The Hurwitz Complex Continued Fraction

The Hurwitz complex continued fraction algorithm generates Gaussian rational approximations to an arbitrary complex number α by way of a sequence (a0, a1, . . .) of Gaussian integers determined by a0 = [α], z0 = α − a0, (where [u] denotes the Gaussian integer nearest u) and for j ≥ 1, aj = [1/zj−1], zj = 1/zj−1−aj . The rational approximations are the finite continued fractions [a0; a1, . . . , ar]. We establish a result for the Hurwitz algorithm analogous to the Gauss-Kuz’min theorem, and we investigate a class of algebraic α of degree 4 for which the behavior of the resulting sequences 〈aj〉 and 〈zj〉 is quite different from that which prevails almost everywhere. 1 Complex Continued Fractions A complex continued fraction algorithm is, broadly speaking, an algorithm that provides approximations by ratios of Gaussian integers, or integers of another complex number field, to a given complex number. We are faced with an immediate choice: does speed matter, or does approximation quality trump everything else? If quality of approximation is paramount, then the algorithm of choice is due to Asmus Schmidt[7]. The basic idea is to partition (we ignore issues concerning breaking ties at the boundaries) the complex plane into pieces bounded by line segments and arcs of circles, and to successively refine this partition. The partition component to which the target ξ belongs shrinks down to ξ as this process is iterated. Each component of the partition is associated with a linear fractional map z → (az + b)/(cz + d), where the coefficients a, b, c, d are Gaussian integers, as well as with the matrix ( a b c d )