When are sequences of Boolean functions tame?

TL;DR

This paper investigates conditions under which sequences of Boolean functions are considered tame, refuting a previous conjecture in certain regimes and confirming it under different assumptions about the parameters.

Contribution

It demonstrates that the conjecture by Jonasson and Steif is false in some cases but holds true under alternative assumptions on the sequence parameters.

Findings

Counter-example provided for the conjecture when np_n diverges and n^α p_n tends to zero

The conjecture holds when p_n is bounded away from zero and one under certain conditions

Abstract

In \cite{js2006}, Jonasson and Steif conjectured that no non-degenerate sequence of transitive Boolean functions $ (f_n)_{n \geq 1}$ with $ \lim_{n \to \infty} I(f_n)= \infty $ could be tame (with respect to some $ (p_n)_{n \geq 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} np_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 $(p_n)_{n \geq 1}$ is bounded away from zero and one.