Diophantine Approximation

Diophantine approximation studies how well real numbers can be approximated by rational numbers. A central question is: how small can

$$\left|\alpha - \frac{p}{q}\right|$$

be, in terms of the denominator $q$?

Dirichlet's approximation theorem

For any real number $\alpha$ and any integer $N>1$, there exist integers $p,q$ with $1 \le q < N$ such that

$$\left|\alpha - \frac{p}{q}\right| < \frac{1}{qN} \le \frac{1}{q^2}.$$

In particular, every irrational $\alpha$ has infinitely many rational approximations $p/q$ satisfying

$$\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^2}.$$

Continued fractions

The “best” rational approximations to $\alpha$ come from its continued fraction convergents.

If $\alpha = [a_0; a_1, a_2, \dots]$ and $p_n/q_n = [a_0; a_1, \dots, a_n]$ is the $n$-th convergent, then

$$\left|\alpha - \frac{p_n}{q_n}\right| < \frac{1}{q_n q_{n+1}} < \frac{1}{q_n^2}.$$

Limits for algebraic numbers

Liouville's theorem gives a first limitation: if $\alpha$ is algebraic of degree $d>1$, then there exists $c(\alpha)>0$ such that for all rationals $p/q$,

$$\left|\alpha - \frac{p}{q}\right| > \frac{c(\alpha)}{q^d}.$$

Roth's theorem is a deep strengthening: for any algebraic $\alpha$ and any $\varepsilon>0$, the inequality

$$\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^{2+\varepsilon}}$$

has only finitely many solutions in coprime integers $p,q$. The exponent $2$ is essentially best possible by Dirichlet's theorem.