An Introduction to the Geometry of Numbers

As its name suggests, the area of mathematics known as “the geometry of numbers” involves using geometric methods to answer questions arising in number theory. These methods often give very elegant solutions to problems that seem intractable without them. The origin of the field goes back to Minkowski, but the first comprehensive book on the subject was J. W. S. Cassels’ An Introduction To The Geometry of Numbers, originally published in 1959 and now available in Springer Verlag’s Classics in Mathematics series.

In the prologue of his book, Cassels gives several of the concepts and results that he considers at the core of this area of mathematics, including the following question: Let f(x₁,…,x_n) be a real-valued function of real variables x_i. How small can we make |f(x₁,…,x_n)| if we impose the additional restriction that the x_i be integers? Often we want to consider cases in which f is a homogeneous polynomial, in which case it is trivial that f(0,…,0)=0, so we exclude that choice of x_i. How small can we make the value of |f|?

One approach to this kind of question involves looking at the geometric regions defined by considering the values where f(x₁,…,x_n) ≤ k for various values of k. Often, we can understand the geometry of these regions and then invoke results such as a classical theorem of Minkowski which states that any convex, symmetric region which has large enough area must include points with integer coordinates. In this way, the geometry allows us to answer what originally appeared to be a purely number theoretic question.

These types of Diophantine questions turn up throughout mathematics and Minkowski’s theorem has found particularly important applications in computing ideal class groups in algebraic number theory and in addressing the types of sphere-packing and sphere-covering problems that coding theorists ask. As is common in mathematics, many of these applications were discovered long after the original theory was developed, and some of them were developed long after Cassels wrote this book. This is perhaps the biggest drawback in reading Cassels’ book in 2011 — there have been many advances in the field in the half century since he wrote it, and several other good books have been written in the area that cover these advances and put the work in a more modern context. See, for example, books by Burger, Conway, Olds, Lax, & Davidoff, or Beck & Robins, each of which touches on similar topics.

That said, it is Cassels’ book that I have found myself returning to over the years. It is very clearly written, and assumes little in the way of prerequisites. In particular, it is accessible to an undergraduate who is willing to work a bit, and I speak from experience as I first read the book the summer before I started graduate school. At the same time, it is a serious work giving an exhaustive (and not at all watered down) account of Minkowski’s theory. The questions that Cassels discusses will be interesting to anyone who has thought about Diophantine questions in number theory, and the techniques he uses to discuss them are simultaneously sophisticated and accessible. This book certainly earns its place in a series on the “Classics in Mathematics.”

Darren Glass is an Associate Professor of Mathematics at Gettysburg College whose research interests include number theory, algebraic geometry, and cryptography. He can be reached at [email protected].