*(English)*Zbl 1277.11010

The theory of higher order Fourier analysis is very recent, and the present book is a very good introduction to this nascent field. Whereas classical Fourier analysis uses the sequence of functions $n\mapsto {e}^{2\pi i\alpha n}$, quadratic Fourier analysis uses functions of the form $n\mapsto {e}^{2\pi i\alpha {n}^{2}}$, and so forth for higher order Fourier analysis. Higher order Fourier analysis has many applications, in areas that include combinatorics, ergodic theory, theoretical computer science and for finding asymptotics for various linear patterns in the growth of prime numbers. The famous precursors of the theory include Weyl, in his study of equidistribution, and Furstenberg, in his structural theory of dynamical systems. The modern theory started at the end of the 1990s in the work of Gowers.

The book is divided into two chapters. The first chapter constitutes a basic introduction to the subject, and it can be used as text for a graduate or post-graduate course. The sections in that chapter include the equidistribution theory in tori, Roth’s theorem, linear patterns, equidistribution in finite fields, the inverse conjecture over finite fields, the inverse conjecture over the integers, and linear equations in primes. The second chapter contains more specialized material, and it contains sections on ultralimit analysis, higher order Hilbert spaces and the uncertainty principle.

Although it is difficult to read, in part because the subject matter and the methods are new, the book is very well written and it should serve as a useful and good introduction to the research papers on the subject.

##### MSC:

11B30 | Arithmetic combinatorics; higher degree uniformity |

11-02 | Research monographs (number theory) |

11L07 | Estimates on exponential sums |

11U07 | Ultraproducts in number theory |

37A45 | Relations of ergodic theory with number theory and harmonic analysis |