Additive Combinatorics
Terence Tao, Van Vu
Cambridge University Press
Hardback, 530 pages (ISBN-13: 9780521853866; ISBN-10: 0521853869)

This book covers the basic tools in additive combinatorics: sum set estimates, inverse theorems, graph theory techniques, crossing numbers, algebraic methods, Szemerédi’s theorem.

  • Sample chapters (contents, probabilistic method, sum set estimates, additive geometry)

— Deleted scenes —

— Precursor material —

— Errata —

  • p. xi: Reference [116] should be [113].
  • p. xiv: “Chevalley-Waring” should be “Chevalley-Warning”.
  • p. 3, after Lemma 1.2: “with probability n” should be “with probability 1″.
  • p. 4, after Question 1.4, “Kleiman” should be “Kleitman”.
  • p. 5: In Q. 1.1.3, (-1)^k should be (-1)^{|A|+1} (two occurrences). In Q. 1.1.5, A \cap (B+x) and A \cap (B+y) should be A \cup (B+x) and A \cup (B+y) respectively.
  • p. 6: Exercise 1.1.8 is missing. This exercise reads: “Let A be an additive set. Show that there exists a subset X \subset 2A-A of cardinality |X|= O( \frac{|2A-A|}{|A|} (1 + \log |A| ) ) such that 2A \subset A+X. (Hint: translate A randomly by independent elements of 2A-A, and use the first moment method.)”
  • p. 8: In the fourth display, \ln should be \log.
  • p. 9: In Exercise 1.2.4, the inequality should be reversed.
  • p. 11: In the paragraph after (1.17), \ln should be \log.
  • p. 14: In the 9th line from below, \ln should be \log (two places); similarly before (1.23) on P. 15. In the final display, \kappa/2 should be \kappa/16.
  • p. 15: In the first display, the first \leq sign should be \geq.
  • p. 17: In the first line, Corollary 1.8 should be Corollary 1.9. In Exercise 1.3.1, n needs to be even and \lambda needs to be odd. In Exercise 1.3.4, the first expectation should be a probability. In Exercises 1.3.5 and1.3.6, \lambda needs to be positive.
  • p.18: In Exercise 1.3.8, the expectation should be 2^{n-1} - 1/2 and the variance should be 2^{n-2} - 1/4. In Exercise 1.3.11, E( |X|^p )^{1/p} should be E( | X-E(X) |^p )^{1/p}.
  • p. 21: In Definition 1.21,”we say that X is a (1-\epsilon)-complementary base of A” should be
    “we say that X is a (1-\epsilon)-complementary base of order k of A“.
  • p. 22, before second display: Lemma 1.53 should be Theorem 1.53.
  • p. 23: Near top: “the event that x_j is good” should be “the event that j is good”. In the last line of the proof of Lemma 1.24, \ln should be \log.
  • p. 24: In the last display, the third \overline{A_i} should be \overline{A_j}.
  • p. 25: In the first display, \geq should be \leq.
  • p. 36: In the first line of the proof of Theorem 1.39, add “Without loss of generality we may assume \epsilon <1/h.   In the third display on the last line, O_h(n^{-he}) should be O_h(m^{-he}).
  • p. 37: In the first display, F_h(g,N) should be |A|.
  • p. 38: In the RHS of the display in Corollary 1.42 RHS of the formula, D_{k,n} should be (D_{k,n}-1)^k.
  • p. 40: In the RHS of the last display D_n should be (D_n-1)^k.
  • p. 41: In the first display, D_n + 1 should be (D_n + 1)^k.
  • p. 44: One line before the third display, m=[n^{1/k}] should be m=[n^{1/r}].
  • p. 47: In the last sentence in the proof of Proposition 1.51, \log t \frac{t}{\log^2 t} should be \log t \frac{1}{t\log^2 t}, and “is the antiderivative” should be “is the derivative”.
