Higher Order Fourier Analysis

Higher order Fourier analysis is a subject that has become very active only recently. Gowers developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemeredi's theorem on arithmetic progressions. However, there are also precursors to this theory in Weyl's classical theory of equidistribution, as well as in Furstenberg's structural theory of dynamical systems. This book examines all of these topics in a unified manner, as well as surveys some of the most recent developments, such as the application of the theory to count linear patterns in primes.