Old URL: https://wesscholar.wesleyan.edu/etd_mas_theses/121
In this paper we explore the relationship between continued fractions and Diophantine approximation using an alternative geometric view developed by Caroline Series in her 1985 paper The Modular Surface and Continued Fractions. Continued fractions provide many useful tools for answering problems in Diophantine approximation, and we begin by giving an overview of this relationship from the classical perspective. We then explore ways to bring dynamical systems into the discussion, specifically the action of PSL(2, R) on the space L of unimodular lattices in R2. This lays a foundation for Series’ work which also deals with certain dynamical flows.
We let H be the hyperbolic plane, and instead of tesselating H in the usual way by copies of the fundamental region {|<(z)| ≤ 1/2, |z| ≥ 1} of PSL(2, Z), we construct an alternate tesselation using Farey fractions. Then Series showed that the way in which a geodesic γ cuts across the triangles of this tesselation is intimately linked to the continued fraction expansions of the endpoints of γ on the real line. We utilize this connection to provide nice visual observations of known properties of continued fractions and their relations to problems in Diophantine approximation that we saw earlier in the paper.