Liouville's Theorem

  • We prove Liouville’s Theorem for the order of approximation by rationals of real algebraic numbers.

  • We construct several transcendental numbers.

  • We define Poissonian Behavior, and study the spacing’s between the ordered fractional parts of {nkα}.


Proof For Liouville’s Theorem 1:


Approximated by rationals to order n :

          A real number x is approximated by rationals to order n if there exist a constant k(x) (possibly depending on x) such that there are infinitely many rational p/q with

|x- p/q | < `(k(x)) / (qn)`

Recall that Dirichlet’s Box Principle gave us:

|x- p/q | <` 1/(q^2)`

for infinitely many fractions p/q . This was proved by choosing a large parameter Q, and considering the Q + 1 fractionary part {q x} C [0, 1] for q c {0… Q}.

The box Contitions ensures us that there must be two different q’s, say:

0 `lt=` q1 < q2 `lt=` Q

Such that both {q1x} and {q2x} belong to the same interval [ `a/Q` , `(a+1)/Q` ), for some

0 `lt=` a` lt=` Q − 1


Liouville’s Theorem Note: There are exactly Q such intervals partitioning [0, 1], and Q + 1 fractionary parts! Now, the length of such an interval is `1/Q` so we get

| {q2x} – {q1x} | < `1/Q`

There exist integers p1 and p2 such that

{q1x} = q1x − p, {q2x} = q2x − p

Letting p = p2 − p1 we find

| (q2 − q1) x − p| `lt=` ` 1/Q`

Let q = q2 − q1, so 1 `lt=` q `lt=` Q, and the previous equation can be rewritten as

| x – `p/q` | < `1/(qQ)`` lt=` ` 1/q2`

Now, letting Q ->`oo` , we get an infinite collection of rational fractions p/q satisfying the above equation. If this collection contains only finitely many distinct fractions, then one of these fractions, say p0/q0, would occur for infinitely many choices Qk of Q, thus giving us:

| x- `(p0)/(q0)` | < `1/(qQk)` -> 0 as k -> `oo ` .

This implies that x = `(p0)/(q0)` C Q. So, unless x is a rational number, we can find infinitely many distinct rational numbers p/q satisfying the before equation. This is mean that real numbers, irrational number can be approximated to order n = 2 by rational numbers.

The Liouville’s Theorem 1 is proved.