Last updated: June 7, 2020

Higher order Fourier analysis
Terence Tao

Graduate Studies in Mathematics, 142

American Mathematical Society, 2012

ISBN-10: 0-8218-8986-9
ISBN-13: 978-0-8218-8986-2

This continues my series of books derived from my blog. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, “An epsilon of room“, “An introduction to measure theory“, and “Topics in random matrix theory“.

An online version of the text can be found here.  It is based primarily on these lecture notes.


  • Page ???: In Exercise 1.1.17, 1/\sqrt{N_0} should be 1/\sqrt{N}.
  • Page ???: In Exercise 1.1.20, \delta^{-C_d} N should be \delta^{C_d} N.
  • Page ???: after (1.9), “left-hand side of (1.8)” should be “left-hand side of (1.9)”.
  • Page ???: Near the start of Section 1.3.2: the c_i should be elements of G rather than {\bf Z}.  “equilently” should be “equivalently”.
  • Page ???: In Exercise 1.3.8, a_1=-a_2=-a_3 should be a_1=a_2=-a_3.
  • Page 98: In the definition of a nilpotent filtered group G_\bullet, the additional hypothesis that G is also nilpotent is required (this is automatic in the most important case G_{\geq 0} = G_{\geq 1} = G, but not in general).
  • Page 100: In the first paragraph, replace “starting from a point” with “starting with the base group G/G_{\geq 1}, which is a point in the most important case G_{\geq 0} = G_{\geq 1} = G“.  Similarly, after Exercise 1.6.14, replace “starting from a point” by “starting from the base space G/(G_{\geq 1}\Gamma), which is a point in the most important case G_{\geq 0} = G_{\geq 1} = G“.
  • Page 109: In Exercise 1.6.22, the argument indicated only works under the stronger hypothesis that \alpha, \beta, \alpha \beta are linearly independent modulo 1 over the integers.  (To handle the general case, one either needs the more complicated quantitative (single-scale) relative van der Corput lemma in my paper with Green, or else rely on the ergodic theorem as was done in the paper of Leibman.)

Thanks to farlabb, Ben Green, Abishek Khetan, and Pavel Zorin for corrections.