An introduction to the geometry of numbers. Repr. of the 1971 ed. (German) Zbl 0866.11041

Cassels, J. W. S.

Classics in Mathematics. Berlin: Springer. x, 344 p. (1997).

The first edition of J. W. S. Cassels’ book appeared in 1959 (see the review in Zbl 0086.26203), and it was the first advanced textbook in the geometry of numbers since H. Minkowski’s basic work in 1896 (JFM 27.0127.09). The second edition in 1971 (Zbl 0209.34401) contained some minor changes and corrections, and now, 38 years after the first edition, the 1971 edition is reissued, which indicates the timeless quality of this book.
Of course, there has been much progress in various parts of the geometry of numbers during the last three or four decades. So the reprint does not contain the results on dense sphere packings, based on coding theory, no lattice point inequalities and nothing about lattice polytopes including Ehrhart’s polynomial and his reciprocity law. Further, recent progress in reduction theory and its application in optimization is missing.
But the book still contains the main stream of the geometry of numbers including the British school, which is characterized by the names of Mordell, Davenport, Rankin, Rogers and Swinnerton-Dyer. And there is no doubt about this book remaining a standard reference, and it still can serve as an additional advanced textbook for all types of courses in the geometry of numbers.

Reviewer: J.M.Wills (Siegen)

MSC:

11Hxx	Geometry of numbers
11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
01A75	Collected or selected works; reprintings or translations of classics
52C05	Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
52C07	Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
52C15	Packing and covering in \(2\) dimensions (aspects of discrete geometry)
52C17	Packing and covering in \(n\) dimensions (aspects of discrete geometry)
11E25	Sums of squares and representations by other particular quadratic forms
11H55	Quadratic forms (reduction theory, extreme forms, etc.)
11H06	Lattices and convex bodies (number-theoretic aspects)
11H31	Lattice packing and covering (number-theoretic aspects)
11J20	Inhomogeneous linear forms
11H46	Products of linear forms
11H56	Automorphism groups of lattices
11H60	Mean value and transfer theorems
11H50	Minima of forms

Keywords:

lattices; reduction; Blichfeldt theorem; Minkowski theorem; distance functions; Mahler’s compactness theorem; Minkowski-Hlawka theorem; successive minima; packings; automorphs; inhomogeneous problems; transference theorem; isolation theorems; critical lattices; star bodies; lattice constants; quadratic forms; polar lattice; geometry of numbers

Citations:

Zbl 0209.34401; Zbl 0131.29003; Zbl 0086.26203; JFM 27.0127.09