Axel Thue

Axel Thue (1863 – 1922) was a Norwegian mathematician, best known for his work in number theory and combinatorics. He made significant contributions to Diophantine approximation and to the study of patterns in strings.

Thue's work on Diophantine approximation concerned finding rational approximations to algebraic numbers. He proved what is now known as Thue's Theorem in 1909, which states that for an algebraic number α of degree n ≥ 2, and for any k > n/2 + 1, the inequality |α - p/q| < 1/|q|^k has only finitely many solutions in rational numbers p/q. This theorem was a major step towards Roth's theorem, which provided the final answer to the problem of Diophantine approximation of algebraic numbers.

In combinatorics, Thue is known for his work on square-free words. A square-free word is a word that does not contain any subword of the form xx, where x is a non-empty word. Thue proved in 1906 that there exist infinitely long square-free words over a three-letter alphabet. This result has applications in diverse areas such as symbolic dynamics, formal language theory, and computer science. He also investigated cube-free words and other related pattern avoidance problems in strings. His contributions laid the groundwork for the field of combinatorics on words.