[FOM] Roth's Theorem; Liouville numbers

[Stephen G Simpson]

Bill Taylor writes:
 > This seems to say, (loosely), that any algebraic irrational real
 > cannot be approximated by rational p/q to within anything better
 > than 1/q^2.
 > 
 > I presume it is well-known that almost all numbers are like this,
 > in some appropriate sense of almost all, (measure? category?).
 > 
 > Is this so, in fact?

Earlier this semester, in an introductory course on foundations of
mathematics, I naively assigned students the following problem:

   Prove that the function f(n) = the nth digit of pi is primitive
   recursive.

This turned out to be harder than I expected.  The proof that I
eventually came up with uses a result due to K. Mahler in the 1950s.
Can anyone here supply an alternative proof that doesn't involve such
heavy number theory?

Here is some of what I learned.

1. A Liouville number is defined to be an irrational number x which
   can be closely approximated by rational numbers, in the following
   sense: for all positive integers n there exist integers a, b with b
   > 0 such |x - a/b| < 1/b^n.

2. All Liouville numbers are transcendental.

3. The Liouville numbers are comeager and of Lebesgue measure 0 in the
   real line.  This is mentioned in Oxtoby's book on measure and
   category.  (I thank my colleague John Clemens for telling me of
   this reference.)

4. pi is a non-Liouville number.  This result is from

   Mahler, K. "On the Approximation of pi", Nederl. Akad. Wetensch.
   Proc. Ser. A. 56/Indagationes Math. 15, 30-42, 1953.

By 4, there exists a positive integer k such that |pi - a/b| > 1/b^k
for all positive integers a,b.  According to Mahler's paper, one can
take k = 42.  More recent research gives something like k = 8 or 9.

Best wishes,
-- Steve

----

Stephen G. Simpson

Professor of Mathematics, Penn State University

Research interest: foundations of mathematics

http://www.math.psu.edu/simpson/