Preface

Summary

This book is about metric Diophantine approximation on smooth manifolds embedded in Euclidean space. The aim is to develop a coherent body of theory on the lines of that which already exists for the classical theory, corresponding to the manifold being Euclidean space. Although the functional dependence of the coordinates presents serious technical difficulties, there is a surprising degree of interplay between the very different areas of number theory, differential geometry and measure theory.

A systematic theory began to emerge in the mid–1960's when V. G. Sprindžuk and W. M. Schmidt established that certain types of curve were extremal (an extremal set enjoys the property that, in a sense that can be made precise, Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved for almost all points in the set; thus the real line is extremal). Sprindžuk conjectured that analytic manifolds satisfying a necessary nondegeneracy condition are extremal. Over the last 30 years, there has been considerable progress in verifying this conjecture for manifolds satisfying various arithmetic and geometric constraints, culminating in its recent proof by D. Y. Kleinbock and G. A. Margulis using ideas of flows on homogeneous spaces of lattices. The greater part of this book is concerned with establishing the counterparts of Khintchine's theorem for manifolds and with the Hausdorff dimension of the associated exceptional sets.