 Tribes function

A tribes function is an OR of AND blocks: partition $mw$ coordinates into $m$ blocks of size $w$, and let $f=1$ when at least one block consists entirely of ones. For independent random variables with each bit uniform, $\mathbb{P}(f=0)=(1-2^{-w}{)}^m$. The influence of a variable is $2^{-(w-1)}(1-2^{-w}{)}^{m-1}$, because every other bit in its block must be one and every other block must fail. Taking $m$ approximately $(\log 2)2^w$ keeps both output probabilities bounded away from zero and gives influences of order $\log (mw)/(mw)$. This shows the order in the Kahn-Kalai-Linial theorem is sharp.