A Note on the Entropy/Influence Conjecture

TL;DR

This paper extends the entropy/influence conjecture to biased measures on the discrete cube and proves a variant for functions with very low Fourier weight on high levels, advancing understanding of Fourier coefficient concentration.

Contribution

It generalizes the entropy/influence conjecture to biased product measures and proves a new variant for functions with minimal high-level Fourier weight.

Findings

Generalization of the conjecture to biased measures

Proof of a variant for functions with low high-level Fourier weight

Enhanced understanding of Fourier spectrum concentration

Abstract

The entropy/influence conjecture, raised by Friedgut and Kalai in 1996, seeks to relate two different measures of concentration of the Fourier coefficients of a Boolean function. Roughly saying, it claims that if the Fourier spectrum is "smeared out", then the Fourier coefficients are concentrated on "high" levels. In this note we generalize the conjecture to biased product measures on the discrete cube, and prove a variant of the conjecture for functions with an extremely low Fourier weight on the "high" levels.