Last updated: Dec 31, 2023
Higher order Fourier analysis
Terence TaoGraduate 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 ” should be “asymptotically equidistributed in “.
- Page 17: In Exercise 1.1.17, should be .
- Page 18: In Exercise 1.1.20, should be .
- Page 29: after (1.9), “left-hand side of (1.8)” should be “left-hand side of (1.9)”.
- Page 30: All appearances of here should be .
- Page 49: Near the start of Section 1.3.2: the should be elements of rather than . “equilently” should be “equivalently”.
- Page 52: In Exercise 1.3.8, should be . After (1.29), should lie in rather than .
- Page 69: and should be and throughout.
- Page 75: In Proposition 1.5.1, should be bounded by 1. In Exercise 1.5.1, should be , and in Lemma 1.5.2, should be $latrex V$.
- Page 77: In the penultimate display, should be .
- Page 93: In Section 1.6.1, should be .
- Page 95: In Exercise 1.6.7, should be .
- Page 98: In the definition of a nilpotent filtered group , the additional hypothesis that is also nilpotent is required (this is automatic in the most important case , but not in general).
- Page 99: In Exercise 1.6.13, it is rather than which is a quadratic polynomial.
- Page 100: In the first paragraph, replace “starting from a point” with “starting with the base group , which is a point in the most important case “. Similarly, after Exercise 1.6.14, replace “starting from a point” by “starting from the base space , which is a point in the most important case “. 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 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 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: should be in several places.
- Page 123: exercise 1.7.5: The bound should be , and the supremum should be over non-zero frequencies .
Thanks to farlabb, Ben Green, Zach Hunter, Abishek Khetan, James Leng, Minghao Pan, Killua Zoldyck, and Pavel Zorin for corrections.
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30 March, 2011 at 9:34 am
Higher order Fourier analysis « What’s new
[…] finished writing the first draft of my thirdbook coming out of the 2010 blog posts, namely “Topics in random matrix theory“, which was based primarily on my graduate course in the topic, though it also contains […]
30 March, 2011 at 5:31 pm
eigenlambda
Thanks! I’m a little past graduation, not yet in grad school, so I’ve been trying to figure stuff out on my own. Your previous measure theory book was incredibly helpful, and also all the lecture notes. And now something new! Christmas has come early this year indeed.
1 March, 2012 at 4:02 am
Non-Uniform Random Variate Generation and Fourier Analysis Textbooks | Multiply Leadership
[…] the engineer looking for some technical background on Fourier analysis, see these free resources from Fields Medal winner Terry […]
29 September, 2015 at 11:08 am
Purple
Hi Terence ! I am an undergraduate student studying physics. I have been figuring out stuff on my own .. But sometimes you get stuck :'( . I have been studying Fourier analysis.. Could you recommend me some really good material and how to go about understanding it .
21 August, 2019 at 6:28 am
mathématicien jeune
Dear Prof. Tao,
I’m in the process of re-reading your textbook on higher-order Fourier analysis, and I have an issue with the second paragraph. Namely, you write that the knowledge of the behaviour of exponential sums was a necessary prerequisite for the study of functions on arithmetic progressions of greater length. Yet, I do not accept this as a fact unless proof is provided. We don’t know whether there exists a completely different way of understanding functions on arithmetic progressions. In fact, this seems to be one of the greatest problems of mathematics: It is difficult to exhaust all possibilities.
I very much hope that my critique will not be discarded, and a rewording will be carried out. A mathematician with great influence on the community must choose his wording carefully, lest he creates misunderstandings in numerous individuals!
21 August, 2019 at 8:32 am
Terence Tao
The fact that exponential sums are intimately related to the counting of longer arithmetic progressions was a highly non-trivial insight of Gowers in his 1998 paper on the subject. Roughly speaking, he demonstrates in that paper that one can count the number of length four progressions in that paper accurately unless certain exponential sums are large; conversely, if these exponential sums are large, he provides examples to show that the number of length four progressions can deviate significantly from a “naive” prediction of the count. The modern formulation of these claims is the inverse conjecture for the Gowers norms (now a theorem), discussed in Section 1.6 of the text.
18 December, 2019 at 10:55 pm
anonymous
Hi Terry and other users:
If you want the parenthetical corrections done, as a last resort you can email me with my nominated address related to this anonymous post if you ever need to release a new book edition, and if you think I would do a fun and free and good job for you. Definitely not hurt nor offended if you think someone else can do the work as well, in the meantime though, will go ahead and finish parenthetical on Higher Fourier Analysis. One reason I sifted through Analysis 1,2 , Measure theory and Additive Combinatorics is because they are parenthesized heavily and are sought after publications hence likely to be continued. Used the books list on these pages to find these titles. I think the fame of the books is unselfish because it leads to a need to be well/meticulously punctuated yet because I am anonymous I can help out with the punctuation and learn about such books, it’s clear it isn’t that bad and helps my math a lot. I’ve invested about 25 hours of time in total (although didn’t keep formal count) in the four books so far and it was giving me an opportunity I could never get before, so think nothing of it. It protects me, my home uni and Terry and other users to stay anonymous.
So to get things started, here is what I got from sifting through the first 30 pages of the preliminary copy of Higher Fourier Analysis, looks well bracketed. Just want to see if you can find this from the preliminary copy and add it to the official errata. If that’s possible, then makes sense to post again a complete list of any further errata.
From first 30 pages one erratum was found:
P11: Hint for exercise 1.1.5 needs the period inside the final parenthesis, not outside – parenthesized sentence.
9 January, 2022 at 3:15 am
Anonymous
Page 3 of the online version :
asymptotically equidistributed on N
should be
asymptotically equidistributed on X ?
[Correction added, thanks – T.]
29 December, 2023 at 3:27 pm
Zach Hunter
On page 69, ‘‘ and ‘‘ are written often, when the superscript should have ‘‘ in place of ‘‘.
[Erratum added, thanks – T.]