We study the notion of the influence of a set of variables on a Boolean function, which was recently introduced by Tal. We show that for an arbitrary fixed $d$, every Boolean function $f$ on $n$ variables admits a $d$-set of influence at least $\frac{1}{10} \mathbf{W}^{\geq d}(f) (\frac{\log n}{n})^d$, which is a direct generalisation of the Kahn-Kalai-Linial theorem. We give an example demonstrating essential sharpness of this result. Further, we generalise a related theorem of Oleszkiewicz regarding influences of pairs of variables.

翻译：我们研究了Tal最近引入的变量集合对布尔函数影响的概念。我们证明，对于任意固定的$d$，每个定义在$n$个变量上的布尔函数$f$都存在一个至少具有$\frac{1}{10} \mathbf{W}^{\geq d}(f) (\frac{\log n}{n})^d$影响的$d$元集合，这是Kahn-Kalai-Linial定理的直接推广。我们通过一个例子证明了该结果的本质紧性。此外，我们推广了Oleszkiewicz关于变量对影响的一个相关定理。