Geometric number theory is an increasingly important topic with growing connections to combinatorics, operations research, and post-quantum cryptography. This book provides an accessible introduction to this area, with an emphasis on the arithmetic applications of Minkowski's geometry of numbers. Linked arithmetic and geometric thinking unifies such diverse topics as Diophantine approximation, quadratic forms, algebraic number theory, Euclidean lattices, discrete optimization, and transcendence; the book is thus appropriate for courses in geometric number theory, diophantine approximation and transcendental number theory, and algebraic number theory. Complete with color figures and more than 400 exercises, the text is accessible to advanced undergraduate and beginning graduate students with basic knowledge of linear algebra, abstract algebra, and real analysis.

‘Seemingly every seasoned mathematician has an appreciation for the geometry of numbers, whose beautiful foundations were laid by Minkowski, with its myriad of connections to so many areas: number theory, discrete & computational geometry, combinatorics, optimization, operations research, cryptography, and beyond. Fukshansky and Ramon Garcia’s excellent book brings both foundations and modern applications of geometric number theory to life, in an engaging and comprehensive style, including numerous problems and historical notes. It is suitable for more than one type of course on these topics, on the advanced undergraduate/beginning graduate level.’

Matthias Beck - San Francisco State University

‘This book is a clear, friendly entry point to geometric number theory. It shows how convex sets and lattices answer arithmetic questions, from packing problems and the LLL algorithm to Dirichlet and Weyl, and even Minkowski’s finiteness of the class number. The writing is careful, the prerequisites light, and the chapters are course-ready. With direct links to combinatorics, optimization, and lattice-based cryptography, it gives advanced undergraduates and beginning graduate students a practical, geometric way to see number theory at work.’

Camilla Hollanti - Aalto University

‘This wonderful book brings together the classical geometry of numbers, arithmetic, lattices, convexity, Diophantine approximation, and modern applications in a coherent way. It is rare to find a text that is simultaneously rigorous, accessible, and genuinely teachable. I look forward to using it in my own courses, and recommend it warmly to students and instructors alike.’

Sinai Robins - University of São Paulo

‘A masterfully crafted introduction to geometric number theory for advanced undergraduates and beginning graduate students. In Minkowski’s way, arithmetic questions are treated geometrically, with excursions into algebraic number theory and Diophantine approximation. Each chapter concludes with problems making the theory concrete and accessible, notes with historical remarks and pointers to further reading, and a concise summary of key concepts.’

Martin Henk - Technische Universität Berlin

‘This fantastic book gives a highly readable, modern perspective on the Geometry of Numbers and its many connections throughout mathematics. Both authors are great, dedicated teachers, and I expect the book to become an inspiring entry point to the subject for students, as well as an invaluable reference for researchers and university teachers.’

Achill Schürmann - University of Rostock

‘This book provides a masterful, accessible introduction to the geometry of numbers, highlighting its utility and fundamental role in a variety of arithmetic applications. It delivers a modern presentation of simple yet powerful classical ideas, proving their continued relevance at the cutting edge of current research.’

Alan Haynes - University of Houston

‘This timely volume offers an excellent introduction to geometric number theory for well-prepared undergraduates and postgraduates. Beginning with Minkowski’s classical geometry of numbers, it develops further geometric methods alongside algebraic techniques. Throughout, the authors select results of significant mathematical interest and elegance while maintaining accessibility.’

Iskander Aliev - Cardiff University