Liouville’s theorem on diophantine approximation

Liouville’s theorem on diophantine approximation Let $\alpha$ be an irrational number that is algebraic of degree $d$. Then for any real number $e>d$, there are at most finitely many rational numbers $\frac{p}{q}$ such that $|\frac{p}{q}-\alpha |<\frac{1}{q^e}$.

If there is no such rational number $\frac{p}{q}$ then the number of solution is clearly finite.

Now assume there exists at least one rational number $\frac{p}{q}$ such that $|\frac{p}{q}-\alpha |<\frac{1}{q^e}$.

Let $P(x)$ be a polynomial of degree $d$, with integers coefficients, such that $P(\alpha )=0$. Choose $ϵ$ such that $P(x)$ has no roots other that $\alpha$ on the interval $[\alpha -ϵ,\alpha +ϵ]$

Write $P(x)=(x-\alpha )\cdot Q(x)$ where $Q(x)$ is a monic polynomial with real coefficients of degree $d-1$. Since $Q(x)$ is continuous, there exists $K>0$ such that $|Q(x)|\le K$ on the interval $[\alpha -ϵ,\alpha +ϵ]$

For all rational number $\frac{p}{q}$ such that $|\frac{p}{q}-\alpha |<\frac{1}{q^e}$, $|\frac{p}{q}-\alpha |>ϵ$ or $|\frac{p}{q}-\alpha |\le ϵ$

For all rational number $\frac{p}{q}$ such that $ϵ<|\frac{p}{q}-\alpha |<\frac{1}{q^e}$. $q^e<1/ϵ$ and $p\in [q(\alpha -\frac{1}{q^e}),q(\alpha +\frac{1}{q^e})]$, it is not difficult to tell the number of such rational numbers will be finite.

For all rational number $\frac{p}{q}$ such that $|\frac{p}{q}-\alpha |<\frac{1}{q^e}\wedge |\frac{p}{q}-\alpha |\le ϵ$. Note

Since $P$ has degree $d$ and integer coefficients,

where both the nominator and the denominator are integers. Since $\alpha$ is irrational, $P(\frac{p}{q})\ne 0$. Then $\left|P\left(\frac{p}{q}\right)\right|\ge \frac{1}{q^d}$. Note

Similarly, the number of rational numbers $\frac{p}{q}$ such that $|\frac{p}{q}-\alpha |<\frac{1}{q^e}\wedge |\frac{p}{q}-\alpha |\le ϵ$ will be finite.

Therefore, there are at most finitely many rational numbers $\frac{p}{q}$ such that $|\frac{p}{q}-\alpha |<\frac{1}{q^e}$.