Hilbert’s fifth problem and related topics.

Terence Tao

Graduate Studies in Mathematics, Vol. 153

American Mathematical Society, 2014

338 pp., hardcover

ISBN-10: 1-4704-1564-X

ISBN-13: 978-1-4704-1564-8

Last updated: Nov 26, 2015

This continues my series of books derived from my blog, and is based on the lecture notes for my graduate course of the same name. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, “An epsilon of room“, “An introduction to measure theory“, “Topics in random matrix theory“, “Higher order Fourier analysis“, and “Compactness and contradiction“.

An online version of the MS can be found here.

This book received the 2015 PROSE award in the mathematics category.

Pre-errata (to be corrected in the published version):

- Page 5: In the proof of Theorem 1.1.2, should be taken to be , rather than .
- Page 10: In Exercise 1.1.7, should be , and should be .
- Page 20: In Problem 1.1.14, the hypothesis that G has polynomial growth is missing and should be inserted.
- Page 79: In the last paragraph, a right parenthesis is missing after “Exercise 1.4.3”.

Errata:

- Page 16: In Example 1.2.5, the series should be .
- Page 18: In Definition 1.2.8, all occurrences of G should be replaced with M.
- Page 181: In remark 8.2.3, it should be added that in the more general non-symmetric case discussed here, needs to be replaced by . Also, in the analogue of Exercise 8.2.2 in this more general case, needs to be replaced by an arbitrary translate of .

Thanks to Michael Cowling, Mikhail Katz, Lam Pham, Fan Zheng, and an anonymous contributor for corrections.

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27 March, 2012 at 11:00 am

Hilbert’s fifth problem and related topics « What’s new[…] finished the first draft of the the first of my books based on my 2011 blog posts, entitled “Hilbert’s fifth problem and related topics“, based on the lecture notes for my graduate course of the same name. The PDF of this draft […]

29 March, 2012 at 11:54 pm

Mads SørensenTypographical suggestions:

P. 5, l. 5 in the proof:

The ‘s in shouldn’t be in italic mode.

P. 6, l. 2 in exercise 1.1.4:

The ‘s in shouldn’t be in italic mode.

P. 8, l. 1 after the two times two matrix:

Is the ‘s in supposed to be in italic mode?

P. 17, footnote 4:

Use \dots instead of “…” and remember a comma after the definition of the group G_{3}.

In general:

When typing a map, use \colon instead of a normal colon. (The spacing is incorrect.) Also, use \coloneqq from the mathtools package instead of “:=”.

[Thanks, this will be incorporated into the next revision of the ms. -T.]1 February, 2013 at 10:19 am

Small doubling in groups « What’s new[…] abelian case is discussed in this book of mine with Vu, and the nonabelian case discussed in this more recent book of mine), but instead focuses on the statements of the various known results, as well as some remaining […]

7 December, 2013 at 4:06 pm

Ultraproducts as a Bridge Between Discrete and Continuous Analysis | What's new[…] of Hilbert’s fifth problem (which classifies the latter) to study approximate groups; see this text of mine for more […]

8 July, 2014 at 11:34 am

Hilbert’s fifth problem and approximate groups | What's new[…] The slides cover essentially the same range of topics in this series of lecture notes, or in this text of mine, though of course in considerably less detail, given that the slides are meant to be presented in […]

18 February, 2016 at 1:38 pm

LamDear Terry,

I have a few questions on Section 8.2 (Sanders-Croot-Sisask-theory), related to remark 8.2.3.

– p.180: exercise 8.2.1. S^m is contained in A^2A^{-2}=A^4 if A is assumed either symmetric, or an approximate subgroup, but in remark 8.2.3, you mention that small doubling+finite+non-empty suffices. But in that case, we can’t get S^m contained in A^4, can we?

– p.180: exercise 8.2.2. If f is supported on A^2, Sf, Tf would be supported on A^3, in which case the small doubling assumption is not quite enough to guarantee the variance bound (we would need small tripling in full generality, of course if A is assumed to be an approximate group, that should be ok). It seemed that it is important that f be supported on gA (which is the case if f=1_{gA}), since then the support of Tf would be of size |A^2|<K|A|.

– p.181: you say that |A^2|/M 1_{y_iA} has an l^2-norm of O_K(|A|^3/2), but I found a bound depending on both K and M.

Is there a natural way that one can, without any symmetry assumption on A, have A^2A^{-2} contained in some higher iterated power A^k?

Thanks!

-L

18 February, 2016 at 3:55 pm

Terence TaoThanks for the corrections. In the non-symmetric case one indeed has to replace by . This need not be contained in for moderately large , even in the abelian case, consider e.g. the case when for some finite group and some , all of whose multiples lie outside of .