The Szekeres multidimensional continued fraction
Authors: Cusick T. W. 
Published: June 2010

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a left-parenthesis n right-parenthesis equals left-parenthesis upper A Subscript n Superscript left-parenthesis 1 right-parenthesis Baseline slash upper B Subscript n Baseline comma ellipsis comma upper A Subscript n Superscript left-parenthesis k right-parenthesis Baseline slash upper B Subscript n Baseline right-parenthesis left-parenthesis n equals 1 comma 2 comma ellipsis semicolon upper A Subscript n Superscript left-parenthesis i right-parenthesis Baseline greater-than-or-slanted-equals 0">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c left-parenthesis n right-parenthesis equals left-parenthesis c Subscript n Baseline 0 Baseline comma c Subscript n Baseline 1 Baseline comma ellipsis comma c Subscript n k Baseline right-parenthesis left-parenthesis n equals 1 comma 2 comma ellipsis right-parenthesis">

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}

.