On the Convergence of the Density of Terras' Set

The Collatz Conjecture's connection to dynamical systems opens it to a variety of techniques aimed at recurrence and density results. First, we turn to density results and strengthen the result of Terras through finding a strict rate of convergence. This rate gives a preliminary result on the Triangle Conjecture, which describes a set nodes that would dominate $L^{C} = \{y \in \N \, | \, T^{k}(y) > y , \,\forall k \in \N \}$. Second, we extend prior arguments to show that the construction of several classes of measures imply the bounded trajectories piece of the Collatz Conjecture.