A unifying theory for metrical results on regular continued fraction convergents and mediants

Abstract

We revisit Ito's (\cite{I1989}) natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results -- including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of Lévy's Theorem to subsequences of convergents and mediants -- are presented as corollaries within this unifying theory.

35 pages, 8 figures. Renumbering reflects published version. Mistake found after publication in Thm. 4.8; discussion and counterexample to appear in corrigendum in Math. Comp. Can be fixed by assuming $\bar{\hat{I}}¼(\partial S)=0$ and $(0,0)\notin \bar{S}\cup \bar{R}$. Cor. 4.9, Thms. 5.2, 5.29 should also assume $(0,0)\notin \bar{R}$. Original applications are unaffected