  Zaremba's Conjecture concerns the formation of continued fractions with
partial quotients restricted to a given alphabet. In order to answer the
numerous questions that arrive from this conjecture, it is best to consider a
semi-group, often denoted $\Gamma_{A}$, which arises naturally as a subset of
$SL_2(\mathbb{Z})$ when considering finite continued fractions. To translate
back from this semi-group into rational numbers, we select a projection mapping
satisfying certain criteria to recover the numerator and denominator of the
continued fractions in rational form. The central question of our research is
to determine the multiplicity of a given denominator. To this end, we develop a
heuristic method similar to the Hardy-Littlewood Circle Method. We compare this
theoretical model to the exact data, gleaned by simulation, and demonstrate
that our formula appears to be asymptotically valid. We then evaluate different
aspects of the accuracy of our formula.
