← All books

Higher order Fourier analysis

Terence Tao · original page

This continues my series of books derived from my blog. It is based primarily on my 254B higher-order Fourier analysis lecture notes.

• Online draft
• Publisher (AMS)
• Lecture notes (254B)

Errata

20 corrections · 1 awaiting a page number

1. p. 3?: “asymptotically equidistributed in ${\bf N}$” should be “asymptotically equidistributed in $X$“.
2. p. 17In Exercise 1.1.17, $1/\sqrt{N_0}$ should be $1/\sqrt{N}$.
3. p. 18In Exercise 1.1.20, $\delta^{-C_d} N$ should be $\delta^{C_d} N$.
4. p. 29after (1.9), “left-hand side of (1.8)” should be “left-hand side of (1.9)”.
5. p. 30All appearances of $\delta$ here should be $\delta'$.
6. p. 49Near the start of Section 1.3.2: the $c_i$ should be elements of $G$ rather than ${\bf Z}$. “equilently” should be “equivalently”.
7. p. 52In 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 $G$.
8. p. 69$V^d$ and $V^{d+1}$ should be $V^D$ and $V^{D+1}$ throughout.
9. p. 75In 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 $latrex V$.
10. p. 77In the penultimate display, $\phi$ should be $\psi$.
11. p. 93In Section 1.6.1, $H^{\{0,1\}^d}$ should be $H^{\{0,1\}^k}$.
12. p. ?In Exercise 1.6.2, the condition $G'_1 = G$ also needs to be added.
13. p. 95In Exercise 1.6.7, $HK^k(G_\bullet^{+1})$ should be $HK^{k-1}(G_\bullet^{+1})$.
14. p. 98In 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).
15. p. 99In Exercise 1.6.13, it is $y$ rather than $z$ which is a quadratic polynomial.
16. p. 100In 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.”
17. p. 105Proof of Lemma 1.6.13: “where $G_{\ge j} \times_{\ge j + 1} G_{\ge j}$ is the subgroup of $G_{\ge j} \times_{\ge j + 1} G_{\ge j}$” should be “where $G_{\ge j} \times_{\ge j + 1} G_{\ge j}$ is the subgroup of $G \times G$“. “to conclude that F is invariant” should be “to conclude that $\tilde F$ is invariant.”
18. p. 109In 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.)
19. p. 116$e(\xi B)$ should be $e(\xi m/N)$ in several places.
20. p. 123exercise 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 $\xi$.

Contributors

Last updated: July 9, 2026.