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Book review

Author(s) Krätzel, E.
Title Lattice Points
Publisher Kluwer Academic Publishers
Year of publication 1988
Reviewed by Dorin Andrica

Lattice point theory deals with the study of points of an Euclidean space, whose coordinates are integers with respect to a Cartesian coordinate system (these points are called lattice points). The estimation of the number of lattice points in closed domains in an important problem of number theory, first studied in some special cases by C.F. Gauss and P.G.L. Dirichlet. It is the main purpose of this book to present, up to the publication year, the basic aspects and results related to this problem. After an introduction in the field the author gives a large presentation of estimates of exponential sums <formula>, where f is a real function of n = (n0, n2, ..., nd) and D is a suitable domain in a p-dimensional Euclidean space. A special attention is paid to the important Van der Corput's method and to its extensions obtained by E.C. Titchmars and I.M. Vinogradov. An extensive part of the book is devoted to the plane additive and multiplicative problems and to the presentation of some many-dimensional generalizations. In the last chapter one gives applications to the study of the distribution of powerful numbers and to the representations of an Abelian group of prime power order as a direct product of cyclic groups. Each chapter ends with some useful historical notes concerning the included results. The book contains a rich and suggestive bibliography (268 important references for the field) as well as an Index of Names and a Subject Index.

The book is written in a very clear manner, the author being a real expert to explain and to organize the material. It is recommended first of all for the specialists in the field, but also for the graduate students and researchers interested in this part of number theory.