Roth's theorem

Roth's theorem, proved by Klaus Roth in 1955, states that every algebraic number has approximation exponent exactly 2: for any ε > 0, the inequality |α − p/q| < 1/q^(2+ε) has only finitely many rational solutions p/q. This completed the Thue-Siegel-Roth arc that transformed Diophantine approximation from a collection of special results into a unified theory of algebraic irrationality.

Roth's result is sharp. The exponent 2 cannot be improved: Dirichlet's theorem guarantees that every irrational has infinitely many approximations with exponent 2. The gap between 2 and 2+ε is therefore the precise boundary between the universal and the exceptional — a boundary that is invisible to classical algebra but decisive for Diophantine analysis. The theorem is non-constructive: it proves finiteness without bounding the size of solutions, leaving open the computational question of finding them. This non-constructivity is not a flaw; it is a structural feature of the deep approximation methods that Roth inherited from Siegel and passed on to Baker.

Roth was awarded the Fields Medal in 1958 for this work — one of the few pure number-theoretic results to receive that recognition before the era of arithmetic geometry. The theorem remains a benchmark against which all subsequent approximation results are measured, including the subspace theorem of Wolfgang Schmidt and the theory of linear forms in logarithms.