  • p. 57: In the bounds for \delta[A], \frac{|A|-1}{2} + \frac{1}{|A|} should be |A|-1 + \frac{1}{|A|} (two occurrences). Similarly, \frac{N-1}{2} should be N-1 (two occurrences).
  • p. 68: In Exercise 2.3.14, A-A should be A+A, and x+A should be x-A. Replace “though the set F may be symmetric around a non-zero origin x/2” with “where by symmetry here we mean that F=x-F”. (The point is that x/2 might not actually exist and be unique for some groups.)
  • p. 69: In Exercise 2.3.20, a square root is missing from the left-hand side.
  • p. 76: In Exercise 2.4.4, Corollary 2.19 should be Corollary 2.20.
  • p. 77: In Exercise 2.4.7, \frac{1}{(KK')^3} should be (KK')^3.
  • p. 78: In, Exercise 2.4.11, Proposition 2.4.11 should be Proposition 2.27.
  • p. 79: In (2.20), K^4 should be K^5.
  • p. 80: In the first paragraph, K^4 should be K^5. In the second display of the proof of Lemma 2.30, the \frac{}{} should be a / (and in particular should not enclose the left-hand side of the inequality). Near top: “the iterated sum and difference sets of A'' and B''” should be the iterated sum and difference sets of A' and B'“.
  • p. 82: In Exercise 2.5.3, add |B''| \geq 2|B|/K after |A''| \geq 2|A|/K. In Exercise 2.5.4, “is can be used” should be “can be used”.
  • p. 85: In, Lemma 2.23, “there exists a set G \subset |S|^2” should be “there exists a set G \subset S^2“.
  • p. 86: In the first display of the proof of Lemma 2.34, \tilde G should be G.
  • p. 87: In the fourth display, the G' on the LHS and RHS should be G_m. In the last sentence in the statement of Theorem 2.35, L^{\epsilon} should be L^{-\epsilon}.
  • p. 95: Lemma 2.41, while correct, does not imply Corollary 2.42 as stated (since d(A,A) does not control |A^{-1} \cdot A|). Thanks to Mei-Chu Chang for pointing out the problem. Replace “As d(A,A) \leq 2d(A,B), this implies” by “A similar argument (see [Proposition 4.5, 362]) gives”. Rename Corollary 2.42 as Proposition 2.42, replace “B” by “A” in the first sentence, replace “|S| \geq \Omega(K^{-O(1)} |A|)” by “|S| \geq |A|/2K” in the second sentence, and replace “O(K)^{O(1+n)}” by “2^n K^{2n+1}” in the display.
  • p. 96: “verstion” should be “version”.
  • p. 97: In Theorem 2.48 (ii), G \subset A \cdot B should be G \subset A \times B.
  • p. 107: before Corollary 2.62, “Corollary 2.60″ should be “Theorem 2.60″.
  • p. 112, bottom: “in Section 4.4″ should be “to Section 4.4″.
  • p. 114, line 3: the last Z should be an X.
  • p. 115, fourth line of Lemma 3.4: {\Bbb R}^d should be {\Bbb R}^k.
  • p. 116, proof of Corollary 3.6: “induce” should be “induct”. In the last two sentences of this proof, w should be v (three occurrences).
  • p. 117, proof of Theorem 3.7: “induce” should be “induct”.
  • p. 118: Exercise 3.1.1 \phi (Z') should be \phi (Z)
  • p. 120: In the second display, (m,m,\ldots, m) should be (m-1, m-1, \ldots, m-1).
  • p. 121, after the proof of Lemma 3.11, “an progression” should be “a progression”.
  • p. 122, after first display: “for all real a,b \geq 0” should be “for all non-negative reals a,b \geq 0“. In the next sentence, “for any integer A” should be “for any integer n“.
  • p. 123, proof of Theorem 3.13: |\det(L)| should be |\det(L)| \hbox{mes}(B_d). On the same line, “whch” should be “which”.
