In this book I describe some of the developments which have taken place in the theory of sets of multiples since Halberstam and Roth's Sequences was published in 1966. My object is twofold: to give a coherent account of the general theory as it exists today, and to encourage others to study an elegant and, perhaps to some persons, surprisingly subtle subject in which I believe much progress is possible. There are still many unsolved problems, some of them arising from the most recent work, and I have attempted to fit these necessarily loose ends into the text in such a way that the reader can see at which point a new idea is required. One of the attractions of the subject is the great variety of techniques which can be employed: thus one chapter (not the easiest) consists entirely of elementary inequalities, another involves Dirichlet series, contour integration and exponential sums, while a third is probabilistic. Where probabilistic methods have been used, I have presented them in an accessible fashion as a non-probabilist writing for (perhaps mostly) non-probabilists.

This tract is a companion volume to Cambridge Tract No. 90, Divisors, written with Gérald Tenenbaum some years ago. Although there are references to Divisors (I refer to this book by its name throughout) at several points, Sets of Multiples is self-contained and can be read by persons unfamiliar with this area.

I have quoted freely from joint papers with two collaborators: Paul Erdös and Gérald Tenenbaum. It is almost automatic that in any long-standing collaboration some of the results will be due, on different occasions, entirely to one or the other author.