An Analytic Heuristic for Multiplicity Computation for Zaremba's Conjecture

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.