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The theory of uniform distribution modulo one (Gleichverteilung modulo Eins, équirépartition modulo un) is concerned, at least in its classical setting, with the distribution of fractional parts of real numbers in the unit interval (0, 1).
This book attempts to summarize the results of these investigations from the beginning to the present, with emphasis on the work done during the last 20 years. Because the literature on the subject is vast, it was inevitable that now and then choices had to be made and selection criteria had to be applied that reflect the personal taste of the authors. As a rule, we have endeavored to produce a comprehensive coverage of the methods used in the theory of uniform distribution. In some instances, we did not present a result in its most general version, but rather tried to describe the underlying principles and ideas, which might otherwise be shrouded in technicalities. The title we have chosen indicates that we have resolved the dilemma of "asymptotic distribution" versus "uniform distribution" in favor of the latter, since it is this aspect to which most of our exposition is devoted. We believe that this book should prove a useful introduction to the subject for students in number theory and analysis and a reference source for researchers in the field.
An important role in our presentation is played by the notes at the end of each section. These not only contain the pertinent bibliographical references, but also provide the reader with a brief survey of additional results relating to the material of that section. The exercises we include range from simple applications of theorems to proofs of propositions more general than, or shedding further light on, results in the text. The reader is encouraged to try his hand at many of these problems in order to increase his understanding of the theory