Expansion in finite simple groups of Lie type.
Terence Tao

Graduate Studies in Mathematics, 164. American Mathematical Society, Providence, RI, 2015.

Last updated: Sep 24, 2016

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“, “Compactness and contradiction“, “Hilbert’s fifth problem and related topics“, and “Spending symmetry“.

Errata:

  • Page 74, Remark 3.3.7, second line: “(pseudo-)randomly sowing” should be “(pseudo-)randomly sewing”
  • Page 89, Lemma 4.1.3, fourth line: “a\in A and b\in B” should be “a\in A' and $b\in B’$”
  • Page 93, Lemma 4.1.7, last line: “g\in H” should be “g\in G”.
  • Page 95, Proof of Lemma 4.1.7, last line: “|gH\cap A|\ll K^{-O(1)}|A|” should be “|gH\cap A|\gg K^{-O(1)}|A|
  • Page 98, Proof of Lemma 4.2.2, eighth line: \mu^{*n} \|)_\infty should be  \|\mu^{*n} \|)_\infty.
  • Page 102, Exercise 5.0.1, seventh line: add right parenthesis after “should not actually depend on d“.
  • Page 103-104, Proof of Theorem 5.0.2, last line of 103 and first two lines of 104: “If h\in H', then A and Ah both lie in H'” should read “If h\in H', then A and Ah both lie in gH'“, and both occurrences of AA^{-1} should be A^{-1} A.
  • Page 109, paragraph 3, first line and third-to-last line: “Lemma 5.1.3” should be “Lemma 5.1.7”
  • Page 122, Proof of Proposition 5.3.3, third line: “|A^2\cap B|\leq K^{-C}|B|” should be “|A^2\cap B|\leq K^{-C}|A|
  • Page 140, Exercise 6.1.3, part (ii): Add the hypothesis that a is not the identity element of \Gamma
  • Page 140, last paragraph, second line: The definition of \mu is missing a right parenthesis.
  • Page 142, second paragraph, fifth line: “sufficiently small \mu” should be “sufficiently small c
  • Page 152,  third line: “in the \mathcal{D}_{+}” and “in the \mathcal{D}_{-} case”, should be interchanged.
  • Page 155, Exercise 7.1.4, part (i), displayed equation: “\pi_2(x,z)” should be “\pi_{*}(x,z)
  • Page 155, Exercise 7.1.4, part (i): \sum_{p\leq x,p\equiv 1\pmod{4}} \frac{1}{p} is \frac{1}{2}\log\log{x}+O(1), not \frac{1}{2}\log{x}+O(1)
  • Page 155, Exercise 7.1.4, part (ii): “infinitely natural numbers” should be “infinitely many natural numbers”
  • Page 157, Theorem 7.2.1, second-to-last line: K_{\phi(i+1)}/K_{\phi(i)} should be K_{\sigma(i+1)}/K_{\sigma(i)}
  • Pages 160-161, after the proof of Corollary 7.2.3: All appearances of \Gamma and \Gamma' should instead be \Lambda and \Lambda'.
  • Page 162, last paragraph, first sentence: “Theorem 7.0.1” should be “Theorem 7.3.1”
  • Page 164, fourth line: “hyopthesis” should be “hypothesis”

Thanks to Sarah Peluse for corrections

A (second) draft version of the MS can be found here.  Note that the section numbering in the published version differs from that in the draft.