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.