  • p. 124: \hbox{arctan}(r^2-1) should be \hbox{arctan}(\sqrt{r^2-1}) (two occurrences). In the first display and two lines further down, \sqrt{d} should be d. In the second-to-last display, L_{\varepsilon,\delta} \ldots \varepsilon e_1 should be \{ L_{\varepsilon,\delta} \ldots \varepsilon e_1: (x_1,\ldots,x_d) \in B_d \}. In the last paragraph, n \cdot B_d should be d \cdot B_d.
  • p. 128: In the proof of Proposition 3.19, \sup_x f(x) = \sup_y f(y) = 1 should be \int_{\Bbb R} f = \int_{\Bbb R} g = 1.
  • p. 129: In the proof of Proposition 3.20, “induce” should be “induct”. Remove the “that” before “(using Fatou’s lemma)”. In the next display, f_d(y_d) should be f_d(x_d).
  • p. 130: In Exercise 3.4.5, “amd” should be “and”.  In Exercise 3.4.6, all occurrences of n should be d.
  • p. 131: Delete the sentence beginning with “The lower bound is trivial…” from the proof of Lemma 3.21.
  • p. 133: Just before the third display, R^d + O_{V,P,d}(R^{d-1}) should be (R^d + O_{V,P,d}(R^{d-1})) \hbox{mes}(B).  In Corollary 3.25, N \theta_1 \dots \theta_d \leq 1 should be N \theta_1 \dots \theta_d \geq 1. In the next display, delete + {\Bbb Z}^d.
  • p. 139: In the last display, |B| should be |B \cap \Gamma|.
  • p. 140: In the proof of Theorem 3.34, $altex \Gamma \cap V_{i-1} +W$ should be \Gamma \cap V_i.
  • p. 141-142: Throughout the statement and proof of Lemma 3.36, r^{2r} should be replaced with O(r)^{3r/2}, and d^{2d} replaced with O(d)^{3d/2}. In the last two equations in the display, replace \hbox{mes}({\Bbb R}^d/\Gamma) and 2d \cdot d! both by O(d)^{3d/2}.
  • p. 144: In the last pragraph of the proof of Theorem 3.38, the superfluous ) should be deleted.
  • p. 145: In Theorem 3.40, replace “If d \geq 2 and P is not proper” with “If d \geq 2, P is not proper, and Z is torsion-free”. In the proof, “We induce on d” should be “We induct on d”. On the last line, “unless Z is torsion-free” should be deleted.
  • p. 147: In the second paragraph, “M_1, \ldots, M_{d-1}” should be “M := (M_1,\ldots , M_{d-1})“.
  • p. 155: In Q 4.1.7, {\Bbb E}_{\xi \in H^\perp} should be \sum_{\xi \in H^\perp}.  In Q 4.1.10, the range of \tilde \bullet should be {\Bbb R}/{\Bbb Z} rather than C.
  • p. 157: In (4.14), E(1_A,1_B) should be E_Z (1_A*1_B)^2.
  • p. 158: In the proof of Lemma 4.10, \xi/a should be \xi a. Between the second-to-last and third-to-last lines of the long display in that proof, insert the lines “= {\Bbb P}_F(A)^3 - |F|^{-1/2} \|f\|_{L^2(F)}^2” and “\geq {\Bbb P}_F(A)^3 - |F|^{-1/2} [ {\Bbb E}_{a \in A} \| 1_{a \cdot A} \|_{L^2(F)} ]^2”.
  • p. 159: In Exercise 4.2.2, “let p' the dual exponent” should be “let p' be the dual exponent”.
  • p. 162: In the proof of Lemma 4.14, F \backslash 0 should be F \backslash \{0\}. In the third line of Corollary 4.15, a_1, \ldots, a_k should belong to A rather than F, and 2^{1-k} should be 2^{-k}(1 + |F|^{-1}). (Remark: the order estimate only gives the qualitative result kA=F for |F| sufficiently large, but the small values of |F| can be done by hand.) In the last word of the last sentence, F should be A.
