# The Szekeres multidimensional continued fraction

Author: T. W. CusickJournal: Math. Comp. 31 (1977), 280-317MSC: Primary 10F20DOI: https://doi.org/10.1090/S0025-5718-1977-0429765-5MathSciNet review: 0429765Full-text PDF Free AccessAbstract | References | Similar Articles | Additional InformationAbstract: In his paper “Multidimensional continued fractions” (Ann. Univ. Sci. Budapest. Eötvös Sect. Math., v. 13, 1970, pp. 113-140), G. Szekeres introduced a new higher dimensional analogue of the ordinary continued fraction expansion of a single real number. The Szekeres algorithm associates with each k-tuple $({\alpha _1}, \ldots ,{\alpha _k})$ of real numbers (satisfying $0 < {\alpha _i} < 1$) a sequence ${b_1}, {b_2}, \ldots$ of positive integers; this sequence is called a continued k-fraction, and for k = 1 it is just the sequence of partial quotients of the ordinary continued fraction for ${\alpha _1}$. A simple recursive procedure applied to ${b_1}, {b_2}, \ldots$ produces a sequence $a(n) = (A_n^{(1)}/{B_n}, \ldots ,A_n^{(k)}/{B_n})\;(n = 1,2, \ldots ;A_n^{(i)} \geqslant 0$ and ${B_n} > 0$ are integers) of simultaneous rational approximations to $({\alpha _1}, \ldots ,{\alpha _k})$ and a sequence $c(n) = ({c_{n0}},{c_{n1}}, \ldots ,{c_{nk}})\;(n = 1,2, \ldots )$ of integer $(k + 1)$-tuples such that the linear combination ${c_{n0}} + {c_{n1}}{\alpha _1} + \cdots + {c_{nk}}{\alpha _k}$ approximates zero. Szekeres conjectured, on the basis of extensive computations, that the sequence $a(1),a(2), \ldots$ contains all of the “best” simultaneous rational approximations to $({\alpha _1}, \ldots ,{\alpha _k})$ and that the sequence $c(1),c(2), \ldots$ contains all of the “best” approximations to zero by the linear form ${x_0} + {x_1}{\alpha _1} + \cdots + {x_n}{\alpha _n}$. For the special case k = 2 and ${\alpha _1} = {\theta ^2} – 1,{\alpha _2} = \theta – 1$ (where $\theta = 2\cos (2\pi /7)$ is the positive root of ${x^3} + {x^2} – 2x – 1 = 0$

Source: Mathematics of Computation

# Diophantine approximation  the American Mathematical Society

Abstract: A modification of the Ford geometric approach to the problem of approximation of irrational real numbers by rational fractions is developed. It is applied to find an upper bound for the Hurwitz constant for a discrete group acting in a hyperbolic space. A generalized Khinchine’s approximation theorem is also given.