A canonical form for planar Farey sets

Abstract

In a 1975 Acta Mathematica paper [3], Asmus Schmidt developed a new method for generating Gaussian rational approximations to complex irrational numbers. The approximations correspond to decreasing sequences of planar sets, which can be viewed as converging to a number (. These sets, which are called Farey sets, in turn correspond to complex unimodular transformations. Each Farey set has a canonical representation in terms of a composition of elements of a special set of seven unimodular maps. Schmidt's method can be viewed as analogous to the classical continued fraction algorithm for real numbers, with a sequence of the seven special maps corresponding to the usual sequence of partial quotients. Each complex irrational number has at most 2 sequences of maps associated with it. These sequences (called chains) are periodic if ( is quadratic and can be used to find the fundamental solution to the Pellian equation X2 Dy2 ? 1, to mention but two analogies to continued fractions. This paper contains a new proof [2] of the canonical form for Farey sets, on which Schmidt's method depends. The relevant definitions are introduced first, together with four fundamental lemmas. Let G be the set of unimodular transformations, i.e. complex bilinear maps of the form m: z -* (az + b)/(cz + d), where a, b, c, d E Z[i] and det m = ad bc E Qt1 = {-+I1,+i}. Let Gp and G, partition G into maps with det m = ?1 and det m = + i respectively. Let the closed upper half-plane be denoted by C, let * denote the set *= {z: O 0, N(z-2) > }