Re: Classes of transcendental number, continued fractions, diophantine approximations

In article <[EMAIL PROTECTED]>,
 "Diana" <[EMAIL PROTECTED]> wrote:

> I am wanting to write a master's thesis on continued fractions. A responder
> to a previous post recommended that I look at them as they relate to
> diophantine approximations.
> 
> There are books about diophantine equations at the San Jose library near me.
> I am wondering if there are articles in MathSciNet or online which might
> give me a starting point. I have studied diophantine equations a little bit
> in my number theory class, but am not sure how to direct my study.
> 
> I am also interested in the fact that Pi doesn't have a continued fraction
> expansion which is an understandable pattern, but E does. There must be more
> that these two transcendental numbers which have interesting continued
> fraction expansions? I am wondering if the "classes" of transcendental
> numbers are outlined anywhere?

Kurt Mahler gave a classification of transcendental numbers via their 
diophantne approximation properties. You can find it, or a discussion 
of it, in any of these papers/books: 

K. Mahler, J. Reine Angew. Math.  166 (1931/32), 118--136; Zbl  3, 151

Mahler, K.
On the order function of a transcendental number.
Acta Arith. 18 (1971), 63--76.

Chapter 8 of A. Baker's  Transcendental number theory, Cambridge Univ. 
Press, London, 1975

Mahler, K.
Fifty years as a mathematician.
J. Number Theory 14 (1982), no. 2, 121--155.

Schmidt, Wolfgang M.
Diophantine approximation.
Lecture Notes in Mathematics, 785.
Springer, Berlin, 1980. x+299 pp. \$19.50. ISBN 3-540-09762-7 

Dozens of papers have been written on topics inspired by Mahler's 
classification. Here are a few. 

Waldschmidt, Michel
Un demi-siècle de transcendance. (French) [A half-century of 
transcendence]
Development of mathematics 1950--2000, 1121--1186,
Birkhäuser, Basel, 2000. 

Amou, Masaaki
On Sprindzuk's classification of transcendental numbers.
J. Reine Angew. Math. 470 (1996), 27--50.

Ki, Haseo
The Borel classes of Mahler's $A$, $S$, $T$, and $U$ numbers. (English. 
English summary)
Proc. Amer. Math. Soc. 123 (1995), no. 10, 3197--3204.

Pollington, Andrew D.
Sum sets and $U$-numbers.
Number theory with an emphasis on the Markoff spectrum (Provo, UT, 
1991), 207--214,
Lecture Notes in Pure and Appl. Math., 147,
Dekker, New York, 1993. 

LeVeque, W. J. On Mahler's $U$-numbers.  J. London Math. Soc.  28,  
(1953). 220--229.

-- 
Gerry Myerson ([EMAIL PROTECTED]) (i -> u for email)