On the Converse of Talagrand's Influence Inequality

TL;DR

This paper proves a converse to Talagrand's generalization of the KKL theorem, showing conditions under which a balanced Boolean function with specified influence bounds exists.

Contribution

It establishes the converse of Talagrand's influence inequality, providing conditions for the existence of Boolean functions with prescribed influence bounds.

Findings

Proves the converse of Talagrand's influence inequality

Constructs Boolean functions with influence bounds under certain conditions

Links influence bounds to a summation condition involving logarithms

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 $0<a_1,a_2,\ldots,a_n\le 1$ 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 $\textrm{Inf}_j[f] < a_j$ for every $1 \le j \le n$.