Uniform Distribution of Sequences

Formal study of the uniform distribution of sequences goes back to a famous 1916 paper of Hermann Weyl. The topic mainly developed around his widely applicable criterion for a sequence of real numbers to be uniformly distributed modulo one. During these years, a lot of research has been done, with some generalizations and applications. The classic reference collecting these results is the book under review, by Kuipers and Niederreiter, recently reprinted by Dover.

The remarkable list of references at the end of book contains hundreds of items; it shows that the book collects all of the main topics and results in the area until the original publication date (1974). The Dover reprint includes a list of typos and some updated references. In the preface to the Dover edition, the second author introduces a list of main activities and developments of the area from 1974 to 2005. Of course, from 2005 to 2009, activity has continued. For example, the journal of Uniform Distribution Theory began publishing from 2006.

The authors try to give a friendly volume to researchers, but that is also useful for teachers and students. All sections contain some exercises and at the end of each chapter historical notes around references have been introduced. For a graduate course in the theory of uniform distribution of sequences, this book is a perfect choice. Also, it is useful for students of complex analysis and measure theory in order to find some applications of what they study in a formal course.

After over 30 years, this book is still fresh for beginners, and it is still the main reference for researchers, For this reason, I recommend it to people working in the area or on related topics.

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.