DIOPHANTINE APPROXIMATION OF COMPLEX NUMBERS

The classical continued fraction algorithm for real numbers associates with each real number a unique sequence of integers. This sequence has many interesting properties. For example, the sequence corresponding to a real number (xi) defines a sequence {p(,n)/q(,n)} of rational numbers which coverge to (xi) as n (--->) (INFIN). Additionally, p(,n)/q(,n) is the best rational approximation to (xi) whose denominator is no larger than q(,n). Also, rational numbers themselves have terminating continued fraction expansions, and quadratic irrationals have periodic expansions. A useful property of continued fractions is that the expansions associated with pairs of number (xi),(eta) which are equivalent under a unimodular transformation are eventually identical.;This dissertation deals with a generalization to the complex numbers of the continued fraction algorithm for real numbers. Thus, it deals with the approximation of complex irrational numbers by Gaussian rationals; that is, ratios p/q where p,q (ELEM) Z{i}, q (NOT=) 0.;Many authors have sought to generalize continued fractions to the complex numbers, yet none of these generalizations preserves all of the desirable features which obtain in the real case. Most attempts at generalization have been aimed at retaining best approximation properties. However, a paper published by Asmus Schmidt in 1975 in Acta Mathematica uses a rather different approach. It focuses on the more geometric aspects of the real continued fraction algorithm. The primary focus of this dissertation will be the exposition of Schmidt's paper. In addition there will be some new material arising from his ideas, and also discussion of the implementation of his algorithm on the computer, and the results of computation.;In the exposition it is hoped that Schmidt's work can be brought to the attention of a wider audience. It is believed that the new algorithm is perhaps more amenable to computation and theorem-proving than previous work with complex approximation. Schmidt was able to use his algorithm to find the entire set of values less than 2 of a certain approximation constant. Only the lower bound of this spectrum has been previously known. The use of his algorithm is quite essential to his argument. On the other hand, the details of his algorithm can be formidable. Examples and explanation are given to help clarify the basic structure. Schmidt's method results in a chain of unimodular maps (possibly two chains) corresponding to a number. Among the attractive properties of this chain are uniqueness, periodicity for quadratics, and chains eventually the same for equivalent numbers. His proof that numbers give rise to unique chains is modified somewhat in the dissertation to clear up some difficulties. His theorem on equivalent numbers is modified extensively to provide for certain cases not considered.;New material related to the convergents of his method and best approximations is presented.;Several BASIC computer programs have been written to perform various computations associated with the method: the multiplication of the chains of 2 x 2 complex matrices, the computation of centers and radii of certain circles in the complex plane, the computations of chains corresponding to given numbers, and the calculation of distances between certain plane sets (Farey sets). Discussion of the programs, listings, and data generated by them are included in this dissertation.