Lattice points. (English) Zbl 0683.10025

Pitman Monographs and Surveys in Pure and Applied Mathematics, 39. Harlow: Longman Scientific & Technical; New York etc.: John Wiley & Sons. viii, 184 p. £30.00 (1989).
This excellent book collects together many fascinating ramifications of the concept of lattice points.
Historically lattice point theory originated as a central subject in the geometry of numbers. Accordingly the authors give a quite detailed exposition of the classical and modern contributions of the geometry of numbers. Besides they touch on many other topics dealing with dissection problems; lattice polytopes; packing, covering, and tiling problems; quadratic forms; crystallography; visibility; connections with integral geometry; and applications to numerical integration, combinatorics, graph theory, and others.
This book is highly recommended to anyone interested or working in lattice point theory. It provides an exposition of the subject with only a few proofs but in general quite detailed (intuitive) explanations of the results. This way the book becomes (as the authors put it) an “appetizer” for further study. I am convinced that the book will considerably stimulate further research in the area of lattice points.
Reviewer: E.Schulte


11Hxx Geometry of numbers
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
52-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to convex and discrete geometry
05-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics
52B99 Polytopes and polyhedra
11H06 Lattices and convex bodies (number-theoretic aspects)
05B40 Combinatorial aspects of packing and covering
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
05B45 Combinatorial aspects of tessellation and tiling problems
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
11H31 Lattice packing and covering (number-theoretic aspects)