  We consider the intersection of a convex surface $\Ga$ with a periodic
perforation of $\R^d$, which looks like a sieve, given by $T_\e = \bigcup_{k\in
\Z^d}\{\e k+a_\e T\}$ where $T$ is a given compact set and $a_\e\ll \e$ is the
size of the perforation in the $\e$-cell $(0, \e)^d\subset \R^d$. When $\e$
tends to zero we establish uniform estimates for $p$-capacity $1<p<d$ and
discrepancy of distributions of intersection $\Ga\cap T_\e$. As an application
one gets that the thin obstacle problem with the obstacle defined on the
intersection of $\Ga$ and perforations, in given bounded domain, is
homogenizable when $p<1+\frac d4$. This result is new even for the classical
Laplace operator.
