Klaus Roth's pioneering research in the field of number theory has led to important and substantial breakthroughs in many areas, including sieve theory, diophantine approximation, and irregularities of distribution. His work on the Thue-Siegel-Roth Theorem earned him a Fields Medal in 1958 - the first British mathematician to receive the honour. Analytic Number Theory: Essays in Honour of Klaus Roth comprises 32 essays from close colleagues and leading experts in those fields in which he has worked, and provides a great insight into the historical development of the subject matter and the importance of Roth's contributions to number theory and beyond. His influence is also discussed in relation to more recent mathematical advances. Extensive lists of references make this a valuable source for research mathematicians in many areas, an introductory overview of the subject for beginning research students, and a fitting long-awaited tribute to a great mathematician.

• Contains 32 contributions from leading figures in the field of number theory and many related areas • Explores the impact of Klaus Roth's research across a number of mathematical fields, including sieve theory and diophantine approximations • An extensive list of references makes this a valuable source for all research mathematicians

### Contents

Preface; Acknowledgments; Klaus Roth at 80; Numbers with a large prime factor II Roger Baker and Glyn Harman; Character sums with Beatty sequences on Burgess-type intervals William D. Banks and Igor E. Shparlinski; The Hales-Jewett number is exponential: game-theoretic consequences József Beck, Wesley Pegden and Sujith Vijay; Classical metric diophantine approximation revisited Victor Beresnevich, Vasily Bernik, Maurice Dodson and Sanju Velani; The sum-product phenomenon and some of its applications J. Bourgain; Integral points on cubic hypersurfaces T. D. Browning and D. R. Heath-Brown; Binary additive problems and the circle method, multiplicative sequences and convergent sieves Jörg Brüdern; On the convergents to algebraic numbers Yann Bugeaud; Complexity bounds via Roth's method of orthogonal functions Bernard Chazelle; Some of Roth's ideas in discrepancy theory William Chen and Giancarlo Travaglini; Congruences and ideals Harold G. Diamond and H. Halberstam; Elementary geometry of Hilbert spaces applied to abelian groups P. D. T. A. Elliott; New bounds for Szemerédi's theorem II: a new bound for r4(N) Ben Green and Terence Tao; One-sided discrepancy of linear hyperplanes in finite vector spaces Nils Hebbinghaus, Tomasz Schoen and Anand Srivastav; How small must ill-distributed sets be? H. A. Helfgott and A. Venkatesh; On the power-free values of polynomials in two variables C. Hooley; On a question of Browning and Heath-Brown Nicholas M. Katz; Good distribution of values of sparse polynomials modulo a prime Sergei Konyagin; Diophantine approximation and continued fractions in power series fields A. Lasjaunias; On transfer inequalities in diophantine approximation Michel Laurent; On exponential sums with multiplicative coefficients Helmut Maier; Multiplicative dependence of values of algebraic functions David Masser; Linear forms in logarithms, and simultaneous diophantine approximation Bernard de Mathan; The Caccetta-Häggkvist conjecture and additive number theory Melvyn B. Nathanson; L2 discrepancy and multivariate integration Erich Novak and Henryk Wózniakowski; Irregularities of sequences relative to long arithmetic progressions A. Sárközy and C. L. Stewart; The number of solutions of a linear homogeneous congruence II A. Schinzel, with an appendix by Jerzy Kaczorowski; The diophantine equation α1x1 . . . α1xn = f (x1,…,xn) Wolfgang M. Schmidt; Approximation exponents for function fields Dinesh S. Thakur; On generating functions in additive number theory I R. C. Vaughan; Words and transcendence Michel Waldschmidt; Roth's theorem, integral points and certain ramified covers of Ρ1 Umberto Zannier.