This book covers the basic tools in additive combinatorics: sum set estimates, inverse theorems, graph theory techniques, crossing numbers, algebraic methods, and Szemerédi’s theorem.
p. 6In Exercises 1.1.6, 1.1.7, $[0,1]^d$ should be $\{0,1\}^d$.
p. 7in the display after (1.10), add an additional right parenthesis before the equality sign.
p. 12In Example 1.12, “Legendre’s theorem” should be “Lagrange’s theorem”.
p. 15In the final display, $\kappa/2$ (or $\kappa/16$) should be $16$ (this supersedes the previous erratum for this line).
p. 16In the final display, $8 \log n$ should be $32 \log n$. On the next line, “Corollary 1.10 (or Corollary 1.8)” should read “Corollary 1.9”.
p. 17In Exercise 1.3.4, $e^{\lambda \sigma/2\sqrt{2}}$ should be $e^{-\lambda \sigma/2\sqrt{2}}$. Also add “The variance of a complex variable $X$ is defined to be ${\bf E}(|X|^2) - |{\bf E} X|^2$.
p. 18In Exercise 1.3.10, add “with $k, m$ not both equal to $1$” before the first comma.
p. 23In (1.30), third summation, $v \in V$ should be $i\ in V$.
p. 26In the penultimate paragraph, $x \in S'$ should be $x+S'$.
p. 30In the third display, $p \geq n-2n^{2/3}$ should be $p \geq n-n^{0.9}$ (say), and the third equality should be a $\geq$ sign.
p. 35first line: a right parenthesis ) missing before ${\bf E}(Y)$.
p. 36-37The proof of Theorem 1.39 requires a number of significant changes. After the first paragraph, add “We will show first that with probability 1, that any natural number $n$ has at most a bounded number of representations as the sum of $k$ elements of $A$ between $n^\varepsilon$ and $n$; the treatment of the remaining sums in which at least one term is less than $n^\varepsilon$ is left as an exercise.” Then, in the definition of $Y_m$ and also in the computation of ${\bf E}(Y_m)$, insert the lower bound $n^\varepsilon \leq n_1$ in the sum. On the first line of page 37, and in Exercise 1.7.2, replace $n^{-1/h}$ with $n^{-\delta}$ for some $\delta > 0$. Extend Exercise 1.7.2 by adding “Then complete the proof of Theorem 1.39 by using similar arguments to treat the contribution of sums in which some of the summands have size less than $n^\varepsilon$ for some sufficiently small $\varepsilon$, by using an induction hypothesis to bound the number of representations by sums of fewer than $k$ elements of size less than $n^\varepsilon$.”
p. 42Exercise 1.8.1 contains a number of inaccuracies (namely, “$A \in {\bf Z}^+$” should be “$A \subset {\bf Z}^+$, and “no two elements” should be “no two distinct elements” (and so $2B$ should really be $2^* B$ here)), and is in any case a consequence of Theorem 6.2, and so should probably be deleted (it is rather difficult to establish this exercise with the tools developed up to that point).
p. 45After equation (1.45), the assertion $p_k = k \log k + O_\epsilon(k^{1/2+\epsilon})$ should instead be $p_k = Li^{-1}(k) + O_\epsilon(k^{1/2+\epsilon})$, where $Li(x) := p.v.\int_0^x \frac{dt}{\log t}$ is the logarithmic integral.
p. 48in the first bound of Exercise 1.10.3, the error term of $O(n^{1/2})$ should be $O(\log n)$.
p. 54After the proof of Lemma 2.1, “Corollary 5.13” should be “Lemma 5.13”.
p. 57In the example after Definition 2.7, $|A-A|$ should be $N(N-1)+1$ rather than $\frac{N(N-1)}{2}+1$.
p. 58In Exercise 2.2.6, $2\sqrt{\sigma(A) |A|}$ should be $\sqrt{2\sigma(A) |A|}$. (This error is not present in the first edition.)
p. 61In the penultimate bullet point of Proposition 2.7, add the requirement $n+m \geq 1$. After Proposition 2.7, “$0$, $0$, or $1$ respectively” should be “$0$, $1$, or $1$ respectively”.
p. 62In (2.7), after Definition 2.8, $\hbox{min}(|A| |B| )$ should be $\hbox{min}(|A|, |B| )$.
p. 63In the sixth line of the first display, $a-b'=a'-b$ should be $a-b'=y$.
p. 66In Exercise 2.3.2, $\frac{1}{2} \log 2$ should be $\log 2$. In Exercise 2.3.7, replace the first bound with $\sigma[A \cup B] \leq O( \exp^{4d(A,B)})$ and the second bound with $d(A,B) \leq O( \log \sigma[A \cup B] + \log \frac{|A \cup B|}{|A|} + \log \frac{|A \cup B|}{|B|}$.
p. 70In the statement of Ruzsa’s covering lemma, insert “at most” after “In particular, $B$ can be covered by”.
p. 71In the proof of the Green-Ruzsa covering lemma, “the size of $|X+A|$” should be “the size $|X+A|$ of $X+A$“.
p. 73In Corollary 2.22, replace “covered by $\delta[A]^5$” with “covered by at most $\delta^5$“. Similarly in the proof of the corollary.
p. 74In Corollary 2.23, add the hypothesis $n_1+n_2, n_3+n_4 > 0$, and replace the exponent $10$ by $20$.
p. 76In Exercise 2.4.5, $\binom{K+n-1}{n}$ can be improved to $\binom{K+n-2}{n-1}$. In the final inequality in Exercise 2.4.8, the quantities $d(G_1,G_2)$ and $d(G_1+G_3,G_2+G_3)$ should be swapped.
