Expansion in finite simple groups of Lie type.
Terence Tao

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

Last updated: Jun 23, 2017

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 51: Before Exercise 2.3.2, “total mass” should be “total variation”,
• 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 121, proof of Lemma 5.3.2:”$A^2$ can be covered by $K$ copies of $A$” should be “$A^2$ can be covered by $K$ translates $aA$ of $A$, and we can restrict to those $a$ in $A^3$ since otherwise $aA$ will not intersect $A$“.  Then, later in the proof, $a \in A$ should be $a \in A^3$, $A^7$ should be $A^{10}$, $K^6 |A|$ should be $K^9 |A|$, and $K^{-2C-11}$ should be $K^{-2C-14}$ (three occurrences). In Exercise 5.3.2, add the hypothesis that $V$ is not the one-dimensional space spanned by the identity element.
• 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 142: In (1.66), $D_+$ should be ${\mathcal D}_+$.
• 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”
• Page 208: In Exercise 10.1.2, “defined as the closure” should be “defined as the completion”.
• Page ???: In Exercise 10.2.1(i), “Liouville’s theorem” should be “the maximum principle”, and the phrase “viewed as a function of $x$ and $t$ should be added.
• Page 252: In Corollary 11.7.2, the conclusions about certain combinations of $\alpha,\beta$ not being roots is incorrect and should be deleted.

Thanks to Claudeh5, Pietro Gheri, Sarah Peluse, Doron Puder, and an anonymous commenter 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.