Threshold phenomena on product spaces: BKKKL revisited (once more)

TL;DR

This paper revisits and refines the understanding of threshold phenomena in Boolean functions on product spaces, extending previous results to more general settings and providing a clearer, more precise statement of the phenomena.

Contribution

It offers a generalized and more precise formulation of threshold phenomena on product spaces, building upon and extending the classical BKKKL theorem.

Findings

Generalized threshold phenomena for product spaces

Extended BKKKL influence bounds

Refined understanding of Boolean function influences

Abstract

We revisit the work of Bourgain, Kahn, Kalai, Katznelson and Linial (1992) -- referred to as ``BKKKL'' in the title -- about influences on Boolean functions in order to give a precise statement of threshold phenomenon on the product space $\{1,..., r\}^\NN$, generalizing one of the main results of a paper by Talagrand (1994).