This 1999 book is concerned with Diophantine approximation on smooth manifolds embedded in Euclidean space, and its aim is to develop a coherent body of theory comparable with that which already exists for classical Diophantine approximation. In particular, this book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. After setting out the necessary background material, the authors give a full discussion of Hausdorff dimension and its uses in Diophantine approximation. A wide range of techniques from the number theory arsenal are used to obtain the upper and lower bounds required, and this is an indication of the difficulty of some of the questions considered. The authors go on to consider briefly the p-adic case, and they conclude with a chapter on some applications of metric Diophantine approximation. All researchers with an interest in Diophantine approximation will welcome this book.

'This book is an important addition to the literature from authors who are leading experts in this field.' Glyn Harman, Bulletin of the London Mathematical Society
'… carefully written, with an extensive bibliography, and will be of lasting value …'. Thomas Ward, Zentralblatt für Mathematik
'This book can be recommended not only to those interested in number-theoretic aspects, but also to those interests lie in topics related to dynamical systems … self-contained and very readable.' EMS