  • p. 164, in Q. 4.3.10, “A\cdot 2A = A” should be “A \cdot 2A=2A“.
  • p. 165, bottom paragraph: “combinatorialinformation” should be “combinatorial information”.
  • p. 170, proof of Lemma 4.25: \log_2 5 should be 2, and the condition |a| \leq 0.1 should be |a-a'| \leq 0.1.
  • p. 171, in Q 4.4.6, \sqrt{2} should be 1/4.
  • p. 173: In the proof of Proposition 4.29, (4.31) should be (4.30).  In the second and last displays, e(x,\xi) should be e(\xi \cdot x).
  • p. 174: In the third display, \|c_\xi\|_H^4 should be |c_\xi|^4.
  • p. 175: In the four summations involving r_{h_1,h_2}, \xi should range over Z rather than S.  After the second display, e(x,\xi) and e(x,-\xi) should be e(\xi \cdot x) and e(-\xi \cdot x) respectively, and similarly on the third display.
  • p. 176: In the last display, e(x,\xi) should be e(\xi \cdot x).
  • p. 177: In the first display, e(x,\xi) should be e(\xi \cdot x).   In the penultimate display, (1-\varepsilon^2) should be (1-\varepsilon)^2. “If take” should be “Taking”.  In the last display, \leq should be =.
  • p. 179: In Exercise 4.5.4, e(x,\xi) should be e(\xi \cdot x).
  • p. 181: In the proof of Lemma 4.35, “incrementing k+1″ should be “incrementing k”.
  • p. 182, middle: “S' \in S_\theta” should be “S' \subset S_\theta“. After the fourth display, e(x,\xi) should be e(\xi \cdot x).  In the last display, the upper limit of the integral should be infinite.
  • p. 183: In the last four displayed equations, all \geq signs should be \leq. In (4.38), (A) is missing from the RHS.  In the sixth display, e(x,\xi+\xi') should be e( (\xi+\xi') \cdot x)).
  • p. 185: In the proof of Proposition 4.39, e(x,\xi) should be e(x\cdot \xi).
  • p. 187: In Theorem 4.41, {\Bbb Z}_p \backslash 0 should be {\Bbb Z}_p \backslash \{0\}.
  • p. 190: In the first display, 1/2 should be 1/6.
  • p. 191: The second equality in the third line should be an inequality \leq.
  • p. 194: In the first display, e(x+jr,\xi) should be e(\xi \cdot (x+jr)).
  • p. 195, first line: S_{err} should be \Gamma_{err}.
  • p. 196: After the proof of Theorem 4.47, “, non-empty set E” should be “. One then considers a non-empty set E”. A few lines afterwards, “replace” should be “replaces”. In Exercise 4.7.1, A_1+A_1 should be A_1+A_2.
  • p. 200: “induce” should be “induct” (four occurrences).
  • p. 203: the subscript {0} should be (0) (three occurrences).
  • p. 209: In Exercise 5.1.2, Lemma 5.1 should be Lemma 5.2. In Exercise 5.1.6, “Schirelmann” should be “Schnirelmann”.
  • p. 210: In Exercise 5.1.9, A_1+\ldots + A_N should equal {\Bbb Z}_p rather than p.
  • p. 211: In the proof of Lemma 5.13, “induce” should be “induct” (two occurrences). “Frieman” should be “Freiman”.
  • p. 212: In the proof of Lemma 5.14, (A \cup B) \backslash 0 should be (A \cup B) \backslash \{0\}.
  • p. 213: In the proof of Proposition 5.15, A \backslash a_0 should be A \backslash \{a_0\}, B \backslash b_0 should be B \backslash \{b_0\}, and 0 \cup A' should be \{0\} \cup A'.
  • p. 214: In the proof of Corollary 5.16, “induce” should be “induct”.
  • p. 215, middle, “closed unit interval” should be “closed unit ball”. In the proof of Lemma 5.18, “induce” should be “induct”.
  • p. 216, middle: “some origin a=x/2” should be “we see that A \cap H_+ contains a set F symmetric around some origin a=x/2“.
