From: bumby@lagrange.rutgers.edu (Richard Bumby)
Subject: Re: Continued fractions from random numbers
Date: 10 Aug 1999 18:37:55 -0400
Newsgroups: sci.math
Keywords: sets of numbers with good CF approximations (Mahler, Koksma)
kramsay@aol.commangled (Keith Ramsay) writes:
>In article
>, Chris
>Hillman writes:
>|Interestingly enough, you have looked at
>|two extreme examples: the continued fraction expansion of algebraic
>|numbers turn out to converge more slowly than the continued fraction
>|expansion of transcendentals, and the cfe of the golden ratio converges
>|unusually slowly even for an algebraic number, whereas the cfe of e
>|converges unusually quickly even for a transcendental. Keywords:
>|Hurwitz's "hierarchy of irrationality", Liouville's theorem and Roth's
>|theorem on rational approximation. (Generalizations: Dirichlet's theorem,
>|Schmidt's theorem on simultaneous rational approximation.)
>In what sense do the continued fraction expansions of algebraic
>numbers converge more slowly than the expansions of transcendentals?
>If I remember correctly, I'd suggest it's more apt to describe them
>as widely overlapping ranges. The very slowest converging c.f.s are for
>quadratic irrationalities (which are algebraic). The numbers for which
>there is some constant c>0 for which |x-p/q|>=c/q^2 for almost all p/q
>are said to lie on the Markov spectrum at the sup of the c's for which
>this is true. The top part of the Markov spectrum has numbers whose
>continued fraction expansions, in order to converge so very slowly,
>are forced to have only partial quotients of just 1 or 2, and also more
>delicately to be periodic. The top of the spectrum starts with the
>golden ratio [1,1,1,1,...] and follows some pattern I'm not familiar
>with, including such numbers as [1,1,2,2,1,1,2,2,...].
>A little further down, however, there are uncountably many c.f.s like
>[1,2,1,1,2,1,1,1,2,1,1,1,2,...] which aren't periodic but have bounded
>partial quotients, and some of them (because they are uncountably many
>of them) have to be transcendental. Those continued fraction expansions
>converge more slowly than *some* quadratic irrationalities, despite
>the number being transcendental, and not a lot better than the
>golden ratio.
>...
Rational approximations aren't powerful enough to distinguish
algebraic numbers. A clear distinction appears if you look at values
of polynomials of all degrees (Mahler's classification) or
approximation by algebraic numbers (Koksma's classification). The
dependence of the approximant on degree and height needs careful
treatment; details can be found in books on transcendental numbers.
The continued fraction of e has a nice pattern, but it is not
particularly rapidly convergent. If it were, then e would have very
good rational approximations. Takeshi Okano, "A note on
rational approximations to e", Tokyo L. Math, 15 (1992), 129-133 shows
that this is far from the case, and my review (MR 93f:11054 or
RevNumTh J82-677) uses the method of the paper to give a slightly
sharper result.
The Markoff (the usual spelling in this context) Spectrum deals with
the infima of indefinite quadratic forms. The forms can be represented
by *double* *ended* continued fractions. The related questions about
approximation of numbers leads to the slightly smaller Lagrange
Spectrum. The size of the partial quotients is a fairly good measure
of where something fits in these spectra, and this can be refined by
considering the patterns in strings of consecutive partial quotients.
The end of the spectrum where the numbers with no very good
approximations live contains only periodic continued fractions with
very rigid patterns. This was Markoff's contribution. There are many
interrelated characterizations of this part of the spectrum. The
conference proceedings, "number theory with an emphasis on the Markoff
spectrum", edited by Andrew D. Pollington and William Moran (Marcel
Dekker lecture notes, vol. 147, 1993) collects diverse views of the
subject.
--
R. T. Bumby ** Rutgers Math || Amer. Math. Monthly Problems Editor 1992--1996
bumby@math.rutgers.edu ||
Telephone: [USA] 732-445-0277 (full-time message line) FAX 732-445-5530