On the existence of UU-polygons of class c≥4c≥4 in planar point sets

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