The Szekeres multidimensional continued fraction

Abstract:

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 ${x^3} + {x^2} - 2x - 1 = 0$ , Szekeres further conjectured that the 2-fraction ${b_1},{b_2}, \ldots$ is "almost periodic" in a precisely defined sense. In this paper the Szekeres conjectures concerning best approximations to zero by the linear form ${x_0} + {x_1}({\theta ^2} - 1) + {x_2}(\theta - 1)$ and concerning almost periodicity for the 2-fraction of $({\theta ^2} - 1,\theta - 1)$ are proved. The method used can be applied to other pairs of cubic irrationals ${\alpha _1},{\alpha _2}$.