Diophantine Approximation

Last Modified: September 8, 2003
Almost everyone knows that p is nearly 22/7.  Over the years other even better rational approximations like 335/113 have been discovered.  It turns out that for any irrational number, there are infinitely many good rational (or "Diophantine") approximations.  What
is more, if you insist that a number be more that just irrational, for example, that is satisfies no polynomial equation of degre less than 5, then there are infinitely many superbly good rational approxmations.
This fact led to Liouville's construction of the first transcendental number, one that can be approximated so well by rational numbers that it cannot satisfy any polynomial equation. 
Requires
Uncountability and Transcendence
References (both attached)
"The measure theory of Diophantine approximation" from Billingsley, Probability and Measure.
"Liouville's Theorem Constructing Transcendentals," lecture notes by Steven Miller posted on the World Wide Web 
Minilecture 1:The pigeonhole principle 
This is one of those mathematical ideas that is some obvious that for a long time nobody bothered to give it a name.
For the best-known name, "pigeonhole principle," one illustration is this:
If there are N nesting holes in a dovecote and N+1 pigeons move into them, then at least
one pigenhole has more than one pigeon in it.  This would allow you to say, as part of a proof,
"Choose a hole with more than one pigeon in it," even though you have no way of knowing which hole this might be. Billingsley calls this concept "Dirichlet's drawer principle", while Miller calls it "Dirichlet's Box Principle."
For a striking numerical application of this innocent-looking idea, calculate 11 multiples of p

Since there are only 10 possibilities for the digit after the decimal point, the 11 numbers must include one duplication.  Sure enough, p and 8p both have 1 after the decimal point. Thus 7p is within 0.1 of an integer (22) and 22/7 is a good rational approximation for p.

Minilecture 2: Approximating irrational numbers
Following the argument in Billingsley, present the same argument as in the preceding minilecture algebraically.  To make the connection between Billingsley's proof of equation 1.33 and the numerical example above, note that in the example Q=10, which decomposes [0, 1) according to the first digit after the decimal point.  Then q' = 1, q" = 8, and q = q"-q' = 7. Finally, p is the difference between the integer parts of 8p and p, 25 -3 = 22.  So we were sure of finding something
as good as |7p - 22| <  or  in accordance with Billingsley's < .
The rest of the proof is the tricky part, because you have to use proof by contradiction.  Assume that x is rational and there are only finitely many fractions p/q that satisfy < . Of these,
there is one that is best, leading to a minimum possible .  But this minimum is not zero,
otherwise x would be rational, not irrational.  Now just choose Q so large that 1/Q is less than this
minimum and you can find a new p and q such that  < .  This contradicts the assumption that you had found the minimum. The assumption that there are only a finite set of fractions with <  must therefore be wrong; there are infinitely many.
The same proof is presented in section 1.1 of Miller, but the final step of the reasoning is a little harder to follow.
Minilecture 3: Approximatinq the square root of 2
Present the special case of Liouville's Theorem (Miller, section 1.2) where n=2, a2 = 1, a0 is a negative integer -m, and m is not a perfect square. A simple example is where x satisfies
x2 - 2 = 0, so that x =  . It was just shown that there are infinitely many fractions for which
 < . Now the job is to show that you cannot do any better in this case -- that 
finding an infinite number of fractions such that  <  is impossible for any e > 0.
Here are the steps from Miller's proof, specialized to the case where x = .
1. f(x) = x2 - 2. As Euclid once showed,  is an irrational number , and so f(x) is nonzero for any rational x.
2. f'(x) = 2x. In the range , f'(z) is less than  +.  This is Miller's M.
Now suppose that we have a half-decent rational approximation to  ,one for which <1.
By the mean-value theorem,  < M , and since =0,.

 < M .

3. Now . This is not zero, and p and q are integers, so the magnitude
of  is at least 1, and  .
4. Combining results 2 and 3, we have M   or  
5. Finally, suppose that someone claims that they can find infinitely many distinct fractions p/q for which   < . Simply choose q so large that , and you have a contradiction with the result of 4.  

Minilecture 4: Approximating any algebraic number
Present Miller's general proof, concentrating on the differences from the special case just presented.
The important thing to point out is that for a real number x that satisfies an equation of degree n, but not one of any lower degree, you can find infinitely many fractions p/q for which
 <  but not for which < . 
Then announce the stunning conclusion (MIller, Remark 1.2.3).  
If a number x has the property that for any n, there are infinitely many fractions p/q for which
 < , then x is a transcendental number 
Minilecture 5: Liouville's transcendental number
The number is x= 0.1100010000000000000000010000...
with additional scattered 1s in positions 120, 840, 8!, 9!, etc.
As an infinite series, x = 
The proof that x is transcendental is theorem 1.3.1 in MIller. It turns out to be remarkably easy,
since the fractions that approximate x well are just the partial sums of the series. If someone
tries to claim that x is algebraic of degree N, just throw away the first N partial sums.  The remaining ones provide infinitely many fractions for which, as Miller shows,
 < , and this refutes the claim.
It is only fair to point out that the transcendental number p does not have this property, and the proof that p is transcendental proceeds along quite different lines