Diophantine Approximation — Advanced Overview

This page collects more formal statements, refinements, and context. It is intended for readers comfortable with analysis and number theory.

Classical Statements

Dirichlet’s Theorem (Simultaneous Form)

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

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

Hurwitz’s Theorem

There exists a constant \\(c=\\frac{1}{\\sqrt5}\\) such that for any irrational \\(x\\), there are infinitely many rationals \\(p/q\\) with

$$\left|x - \frac{p}{q}\right| < \frac{c}{q^2}.$$

The constant is best possible (attained by the golden ratio and its equivalents).

Continued Fractions and Best Approximations

Convergents p_n/q_n of the continued fraction of x are best approximations: any fraction a/b with denominator b at most q_n cannot achieve a smaller error than p_n/q_n.

$$\left|x - \frac{a}{b}\right| \ge \left|x - \frac{p_n}{q_n}\right|.$$

Metric Results

Khintchine’s theorem relates the measure of numbers satisfying \(|x - p/q| < \psi(q)/q\) infinitely often to a divergence test on \(\sum q\,\psi(q)\). Jarnik–Besicovitch theorems refine to Hausdorff dimension.

Geometry of Numbers

Minkowski's convex body theorem gives clean lattice-based proofs of Dirichlet-type bounds and generalizations to linear forms.