[FOM] Roth's Theorem; Liouville numbers
Stephen G Simpson
simpson at math.psu.edu
Mon Apr 17 09:40:16 EDT 2006
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
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
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.
Stephen G. Simpson
Professor of Mathematics, Penn State University
Research interest: foundations of mathematics
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