  • p. 217: In the first paragraph, P’ should be symmetric around a rather than around a’.  In the proof of Theorem 5.20, “induce” should be “induct”.
  • p. 218, fourth line: \varepsilon/K|A| should be \frac{\varepsilon}{K} |A|. Similarly on the sixth line.  Also, on that line, |A''| \leq \eps/K|A| should be |A''| \geq \frac{\varepsilon}{K}|A|.
  • p. 221: In the first example, \phi(Z) should be \phi(A).  In the fourth line of the fifth example, (P, {\bf Z}) should be (P,Z).
  • p. 225: In the first line, j(\lambda)<8 should be j(\lambda)<n.  A period is missing before “Fortunately” in the sentence after the first display.
  • p. 226: In Exercise 5.3.7, the range of \phi should be A' \uplus B' rather than A' \uplus B.
  • p. 229: In the proof of Theorem 5.30, Corollary 2.23 should be Lemma 5.26.
  • p. 231, first line of proof of Theorem 5.33: “|A| \leq 100 K^2” should be “|A| \geq 100 K^2“.
  • p. 236: In Proposition 5.41, “Frieman” should be “Freiman”. In Corollary 5.42, K should be d.
  • p. 237: In Exercise 5.5.1, Hom should be \hbox{Hom}_k.
  • p. 240: In Theorem 5.44, “Then there a” should be “Then there is a”
  • p. 247: In Exercise 6.1.2, “symmetruc” should be “symmetric”, and one should assume 0 \not \in A.
  • p. 248: In Exercise 6.1.4, the denominator |V_1| |V_2| should be |V_1|^2 |V_2|^2.
  • p. 253: In Remark 6.8, “We say that S” should be “We say that S is”.
  • p. 255: In the proof of Theorem 6.9, “induce” should be “induct”, “vacuoust” should be “vacuous”, and “(with R(1,m)=1)” should be deleted. “G_{red} = G_{red}” should be “G_{red} = G'_{red}“. Shortly afterwards, “latter” should be “former”.
  • p. 256: In the proof of Corollary 6.12, “induce” should be “induct”, and “G contains a” should be “G contains an” (two occurrences). “complete the induction and than the proof” should be “closes the induction and completes the proof”.
  • p. 257: In the first line, “E_r monochromatic” should be “E_r-monochromatic”. In Theorem 6.15, d(|A|,m) should be d(n,m). In Proposition 6.16, {\Bbb Z}^d should be {\Bbb Z}^{\tilde d}, “exists distinct classes” should be “exist distinct classes”, and “v_1,\ldots,v_s” should be “non-zero v_1,\ldots,v_s“. In the proof of Proposition 6.16, “induce” should be “induct”, and “inducing” should be “inducting”.
  • p. 258: In the last paragraph of the proof of Proposition 6.16, v \in [0,1]^{d_2} should be v_* \in [0,1]^{d_2}.
  • p. 259, first paragraph: “the bound…which follow…are” should be “the bound…which follows…is”. “being of growing” should be “growing”.
  • p. 260, in Exercise 6.3.3, \{x+y,xy\} should be \{x,y,x+y,xy\}.
  • p. 263: In the statement of Corollary 6.20, the K^4 in the denominator in the last line should be K^5. In the second paragraph of the proof. the denominator 8K^2 should be 128 K^2.
  • p. 264: In the second paragraph, the denominator 8K^2 should be 128 K^2. In the penultimate display, A \stackrel{G}{+} B should be (A \stackrel{G}{+} B)^3. In the proof of Corollary 6.20, 16 \sqrt{2} K should be 16 \sqrt{2} K^2 (two occurrences), and K^4 should be K^5 (one occurrence). In the proof of Theorem 2.29, K^4 should be K^5 (four occurrences).
  • p. 265: In Theorem 6.21, the first “then” should be an “and”.
