The szekeres multidimensional continued fraction

In his paper "Multidimensional continued fractions" (Ann. Univ. Sei. Budapest. EOtvOs Sect. Math., y. 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 fc-tuple (K-0)of real numbers (satisfying 0 < αI < 1) a sequence b1, b2,… of positive integers; this sequence is called a continued fc-fraction, and for k = 1 it is just the sequence of partial quotients of the ordinary continued fraction for oij. A simple recursive procedure applied to öj, b2,… produces a sequence (An(1)/Bn,…, An(k)/Bn) (n = 1, 2,…; A® > 0 and Bn > o are integers) of simultaneous rational approximations to (aj.ak) and a sequence c(n) = (cnα, c„j,…, cnk) (n = 1, 2,…) of integer (fe + l)-tuples such that the linear combination cng + cnXax + • • • + cnkuk approximates zero. Szekeres conjectured, on the basis of extensive computations, that the sequence a(l), a(2),… contains all of the "best" simultaneous rational approximations to (c«j,…, ak) and that the sequence α (l), α (2),… contains all of the "best" approximations to zero by the linear form x0 + xxax + • • • + xnan. For the special case fe = 2 and aj = α2 - 1, a2 = α - 1 (where 6=2 cos(27r/7) is the positive root of x3 + x2 - 2x - 1 =α), Szekeres further conjectured that the 2-fraction ftj, b2,… is "almost periodic" in a precisely defined sense. In this paper the Szekeres conjectures concerning best approximations to zero by the linear form x0 + Xj(α — 1) + x2(6 - 1) and concerning almost periodicity for the 2-fraction of (α—1, 9-1) are proved. The method used can be applied to other pairs of cubic irrationals oij, c¡2.