When are sequences of Boolean functions tame?

Abstract

In [10], Jonasson and Steif conjectured that no non-degenerate sequence of transitive Boolean functions ${(f_n)}_{n⩾1}$ with ${lim}_{n\to \mathrm{\infty }}I(f_n)=\mathrm{\infty }$ could be tame (with respect to some ${(p_n)}_{n⩾1}$). In a companion paper [5], the author showed that this conjecture in its full generality is false, by providing a counter-example for the case when, at the same time, ${lim}_{n\to \mathrm{\infty }}n{p}_n=\mathrm{\infty }$ and ${lim}_{n\to \mathrm{\infty }}{n}^{\mathit{\alpha }}{p}_n=0$ for some $\mathit{\alpha }\in (0,1)$. In this paper we show that with slightly different assumptions, the conclusion of the conjecture holds when the sequence ${(p_n)}_{n⩾1}$ is bounded away from zero and one.