  • p. 268: In the first line, “smallest” should be “largest”. In the line before Example 6.26, the right parenthesis should be deleted.
  • p. 276: In the first bullet point, “can occur” should be “is”. In the second sentence after these bullet points, “has” should be “have”.
  • p. 278: In the proof of Corollary 7.4, v_i > 1 should be v_i \geq 1. In the proof of Lemma 7.6, “induce” should be “induct”.
  • p. 294: In statement of Proposition 7.21, add “Furthermore, each of the w_i lies in the set \{v_1,\ldots, v_n\}“.
  • p. 308: In the third paragraph, Andrew’s should be Andrews’.
  • p. 309: In the second paragraph, the accent on Szemerédi is misplaced.
  • p. 310: In the second display, p^{-2} should be p^{-3}.
  • p. 312: In the dual formulation of Corollary 8.5, there is a “:” missing in the set notation.
  • p. 330: In the proof of Theorem 9.2, “induce” should be “induct”.
  • p. 331: After the end of the proof, “Combinatorial Nullstelensatz” should be “combinatorial Nullstellensatz”.
  • p. 332: In Exercise 9.1.2, the d_j in \cup_{j=1}^{d_j} should be d_i. In Exercise 9.1.3, the Hilbert nullstellensatz of course only applies when F is algebraically closed.
  • p. 333: In Lemma 9.3, F_p should be F.
  • p. 338: In the proof of Theorem 9.11, “induce” should be “induct”.
  • p. 341: In Question 9.2.6, V should be \Delta_n.
  • p. 342: In Theorem 9.17, |R_i| should be |R_i|+1, and r_i should be |R_i|. In the last sentence of the proof of Theorem 9.16, “s \in S_i” should be “s \in B“.
  • p. 343: In the first display, the right-hand side should just be \prod_{i, j \in [1,k]: i \neq j} \prod_{r \in R_i} (a_i+t_i-a_j-t_j-r). In the next paragraph, x_i should be t_i, and in the paragraph after that, r_i should be |R_i|. In the first line of the last display, (a_{\tau(i)}t_{\tau(i)}^{i-1} should be (a_{\tau(i)}t_{\tau(i)})^{i-1}. In Lemma 9.18, the second sentence should start “Then” rather than “The”.
  • p. 344: In proof of Corollary 9.19, “the multiplicatively invertible” should be “the set of multiplicatively invertible”. After the end of the proof of Lemma 9.18, “additive group” should be “an additive group”. In Theorem 9.20, “be power of p” should be “be a power of p“.
  • p. 345: In Exercise 9.3.3, “Let P_{\pi}” should be “Set P_{\pi}“.
  • p. 348: In the proof of Theorem 9.24, \hbox{deg}(F) should be \hbox{deg}(P_i).
  • p. 352-353: In the proof of Lemma 9.31: all occurrences of g_1 should be a_1, etc. Also, g_{m+1},\ldots,g_m should be a_{m+1},\ldots,a_n.
  • p. 354: Right before Theorem 9.36, “made a significant progress” should be “made significant progress”.
  • p. 355: In Remark 9.38, “combining” should be “combined”.
  • p. 362, bottom line: “as follows” should be “are as follows”.
  • p. 363, middle: delete “and using Exercise 9.4.1″. Somewhat further down, f, g \in {\Bbb Z}[z] should just be {\Bbb Z}[z].
  • p. 364, in the proof of Lemma 9.48, “least common multiple” should be “greatest common divisor”.
  • p. 370: In Examples 10.3, k > n should be k > p. In Theorem 10.5, the phrase “… with |Z| coprime to (k-1)!” should be added at the end of the first sentence.
  • p. 371: In Exercise 10.0.4, Exercise 6.3.7 should be Theorem 6.17. In the third line of Exercise 10.0.5, “color class” should be “color classes”.
  • p. 375: In the second and third displays, \Lambda should be \Lambda_3. In the line afterwards, (4.3) should be (4.2).
  • p. 376: In the second line before the exercises, one of the “that”s should be removed.
