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

An Analytic Heuristic for Multiplicity Computation for Zaremba's Conjecture