Diophantine Approximation — The Simple Version

How closely can a real number be approximated by rationals? Diophantine Approximation studies inequalities of the form $$\left|x - \frac{p}{q}\right| < \varepsilon,$$ where p and q are integers and q is positive. The smaller the error for a given q, the “better” the approximation.

Why care?

• It explains why continued fractions give excellent approximations.
• It underlies results in dynamics, geometry of numbers, and cryptography.
• It powers practical tasks like frequency estimation and synchronization.

Dirichlet’s Guarantee

For any real number x and positive integer Q, there exist integers p and q with 1 \le q \le Q such that

$$\left|x - \frac{p}{q}\right| < \frac{1}{qQ}.$$

This already ensures infinitely many rational approximations with error better than 1/q^2 by letting Q=q and sending q to infinity.

Continued Fractions

The convergents of the continued fraction of x (fractions obtained by truncating the expansion) give some of the best possible approximations. For example, with \(x=\varphi=(1+\sqrt5)/2\):

$$\varphi = [1; 1, 1, 1, \dots],\quad \frac{F_{n+1}}{F_n} \to \varphi$$

where \(F_n\) are Fibonacci numbers.