  Let $H$ be an open subgroup of a profinite group that can be expressed as
intersection of maximal subgroups of $G.$ Given a positive real number $\eta,$
we say that $H$ is an $\eta$-intersection if there exists a family of maximal
subgroups $M_1,\ldots,M_t$ such that $H=M_1\cap\ldots\cap M_t$ and
$|G:M_1|\cdots|G:M_t|\le |G:H|^{\eta}$. We investigate the meaning of this
property and its influence on the group structure.
