On the Converse of Talagrand's Influence Inequality

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 ${\sum }_{j=1}^n{a}_j\mathrm{/}(1-\log a_j)\ge C$ for some constant $C>0$, it is possible to find a roughly balanced Boolean function $f$ such that $$\text{Inf}$_j[f] < a_j$forevery$1 \le j \le n$.