Plane sets invisible in finitely many directions

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 $π$, 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.

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