01 · A life in mathematics
The person behind the work
Klaus Roth was born in Breslau, then Germany and now Wrocław in Poland, in 1925. His family moved to Britain, where he developed a career in number theory. While at University College London he proved his landmark theorem on rational approximation to algebraic irrational numbers, work recognised by the 1958 Fields Medal. He also made important contributions to arithmetic progressions, discrepancy theory and sieve methods. His research exemplified the power of sharp estimates: a seemingly modest improvement in an exponent could settle a major question about the possible solutions of equations.
The contribution, explained
Fractions provide increasingly good approximations to irrational numbers. Roth’s theorem identifies a sharp limit for algebraic irrationals: for every positive ε, only finitely many reduced fractions p/q satisfy |α−p/q|<1/q^(2+ε). The exponent two is the crucial threshold; approximations of order 1/q² occur naturally for every irrational number. Roth showed that an algebraic number cannot repeatedly beat this threshold by any fixed positive amount. The result completed a long sequence of improvements and revealed a fundamental arithmetic difference between algebraic numbers and some extraordinarily well-approximable transcendental numbers.
Why it matters
The theorem settled the central problem of Diophantine approximation for algebraic irrationals and became a foundational restriction in the study of rational and integer solutions.