  • p. 377: In Exercise 10.1.6, “aproper” should be “a proper”.
  • p. 378: Corollary 10.10 should be Proposition 10.10, and similarly in page 380.
  • p. 379: In the third display of the proof of Lemma 10.15, e( \xi \cdot x + \theta ) should be e( \xi \cdot (x+x_0) + \theta ) . In the third line of the proof of Theorem 10.12 (which should be Proposition 10.12), 3/n should be 3|Z|/n. In the second line of the proof of Theorem 10.12, “induce” should be “induct”.
  • p. 380: In the proof of Proposition 10.17, “induce” should be “induct”. In the line after the top display, it should say “n > 3” rather than “n \geq 3“.
  • p. 381: In the second line, 4 should be 8. (In the next two displays, the 4 is correct.) In the third line, add “and x_0 in Z” just before “, such that”. In the third and fourth displays, {\Bbb E}_Z(g) should be {\Bbb E}_{Z'}(g).
  • p. 382: in the paragraph after statement of Theorem 10.20: replace”\delta and M” with just “M“.
  • p. 383: in the third display, the first = sign should be \leq, while \sup_{\xi \in V^\perp \ 0} |\hat \nu(\xi)| should be \sup_{\xi \in V^\perp} |\hat \nu(\xi) - {\Bbb I}(\xi = 0)|.
  • p. 384: in Lemma 10.22, \eta needs to be between 0 and 1.
  • p. 385: in third display, \xi-\xi' should be \xi'-\xi, and similarly on the next two lines. In the next display, the sum \eta + {\Bbb I}(\xi-\xi'=0) should be enclosed in parentheses.
  • p. 387: In the first line, Theorem 10.12 should be Proposition 10.12; in the fourth line, Corollary 10.10 should be Proposition 10.10.
  • p. 388, fifth line of proof of Lemma 10.25: insert “, since” after “On the other hand”. In the third-to-last line, “the sum” should be “the summand”.
  • p. 389: in top line, “f is bounded” should be “f is bounded by 1″. In third line, P should be P'. In Exercise 10.3.1, add the hint “(You may want to first establish the weaker but easier bound r_3(P) = o(|P|).)”
  • p. 394. In the second, third, and fourth display, the indicator functions should be applied to x instead of r. Also, in the fourth equation, \leq should be \geq.
  • p. 398: About half-way down on the page, “to be introduce” should be “to be introduced”.
  • p. 402: in the last two displays, \in a+bi+\alpha \in B should be \in a+bi+\alpha+B).
  • p. 407: In Remark 10.44, “Gowers shown” should be “Gowers has shown”.
  • p. 418: In “choices of f_0,\ldots,f_{k-1})“, the parenthesis should be deleted.
  • p. 439: In the second line, \xi \in W should be x \in W. In the fourth line, Mx \cdot x should be Mx \cdot x = 0. In the proof of Proposition 11.12, “induce” should be “induct”.
  • p. 454: In the line before Theorem 11.27, the brackets \{ \} around v_{k+1} are missing.
  • p. 462: In the last line, “in” should be \in.
  • p. 465: “and some k-pseudorandom measure \nu, then” should be “for some k-pseudorandom measure \nu. Then”.
  • p. 474: In the proof of Lemma 12.6, “induce” should be “induct”.
  • p. 500: In [288], “arithemtic” should be “arithmetic”.
  • p. 503: Reference [352] should be redirected to [380].
  • p. 505: o_n \to \infty should be o_{n \to \infty}, and O(f(n)) needs to be in math mode.
  • p. 506: “Chevalley-Waring” should be “Chevalley-Warning”.

Thanks to Louigi Addario-Berry, Andres Caicedo, Kestutis Cesnavicius, Mei-Chu Chang, Moubariz Garaev, Le Thai Hoang, Tom Koornwinder, Choongbum Lee, Isabel Lugo, Vicky Neale, Olof Sisask, and Sam van Gool for corrections.