Klaus Roth

1925–2015, Breslau, Germany (now Wrocław, Poland), Inverness, Scotland

Arithmetic and number theoryFields Medal 1958

German-born British mathematician, Fields Medalist 1958, famous for Roth's theorem giving the best-possible bound on rational approximations to algebraic numbers, and for fundamental work on irregularities of distribution.

Klaus Friedrich Roth was born in Breslau to a Jewish father and fled Nazi Germany with his family to Britain in 1933. He studied mathematics at Peterhouse, Cambridge, during the war, and after a difficult start in research took his doctorate under Harold Davenport at University College London. There he proved his celebrated theorem (1955): algebraic numbers cannot be approximated by rationals to within $q^{-2-\varepsilon}$ infinitely often, closing a line of improvement from Liouville through Thue and Siegel to Dyson.

Roth's theorem — sometimes called the Thue–Siegel–Roth theorem in its lineage — earned him the Fields Medal at the Edinburgh congress of 1958, jointly with René Thom. He continued in analytic number theory with influential work on the irregularities of distribution of point sequences, where his lower-bound methods remain definitive, and on arithmetic progressions in dense sets, an early ancestor of Roth's theorem on three-term progressions later generalized by Szemerédi.

Roth moved to Imperial College London in 1966, where he remained until retirement. He died in Inverness in 2015 on his ninetieth birthday.

Workplaces: University College London, Imperial College London

United Kingdom, Germany

Award-winning work

Fields Medal 1958Roth received the Fields Medal for his 1955 theorem on the rational approximation of algebraic numbers: if

\alpha

is an algebraic number, then for every

\varepsilon > 0

the inequality

|\alpha - p/q| < q^{-2-\varepsilon}

has only finitely many solutions in rationals

p/q

. This resolved a problem going back to Liouville and improved stepwise by Thue, Siegel and Dyson, and the exponent 2 is now known to be best possible — Dirichlet's theorem shows it cannot be lowered. Roth's theorem completed a chapter of Diophantine approximation and became a model for later effective and higher-dimensional generalizations.

Contributions, linked to the library

Library topicphuong-trinh-diophantineRoth's theorem gives the sharp exponent for rational approximation of algebraic numbers, a cornerstone of Diophantine approximation with consequences for Diophantine equations.