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

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 $\{1,\mathrm{.}\mathrm{.}\mathrm{.},r{\}}^{\text{NN}}$, generalizing one of the main results of a paper by Talagrand (1994).