  We consider the problem of mirror invisibility for plane sets. Given a circle
and a finite number of unit vectors (defining the directions of invisibility)
such that the angles between them are commensurable with $\pi$, for any
$\varepsilon > 0$ there exists a set invisible in the chosen directions that
contains the circle and is contained in its $\varepsilon$-neighborhood. This
set is the disjoint union of infinitely many domains with piecewise smooth
boundary.
