  The focus of this paper is on a thin obstacle problem where the obstacle is
defined on the intersection between a hyper-plane $\Gamma$ in $\mathbb{R}^n$
and a periodic perforation $\mathcal{T}_\varepsilon$ of $\mathbb{R}^n$,
depending on a small parameter $\varepsilon>0$. As $\varepsilon\to 0$, it is
crucial to estimate the frequency of intersections and to determine this number
locally. This is done using strong tools from uniform distribution. By
employing classical estimates for the discrepancy of sequences of type
$\{k\alpha\}_{k=1}^\infty$, $\alpha\in\R$, we are able to extract rather
precise information about the set $\Gamma\cap\mathcal{T}_\varepsilon$. As
$\varepsilon\to0$, we determine the limit $u$ of the solution $u_\varepsilon$
to the obstacle problem in the perforated domain, in terms of a limit equation
it solves. We obtain the typical "strange term" behaviour for the limit
problem, but with a different constant taking into account the contribution of
all different intersections, that we call the averaged capacity. Our result
depends on the normal direction of the plane, but holds for a.e. normal on the
unit sphere in $\R^n$.
