Diophantine approximation asks how closely irrational numbers can be approximated by rationals and how arithmetic structure limits that accuracy. Continued fractions give optimal one-dimensional approximants; geometry of numbers and transcendence methods extend the theory.
Dirichlet's approximation theorem
Theorem (Dirichlet). For every \(\alpha\in\mathbb R\) and \(Q\ge1\), there exist integers \(p,q\) with \[ 1\le q\le Q,\qquad \left|\alpha-\frac pq\right|<\frac1{qQ}. \] If \(\alpha\) is irrational, infinitely many \(p/q\) satisfy \[ \left|\alpha-\frac pq\right|<\frac1{q^2}. \]
Proof. Place the \(Q+1\) fractional parts \[ 0,\{\alpha\},\ldots,\{Q\alpha\} \] into \(Q\) intervals of length \(1/Q\). Two lie in the same interval, so for \(q=|j-i|\) there is an integer \(p\) with \(|q\alpha-p|<1/Q\). Irrationality prevents finitely many denominators from accounting for all \(Q\).
Continued fractions
Every irrational \(\alpha\) has a unique infinite expansion \[ \alpha=[a_0;a_1,a_2,\ldots], \] with convergents \(p_n/q_n\) generated by \[ p_n=a_np_{n-1}+p_{n-2},\qquad q_n=a_nq_{n-1}+q_{n-2}, \] and determinant identity \[ p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}. \]
Theorem. Convergents satisfy \[ \left|\alpha-\frac{p_n}{q_n}\right| <\frac1{q_nq_{n+1}}<\frac1{q_n^2}. \] If \(0<q<q_{n+1}\), then \[ |q_n\alpha-p_n|\le|q\alpha-p| \] for every integer \(p\). Thus convergents are best approximations.
Example. \[ \sqrt2=[1;\overline2], \] whose convergents \[ 1,\ \frac32,\ \frac75,\ \frac{17}{12},\ldots \] come from solutions of \(p^2-2q^2=\pm1\). Periodicity of a continued fraction characterizes quadratic irrationals.
Badly approximable and algebraic numbers
Theorem (Hurwitz). Every irrational \(\alpha\) has infinitely many approximants satisfying \[ \left|\alpha-\frac pq\right|<\frac1{\sqrt5\,q^2}. \] The constant \(1/\sqrt5\) is optimal.
A number is badly approximable if \[ \left|\alpha-\frac pq\right|>\frac c{q^2} \] for some \(c>0\) and all \(p,q\). This is equivalent to bounded continued fraction coefficients. Quadratic irrationals are badly approximable.
Theorem (Roth). If \(\alpha\) is algebraic irrational, then for every \(\varepsilon>0\), \[ \left|\alpha-\frac pq\right|<\frac1{q^{2+\varepsilon}} \] has only finitely many rational solutions.
Dirichlet shows exponent \(2\) is universal; Roth shows algebraic irrationals cannot improve it by any fixed power. Liouville's earlier, weaker bound already produced the first explicit transcendental numbers.
Simultaneous approximation
The pigeonhole argument in dimension \(n\) gives, for \(\alpha_1,\ldots,\alpha_n\), integers \(1\le q\le Q^n\) and \(p_i\) such that \[ |q\alpha_i-p_i|<Q^{-1}. \] Minkowski's convex-body theorem recasts this as a lattice-point problem and opens the path to approximation by linear forms.