TY  - BOOK
AU  - Tao,Terence
TI  - Higher order Fourier analysis 
T2  - Graduate studies in mathematics
SN  - 9781470425883
U1  - 515.2433 
PY  - 2012///
CY  - Rhode Island
PB  - American mathematical society
KW  - Fourier analysis
KW  - Number theory -- Sequences and sets -- Arithmetic combinatorics; higher degree uniformity
KW  - Dynamical systems and ergodic theory -- Ergodic theory -- Relations with number theory and harmonic analysis
KW  - Number theory -- Connections with logic -- Ultraproducts
KW  - Number theory -- Exponential sums and character sums -- Estimates on exponential sums
N2  - Traditional Fourier analysis, which has been remarkably effective in many contexts, uses linear phase functions to study functions. Some questions, such as problems involving arithmetic progressions, naturally lead to the use of quadratic or higher order phases. Higher order Fourier analysis is a subject that has become very active only recently. Gowers, in groundbreaking work, developed many of the basic concepts of this theory in order to give a new, quantitative proof of Szemerédi’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. The book serves as an introduction to the field, giving the beginning graduate student in the subject a high-level overview of the field. The text focuses on the simplest illustrative examples of key results, serving as a companion to the existing literature on the subject. There are numerous exercises with which to test one’s knowledge
ER  -