On the Existence of $U$-Polygons of Class $c\ge 4$ 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 $c\ge 4$ with all their vertices in certain subsets of the plane and derive explicit results in the case of cyclotomic model sets.