Diophantine Approximation: Liouville’s Theorem

Diophantine approximation deals with the approximation of real numbers by rational numbers.

Liouville’s Theorem

In the 1840’s Liouville obtained the first lower bound for the approximation of algebraic numbers:

Let α ∈ R be an irrational algebraic number satisfying f(α) = 0 with non-zero irreducible (cannot be reduced) f ∈ Z[x] of degree d. Then there is a non-zero constant C such that for every fraction p/q

Screen Shot 2016-12-11 at 10.35.05 AM.png

Proof

The proof utilises the mean value theorem. By this theorem, given p/q, there is a real ξ between α and p/q such that

Screen Shot 2016-12-11 at 10.35.08 AM.png

Since f has integer coefficients and is of degree d, the value of f(p/q) is a rational number with denominator at worst q^d. Since f is irreducible, f(p/q) cannot be equal to 0. Thus

Screen Shot 2016-12-11 at 10.40.22 AM.png

and so

Screen Shot 2016-12-11 at 10.40.58 AM.png

A corollary of this result is that numbers that are well approximable by rational numbers, i.e. in for every d ≥ 1 and positive constant C, there is a rational p/q such that

Screen Shot 2016-12-11 at 10.43.32 AM.png

are transcendental.

Example

Letscreen-shot-2016-12-11-at-10-45-22-am

β is a real, transcendental number.

This is because there is a rational approximation

Screen Shot 2016-12-11 at 10.46.43 AM.png

with

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Analysing this inequality, the ratio

screen-shot-2016-12-11-at-10-48-30-am

is unbounded as N → +∞, and so β is well approximable by rationals.

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