Balanced tribes construction with unused coordinates ($m=\lfloor (\log 2)2^w\rfloor$)

Choose $w\ge 1$ maximal such that $wm\le n$ for the displayed $m$, and take the OR of $m$ AND blocks of size $w$, ignoring unused coordinates. Its failure probability is $(1-2^{-w}{)}^m$, between one quarter and three quarters. Active coordinates have influence $2^{1-w}(1-2^{-w}{)}^{m-1}$ and unused ones have zero influence. Maximality gives $2^{1-w}<4(\log 2)(w+1)/n$; for $n\ge 2$, $w\le {\log }_{2}n$, so every influence is below $8\log n/n$. The construction is therefore a quite fair Boolean function with small influences for every $n\ge 2$. The logarithmic bound cannot hold at $n=1$, where every quite fair function has influence one.