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.

Errata:

- Page ???: In Exercise 1.1.17, should be .
- Page ???: In Exercise 1.1.20, should be .
- 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 should be elements of rather than . “equilently” should be “equivalently”.
- Page ???: In Exercise 1.3.8, 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 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 “.
- 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.)

Thanks to farlabb, Ben Green, Abishek Khetan, 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

eigenlambdaThanks! 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

PurpleHi 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 jeuneDear 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 TaoThe 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

anonymousHi 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.