theorem 5
• Example 2
• Proposition 1
• Proposition 2