Applications of dynamical systems to Farey sequences and continued fractions

This thesis explores three main topics in the application of ergodic theory and dynamical systems to equidistribution and spacing statistics in number theory. The first is concerned with utilizing the ergodic properties of the horocycle flow in SL(2,R) to study the spacing statistics of Farey fractions. For a given finite index subgroup H ⊆ SL(2,Z), we use a process developed by Fisher and Schmidt to lift a cross section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to the finite cover SL(2,R)/H of SL(2,R)/SL(2,Z). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erdős, Szüsz, and Turán. The latter two topics of this thesis establish properties of the Farey map F by analyzing the transfer operators of F and the Gauss map G, well known maps of the unit interval relating to continued fractions. We first prove an equidistribution result for the periodic points of the Farey map using a connection between continued fractions and the geodesic flow in SL(2,Z)\SL(2,R) illuminated by Series. Specifically, we expand a cross section of the geodesic flow given by Series to produce another section whose first return map under the geodesic flow is a double cover of the natural extension of the Farey map. We then use this cross section to extend the correspondence between the closed geodesics on the modular surface and the periodic points of G to include the periodic points of F. Then, analogous to the work of Pollicott, we find the limiting distribution of the periodic points of F when they are ordered according to the length of their corresponding closed geodesics through the analysis of the transfer operator of G. Lastly, we provide effective asymptotic results for the equidistribution of sets of the form F⁻ⁿ([α,β]), where [α,β] ⊆ (0,1], and, as a corollary, certain weighted subsets of the Stern-Brocot sequence. To do so, we employ mostly basic properties of the transfer operator of the Farey map and an application of Freud's effective version of Karamata's Tauberian theorem. This strengthens previous work of Kesseböhmer and Stratmann, who first established the equidistribution results utilizing infinite ergodic theory