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I am trying to read Tao's Higher order Fourier analysis but I would be very happy to find another book on the subject. I would like to learn something about the Gowers norm and about Roth's theorem (density increment and energy increment arguments). Sorry if this question is a bit too open ended . Lecture notes, even separately on each topic I just mentioned are absolutely fine.

Part of the reason is that I am not sure what the natural context for Fourier measurability and complexity is, whatever natural means and I was hoping I could understand these if I had another reference. Thanks!

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    $\begingroup$ You may also find helpful this relatively recent survey of Gowers: arxiv.org/abs/1608.04127 . A published version appeared in the Bulletin of the AMS. $\endgroup$
    – Brad Rodgers
    Commented May 12, 2020 at 16:10
  • $\begingroup$ @BradRodgers Thanks, I'll take a detailed look at this. It seems like the sort of thing I was looking for. $\endgroup$
    – user147650
    Commented May 13, 2020 at 9:12

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You could try "Nilpotent structures in Ergodic theory" by Host and Kra, which covers this topic in greater depth.

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  • $\begingroup$ Thank you for the reference. I shall take a look at this book. $\endgroup$
    – user147650
    Commented May 12, 2020 at 11:55
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There are a couple of references I have found immensely useful, incidentally co-written by the same author.

  1. The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics,167(2008), 481–547.

  2. Green B., Tao T. (2010) An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications. In: Bárány I., Solymosi J., Sági G. (eds) An Irregular Mind. Bolyai Society Mathematical Studies, vol 21. Springer, Berlin, Heidelberg.

Note that the notion of Fourier-measurability in the book listed above coincides with that of s-measurability in the second paper (Def 2.2).

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