The aim of this book is to present the basic theory of diophantine approximation on smooth manifolds embedded in Euclidean space, from a classical perspective. There are two additional chapters – one on diophantine approximation on the \(p\)-adics, and one on applications of ideas from diophantine approximation. A framing notion is that of extremality: an extremal set is one on which Dirichlet’s simultaneous diophantine approximation theorem cannot be strengthened. Dirichlet’s theorem itself asserts that \(\mathbb R\) is extremal. Sprindzhuk’s conjecture that sufficiently nondegenerate analytic manifolds are extremal has recently been proved by D. Y. Kleinbock and G. A. Margulis [Ann. Math. (2) 148, 339–360 (1998; Zbl 0922.11061)] using ideas from the ergodic theory of flows on lattices. Here the approach is more classical, and in particular flows on lattices are not used. Instead, the approach follows on from the monographs of V. G. Sprindzhuk [Mahler’s problem in metric number theory. Am. Math. Soc. Transl. (1969; Zbl 0181.05502) and Metric theory of diophantine approximations, John Wiley (1979; Zbl 0482.10047)], using analysis and geometry to address delicate questions about the Hausdorff dimension of sets defined by diophantine conditions on manifolds (for example, the Hausdorff dimension counterpart to versions of the Khintchine-Groshev theorem).
The material is unavoidably technical, and the authors have gone to great lengths to motivate the material with short discussions about strategy, and to try and extract unifying ideas and techniques (for example, Khintchine’s transference principle, and the mass distribution principle). In addition, each chapter concludes with a detailed set of notes covering the history of the ideas used, related work, and different approaches.
The contents are roughly as follows. Chapter 1 contains a brief review of Dirichlet’s theorem in many variables and Khintchine’s transference principle, and ends with a formulation of the basic questions about diophantine approximation on manifolds (some background material on manifolds is included here). In Chapter 2 the Khintchine-Groshev theorem is extended to certain manifolds. Chapters 3, 4 and 5 consider the delicate problem of computing the Hausdorff dimension of sets defined by diophantine conditions on manifolds. Chapter 6 is a short review of diophantine approximation on the \(p\)-adic numbers, with analogues of Khintchine-Groshev and a study of the Hausdorff dimension of sets associated to integral polynomials with small \(p\)-adic values. Chapter 7 describes several interesting applications of ideas from diophantine approximation to problems in number theory, partial differential equations, dynamical systems, and Kolmogorov-Arnol’d-Moser theory.
The book is carefully written, with an extensive bibliography, and will be of lasting value to graduate students and researchers interested in diophantine approximation.