On the Converse of Talagrand's Influence Inequality

Abstract

In 1994, Talagrand showed a generalization of the celebrated KKL theorem. In this work, we prove that the converse of this generalization also holds. Namely, for any sequence of numbers $0<a_1,a_2,\dots ,a_n\le 1$ such that ${\sum }_{j=1}^n{a}_j/(1-\log a_j)\ge C$ for some constant $C>0$, it is possible to find a roughly balanced Boolean function $f$ such that ${\text{Inf}}_j[f]<a_j$ for every $1\le j\le n$.