🚀 go-pugleaf

In article <6srm44$678$1@news-1.news.gte.net> aardwolf@gte.net writes:
>Here is a problem that has been gnawing at me for years.  I am giving up
>on it and I'm going to share it with the rest of you.  
>
>Consider the recurrence un = an un-1 + bn un-2 , where we are given
>initial values
>u0 and u1 and sequences of an's and bn's .  We all know that if pn and
>qn are solutions of the recurrence, with p0 = 1, p1 = a1, q0 = 0, q1 =
>b1 (and also p-1 = 0, q-1 = 1), then the fractions pn/qn are the
>convergents of the continued fraction :
>
>a1+ b1/a2+ b2/a3+... , which may be terminated at any predetermined
>point. 
>
>There is also a matrix analogy to this process, namely if we multiply
>the 2 X 2 matrices
>
>( ai  1 )                                      (pn  pn-1 )
>( bi  0 )  from i = 1 to n, we get the matrix  (qn  qn-1 )  .
>
>This process can be extended, at least with the matrices and the
>recurrence.  For
>3 dimensions, we would have the recurrence un = an un-1 + bn un-2 + cn
>un-3 , with
>an, bn and cn given.  If pn, qn and rn are solutions of this recurrence,
>with
>p-2 = p-1 = 0, p0 = 1 ; q-2 = 0 , q-1 = 1 , q0 = 0 ; r-2 = 1, r-1 = r-0
>= 0 
>(which implies p1 = a1 , q1 = b1, r1 = c1 ), we can also generate the
>sequences
>pn, qn and rn as the entries in the following matrix product :
>
>(ai  1  0 )						(pn  pn-1 pn-2 )
>(bi  0  1 )  from i = 1 to n , which yields the matrix  (qn  qn-1 qn-2 )
>(ci  0  0 )                                             (rn  rn-1 rn-2 )
>
>Now comes the $64 million question:  what objects comparable to
>continued fractions
>can be conjured from the situation for 3 dimensions?  Same question for
>higher dimensions. The extension to such higher dimensions is obvious,
>as far as the recurrence and the matrix product are concerned.  But not
>so with the continued fraction algorithm.
>The difficulty seems to be that a fraction is inherently a binary
>operation operating
>on two integers, while in the case of 3 or higher dimensions as
>indicated above, there
>does not seem to be a "ternary" (or higher) process involving an
>extension of such
>fractions.  
>
>Any and all assistance with this research problem would be appreciated. 
>I am receptive
>to co-authoring a paper, if this should lead to a productive result. 
>Thanks for your
>time.  Paul Bruckman
>

Your question is a good one. I can only refer you to what I know in
terms of people:

First we have lattice reduction algorithms (best-known is LLL =
Lenstra, Lenstra, Lovasz) - here I would recommend looking at the
work of H.R.P Ferguson.

A person who has axiomatized continued fractions is Rabin at Harvard.

A student of V.I. Arnol'd (at Paris 9) is writing a thesis on a
general continued fraction algorithm using ideas from topology and
geometry.

Continued fractions and their generalizations remain a fecund area for
research.

Good luck!

John McKay

-- 
But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.