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 ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) of real numbers (satisfying 0 > α i > 1 0 > {\alpha _i} > 1 ) a sequence b 1 , b 2 , … {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 α 1 {\alpha _1} . A simple recursive procedure applied to b 1 , b 2 , … {b_1}, {b_2}, \ldots produces a sequence a ( n ) = ( A n ( 1 ) / B n , … , A n ( k ) / B n ) ( n = 1 , 2 , … ; A n ( i ) ⩾ 0 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 {B_n} > 0 are integers) of simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and a sequence c ( n ) = ( c n 0 , c n 1 , … , c n k ) ( n = 1 , 2 , … ) c(n) = ({c_{n0}},{c_{n1}}, \ldots ,{c_{nk}})\;(n = 1,2, \ldots ) of integer ( k + 1 ) (k + 1) -tuples such that the linear combination c n 0 + c n 1 α 1 + ⋯ + c n k α k {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 ) , … a(1),a(2), \ldots contains all of the "best" simultaneous rational approximations to ( α 1 , … , α k ) ({\alpha _1}, \ldots ,{\alpha _k}) and that the sequence c ( 1 ) , c ( 2 ) , … c(1),c(2), \ldots contains all of the "best" approximations to zero by the linear form x 0 + x 1 α 1 + ⋯ + x n α n {x_0} + {x_1}{\alpha _1} + \cdots + {x_n}{\alpha _n} . For the special case k = 2 and α 1 = θ 2 − 1 , α 2 = θ − 1 {\alpha _1} = {\theta ^2} - 1,{\alpha _2} = \theta - 1 (where θ = 2 cos ⁡ ( 2 π / 7 ) \theta = 2\cos (2\pi /7) is the positive root of x 3 + x 2 − 2 x − 1 = 0 {x^3} + {x^2} - 2x - 1 = 0 , Szekeres further conjectured that the 2-fraction b 1 , b 2 , … {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 ( θ 2 − 1 ) + x 2 ( θ − 1 ) {x_0} + {x_1}({\theta ^2} - 1) + {x_2}(\theta - 1) and concerning almost periodicity for the 2-fraction of ( θ 2 − 1 , θ − 1 ) ({\theta ^2} - 1,\theta - 1) are proved. The method used can be applied to other pairs of cubic irrationals α 1 , α 2 {\alpha _1},{\alpha _2} .