From: hillman@math.washington.edu (Christopher Hillman)
Newsgroups: sci.math
Subject: Re: Multidimensional Continued Fractions
Date: 1 Jul 1997 11:49:40 GMT
Organization: "University of Washington, Mathematics, Seattle"

noadd@nowhere.com (No Chance) writes:

|> 	A few months ago, someone posted a reply to an article about the 
|> continued fraction expansion of pi. At the end of the article, the poster wrote 
|> that research was being done on mulitdimensional continued fractions. I was 
|> wondering if anyone could tell me anything about this subject and give me some 
|> refrences.

You are probably thinking of an article I posted (I didn't save a copy).

The ordinary continued fraction algorithm provides a way to expand a real
number in a way quite different from a "decimal" expansion wrt to some base,
one which reveals some algebraic/number theoretic structure much better.
By truncating the expansion after n, n+1, n+2, ... terms we obtain a
sequence of rational approximations.

A multidimensional CFA is just some algorithm which gives a sequence of
rational approximations to a d-tuple of real numbers.  The best known
such algorithm is the Jacobi-Perron algorithm.

One would naturally hope to be able to find an algorithm with a theory
which works out just as nicely as the one dimensional algorithm, and
which yields not only a definite sequence of approximations which is
in some sense optimal but which also yields an "expansion" which
reveals something about the number theoretic properties of the d-tuple,
in particular whether the various components are rationally independent.

Alas, it turns out that in higher dimensions there are MANY competing
algorithms, all equally disappointing ;)  Well, if not all equally
disappointing, certainly disappointing for one reason or another.
Algorithms which are good from one standpoint are often quite bad
according to another way of thinking.  Yet in one dimension there
is essentially only one algorithm which is at all reasonable, and
this one turns out to be good for many purposes.

References: two books

A. J. Brentjes, Multidimensional Continued Fraction Algorithms,
Amsterdam: Mathematisch Centrum, 1981.

Fritz Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory,
Oxford U Press, 1995.

(Has a chapter on multidimensional CFA's and many references.)

Some papers:

Giles Lachaud, Sails and Klein Polyhedra, Contemporary Mathematics, to appear.
(According to Vershik, the notion of a sail is the best to come along in this
field for years.) 

David J. Grabiner, Farey Nets and Multidimensional CFA's,
      Mh. Math. 114 (1992) 35-60

J. C. Lagarias, Geodesic Multdimensional Continued Fractions,
      Proc. Lon. Math. Soc 69 (1994) 464-488.

A. Nogueira, The Three-Dimensional Continued Fraction Algorithm,
      Is. J. Math. 90 (1995) 373-401.

Shunji Ito and Makoto Ohtsuki, Parallelogram Tilings and Jacobi-Perron Algorithm,
      Tokyo J. Math. 17 (1994): 33-58.

(The reason for my interest in this is that these CFA's turn out to be relevant
for studying the combinatorial properties of certain types of tilings which
are idealized models of quasicrystals.)

This should give a quick impression of the scope of current research--- there are
ALOT of ideas out there!

Chris Hillman