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