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This book offers a gentle introduction to the geometry of numbers from a modern Fourier-analytic point of view. One of the main themes is the transfer of geometric knowledge of a polytope to analytic knowledge of its Fourier transform. The Fourier transform preserves all of the information of a polytope, and turns its geometry into analysis. The approach is unique, and streamlines this emerging field by presenting new simple proofs of some basic results of the field. In addition, each chapter is fitted with many exercises, some of which have solutions and hints in an appendix. Thus, an individual learner will have an easier time absorbing the material on their own, or as part of a class.
Overall, this book provides an introduction appropriate for an advanced undergraduate, a beginning graduate student, or researcher interested in exploring this important expanding field.
Sinai Robins: University of São Paulo, São Paulo, Brazil
Chapters Motivational problem: Tiling a rectangle with rectangles Examples nourish the theory The basics of Fourier analysis Geometry of numbers, Part I: Minkowski meets Siegel An introduction to Euclidean lattices Geometry of numbers, Part II: Blichfedt’s theorem The Fourier transform of a polytope via its vertex description: Brion’s theorem What is an angle in higher dimensions? Appendix A. Solutions and hints to selected problems Appendix B. The dominated convergence theorem and other goodies