When are sequences of Boolean functions tame?

Abstract

In \cite{js2006}, Jonasson and Steif conjectured that no non-degenerate sequence of transitive Boolean functions $(f_n{)}_{n\ge 1}$ with ${\lim }_{n\to \infty }I(f_n)=\infty$ could be tame (with respect to some $(p_n{)}_{n\ge 1}$). In a companion paper \cite{f}, 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, ${\lim }_{n\to \infty }n{p}_n=\infty$ and ${\lim }_{n\to \infty }{n}^{\alpha }{p}_n=0$ for some $\alpha \in (0,1).$ In this paper we show that with slightly different assumptions, the conclusion of the conjecture holds when the sequence $(p_n{)}_{n\ge 1}$ is bounded away from zero and one.

arXiv comment12 pages