On the existence of -polygons of class in planar point sets

For a finite set U of directions in the Euclidean plane, a convex non-degenerate polygon P is called a U -polygon if every line parallel to a direction of U that meets a vertex of P also meets another vertex of P . We characterize the numbers of edges of U -polygons of class c ≥ 4 with all their vertices in certain subsets of the plane and derive explicit results in the case of cyclotomic model sets.