problem-solver in mathematics, rather than as a theory-builder. Harold Davenport writes that the "moral in Dr Roth's work" is that "the great unsolved problems of mathematics may still yield to direct attack, however difficult and forbidding they appear to be, and however much effort has already been spent on them". His research interests spanned several topics in number theory, discrepancy theory, and the theory of integer sequences. Diophantine approximation The subject of Diophantine approximation seeks accurate approximations of irrational numbers by rational numbers. The question of how accurately algebraic numbers could be approximated became known as the Thue–Siegel problem, after previous progress on this question by Axel Thue and Carl Ludwig Siegel. The accuracy of approximation can be measured by the approximation exponent of a number , defined as the largest number such that has infinitely many rational approximations with . If the approximation exponent is large, then has more accurate approximations than a number whose exponent is smaller. The smallest possible approximation exponent is two: even the hardest-to-approximate numbers can be approximated with exponent two using continued fractions. Before Roth's work, it was believed that the algebraic numbers could have a larger approximation exponent, related to the degree of the polynomial defining the number. In 1955, Roth published what is now known as Roth's theorem, completely settling this question. His theorem falsified the supposed connection between approximation exponent and degree, and proved that, in terms of the approximation exponent, the algebraic numbers are the least accurately approximated of any irrational numbers. More precisely, he proved that for irrational algebraic numbers, the approximation exponent is always exactly two. In a survey of Roth's work presented by Harold Davenport to the International Congress of Mathematicians in 1958, when Roth was given the Fields Medal, Davenport called this result Roth's "greatest achievement". Arithmetic combinatorics Another result called "Roth's theorem", from 1953, is in arithmetic combinatorics and concerns sequences of integers with no three in arithmetic progression. These sequences had been studied in 1936 by Paul Erdős and Pál Turán, who conjectured that they must be sparse. However, in 1942, Raphaël Salem and Donald C. Spencer constructed progression-free subsets of the numbers from to of size proportional to , for every . Roth vindicated Erdős and Turán by proving that it is not possible for the size of such a set to be proportional to : every dense set of integers contains a