On the Converse of Talagrand's Influence Inequality

In [Tal94], 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 < a1, a2, . . . , an ≤ 1 such that ∑n j=1 aj/(1 − log aj) ≥ C for some constant C > 0, it is possible to find a roughly balanced Boolean function f such that Infj [f ] < aj for every 1 ≤ j ≤ n.