Subrings of $\mathbb{C}$ Generated by Angles

TL;DR

This paper characterizes when certain geometric constructions involving angles generate rings in the complex plane, providing conditions under which the resulting set is closed under multiplication.

Contribution

It establishes criteria for when the inductively defined set $R(U)$ forms a ring, especially when $1 otin U$ and $|U| eq 4$, extending previous understanding.

Findings

$R(U)$ equals the module over $Z[P]$ generated by initial points.

Closure under multiplication occurs when pairwise products of initial points remain in $R(U)$.

Main results apply when $1 otin U$ and $|U| eq 4$.

Abstract

Consider the following inductively defined set. Given a collection $U$ of unit magnitude complex numbers, and a set initially containing just 0 and 1, through each point in the set, draw lines whose angles with the real axis are in $U$. Add every intersection of such lines to the set. Upon taking the closure, we obtain $R(U)$. We investigated for which $U$, $R(U)$ is a ring. Our main result holds for when $1 \in U$ and $|U| \ge 4$. If $P$ is the set of real numbers in $R(U)$ generated in the second step of the construction, then $R(U)$ equals the module over $\mathbb{Z}[P]$ generated by the set of points made in the first step of the construction. This lets us show that whenever the pairwise products of points made in the first step remain inside $R(U)$, it is closed under multiplication, and is thus a ring.