The 3x+1 Problem and Integer Representations

The $3x+1$ Problem asks if whether for every natural number $n$, there exists a finite number of iterations of the piecewise function $$f(2n)=n, \quad f(2n-1)=6n-2,$$ with an iterate equal to the number $1$, or in other words, every sequence contains the trivial cycle $\left\langle {4,2,1}\right\rangle$. We use a set-theoretic approach to get representations of all inverse iterates of the number $1$. The representations, which are exponential Diophantine equations, help us study both the \textit{mixing} property of $f$ and the asymptotic behavior of sequences containing the trivial cycle. Another one of our original results is the new insight that the \textit{ones-ratio} approaches zero for such sequences, where the number of odd terms is \textit{arbitrarily large}.