Diophantine Approximation

Diophantine approximation is the branch of number theory concerned with how well real numbers — especially irrational and transcendental numbers — can be approximated by rational numbers. The field takes its name from Diophantus of Alexandria, whose work on integer solutions to equations prefigured the modern study of approximation by rationals.

The fundamental question is: given a real number α and a bound on the denominator q, how small can |α − p/q| be made? The answer depends on the arithmetic nature of α. For any irrational α, there are infinitely many rationals p/q such that |α − p/q| < 1/q² — this is Dirichlet's approximation theorem, one of the foundational results of the field. But for some numbers, this bound cannot be improved: these are the badly approximable numbers, characterized by bounded partial quotients in their continued fraction expansions.

Diophantine approximation divides numbers into classes based on their approximation properties. A number is Diophantine if it satisfies a certain approximation bound; Liouville numbers are those that can be approximated extraordinarily well, so well that they must be transcendental. The Thue–Siegel–Roth theorem establishes that algebraic irrational numbers cannot be approximated too well: for any algebraic irrational α and any ε > 0, there are only finitely many rationals p/q with |α − p/q| < 1/q^(2+ε). This result, for which Klaus Roth received the Fields Medal in 1958, draws a sharp line between algebraic and transcendental numbers in terms of their rational approximability.

The field connects to dynamical systems through the Gauss map, to geometry through the theory of lattices and geometry of numbers, and to transcendence theory through the approximation properties of special constants like e and π.