Subrings of C Generated by Angles

In [1], Buhler et al. considered the following scenario. Given a collection U of unit magnitude complex numbers and a set S of constructed points initially containing just 0 and 1, through each constructed point draw lines whose angles with the real axis are in U . The intersections of such lines are also constructed points. Upon taking the closure we form a set R(U). They investigated which U result in R(U) being a ring. Our main result holds for when 1 ∈ U and |U | ≥ 4. We classify R(U) as the set of linear combinations of elementary monomials which are the points constructed in the first step. The coefficients are taken from Z[P ] = R(U) ∩ R which is easily calculated. We also show that when |U | ≥ 4, R(U) is dense in the complex plane. Furthermore, we classify R(U) completely for when 1 ∈ U and |U | ≥ 3, showing that R(U) is a ring whenever one of the points constructed in the first step is a quadratic integer. 1 Background Suppose we are given a collection U of unit length elements of C. If we have some collection of points in C, we can draw lines through each of them with every angle in U (with respect to the real axis). In this way we can construct intersections of lines and repeat the process. Specifically, if we start with 0 and 1 in the complex plane and continue this construction forever until it’s completed, when is the resulting collection of points a subring of the complex numbers? Note that even though we are drawing lines, only the intersection points are considered to be constructed. In [1], Buhler et al. motivated