Plane sets invisible in finitely many directions

Abstract

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 epsilon > 0 there exists a set invisible in the chosen directions that contains the circle and is contained in its epsilon-neighborhood. This set is the disjoint union of infinitely many domains with piecewise smooth boundary.