Last updated: Dec 31, 2023

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.

Errata:

  • Page 3?: “asymptotically equidistributed in {\bf N}” should be “asymptotically equidistributed in X“.
  • Page 17: In Exercise 1.1.17, 1/\sqrt{N_0} should be 1/\sqrt{N}.
  • Page 18: In Exercise 1.1.20, \delta^{-C_d} N should be \delta^{C_d} N.
  • Page 29: after (1.9), “left-hand side of (1.8)” should be “left-hand side of (1.9)”.
  • Page 30: All appearances of \delta here should be \delta'.
  • Page 49: 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 52: In Exercise 1.3.8, a_1=-a_2=-a_3 should be a_1=a_2=-a_3. After (1.29), c should lie in {\bf R}/{\bf Z} rather than G.
  • Page 69: V^d and V^{d+1} should be V^D and V^{D+1} throughout.
  • Page 75: In Proposition 1.5.1, \|f\|_{L^\infty(V)} should be bounded by 1.  In Exercise 1.5.1, F should be f, and in Lemma 1.5.2, G should be $latrex V$.
  • Page 77: In the penultimate display, \phi should be \psi.
  • Page 93: In Section 1.6.1, H^{\{0,1\}^d} should be H^{\{0,1\}^k}.
  • Page 95: In Exercise 1.6.7, HK^k(G_\bullet^{+1}) should be HK^{k-1}(G_\bullet^{+1}).
  • 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 99: In Exercise 1.6.13, it is y rather than z which is a quadratic polynomial.
  • 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“.  In the proof of lemma 1.6.15: “if this vertical frequency is non-zero” should be “if this vertical frequency is zero…” instead of “non-zero.”
  • Pages 105: Proof of Lemma 1.6.13: “where G_{\ge j} x_{\ge j + 1} G_{\ge j} is the subgroup” should be “where $G_{\ge j} x_{\ge j + 1} G_{\ge j}$ is the subgroup”.
  • 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.)
  • Page 116: e(\xi B) should be e(\xi m/N) in several places.
  • Page 123: exercise 1.7.5: The bound 0 \leq f \leq 1 should be 0 \leq f \leq \nu, and the supremum should be over non-zero frequencies \xi.

Thanks to farlabb, Ben Green, Zach Hunter, Abishek Khetan, James Leng, Minghao Pan, Killua Zoldyck, and Pavel Zorin for corrections.