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 {n^{k}α}.
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
| {q_{2}x} – {q_{1}x} | < `1/Q`
There exist integers p_{1} and p_{2} such that
{q_{1}x} = q_{1}x − p, {q_{2}x} = q_{2}x − p
Letting p = p_{2} − p_{1} we find
| (q_{2} − q_{1}) x − p| `lt=` ` 1/Q`
Let q = q_{2} − q_{1}, 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 p_{0}/q_{0}, would occur for infinitely many choices Q_{k} 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.