p. 83In Exercise 2.5.5, the small K case is somewhat more difficult than the hint suggests, since if $K = 1 + \varepsilon$ for some small $\varepsilon$ then a direct application of Exercise 2.5.4 will give losses of $1 + O(\varepsilon^{1/2})$ rather than $1 + O(\varepsilon)$. One way to proceed is to first use the lossy argument, apply Exercise 2.4.4 to locate a relevant finite group, and then use ad hoc arguments to refine the error. In Exercises 2.5.6 and 2.5.7, the reference [80] should instead be to “G. Elekes, I. Ruzsa, The structure of sets with few sums along a graph”, J. Combin. Theory Ser. A 113 (2006), no. 7, 1476–1500. Finally, in Exercise 2.5.7, the graph G should be assumed to be symmetric.
p. 88In the definition of $\tilde O, \tilde \Omega$, $\log L$ should be $(1+\log L)$.
p. ?Exercise 2.6.6 is false as stated; the conclusion needs to be weakened, in that the indicated translates of $H$ only cover $1-\varepsilon$ of $A$ rather than all of $A$. Furthermore, the proof requires Exercise 6.5.1. Hence this exercise should be moved to Section 6.5, with a hint to use Exercise 6.5.1 and the Ruzsa covering lemma (Theorem 2.35 is not directly required, though the strategy of proof can still be useful).
p. 100In Corollary 2.52, A should be assumed to be a finite subset, rather than a finite subfield.
p. 104the last paragraph here (from “Now let $x$…” onwards) can be deleted, and replaced by the much shorter “Since $G= Q[A']$ and $A'$ contains both 0 and 1, we have $A' \subset G$“.
p. 116In the proof of Corollary 3.6, $\{ n \in {\bf Z}^n: n \cdot (v_1,\ldots,v_n) = 0 \}$ should be $\{ m \in {\bf Z}^n: m \cdot (v_1,\ldots,v_n) = 0 \}$, and “for some $w \in {\bf Z}^d$” should be “for some $w \in {\bf Z}^n$.
p. 118In the proof of Corollary 3.9, some explanation should be added as to why the torsion group is finite. This follows first from the observation that any subgroup of a finitely generated abelian group is also finitely generated (this follows from viewing the group as the image of a lattice and using Lemma 3.4), and that finitely generated abelian torsion groups are finite (because there are only finitely many distinct ways to combine the generators together).
p. 121After Lemma 3.11, “exponentially in |A|” should be “exponentially in K”.
p. 128In the proof of 3.19, the normalisation $\int_{\bf R} f = \int_{\bf R} g = 1$ should not be used; instead, one should normalise $\sup f = \sup g = 1$ (and the comment that it suffices to show $\int_{\bf R} h \geq 1$ should be deleted). The second display should then be restricted to the range $0 < \lambda < 1$ (note that Lemma 3.18 requires all sets involved to be non-empty), and one should also mention the weighted arithmetic mean-geometric mean inequality at the end of the proof.
p. 131In the second paragraph of the proof of Lemma 3.21, “translates of $\Gamma$” should be “translates of $\Gamma'$“.
p. 132In Lemma 3.23, the hypothesis that the cosets $p+\Gamma$ for $p \in P$ are disjoint should be added.
p. 133The Kronecker approximation theorem should more accurately be called a simultaneous Dirichlet approximation theorem.
p. 134In the first line, “with” should be “within”.
p. 135In the definition of $\lambda_j$, $k$ should be $j$.
p. 136in the first line, “directional basis for A” should be “directional basis for B”.
p. 142In the proof of Lemma 3.36, $L$ and $L_j$ should be $N$ and $N_j$ respectively (for consistency).
p. 143?: Exercise 3.5.3 is false. In its place, one can put the following example (from this reference): Let $A$ be the convex hull of the four points $(N,0,0), (-N,0,0), (0,N,1), (0,-N,1)$ for a natural number $N$. Show that $(A \cap {\bf Z}^3)$ has cardinality $4N+2$, but $2(A \cap {\bf Z}^3)$ has cardinality at least $(N+1)^2$. Thus one cannot remove the symmetry hypotheses from Lemma 3.21.
p. 154In display (4.9) there is a full stop missing.
p. 155In Exercise 4.1.5, “pairwise independent” should be “pairwise uncorrelated”. In Exercise 4.1.7, ${\bf E}_{\xi \in H^\perp} \hat f(\xi)$ should be $\sum_{\xi \in H^\perp} \hat f(\xi)$. In Exercise 4.1.8, $f((\phi^\dagger)^{-1}(x))$ should be $\hat f((\phi^\dagger)^{-1}(x))$. In Exercise 4.1.4, “for all $c$” should be “for all $x$“.
p. 156In the second display (defining $\|f\|_{L^\infty(Z)}$), $=$ should be $:=$.
p. 158In the proof of Lemma 4.10, in the final display, the last equals sign should be a $\geq$ sign. An additional sentence of explanation should be added: “The penultimate inequality follows from Plancherel’s theorem, followed by Minkowski’s inequality to estimate the $L^2$ norm of f.”
p. 162In the last line, “sum of two elements” should be “sum of two squares”.
p. 163In Exercise 4.3.4, “is contained in a coset” should read “is a coset”. In Exercise 4.3.14, Polya should be Pólya (and similarly for the corresponding entry in the index). In Exercise 4.3.2, $T^h A$ should be $h+A$. In the discussion of Lemma 4.16, the term $\tau Z$ in the approximation for $\|B\|_u$ after the display should be deleted, and $O(\sigma^2 \log^{1/2} |Z|)$ should be $O(\sigma \log^{1/2} |Z|)$. In the statement of Lemma 4.16, $e^{-\lambda \sigma/2\sqrt{2}}$ should be $e^{-\lambda \sigma |Z|/2\sqrt{2}}$.
p. 165In Exercise 4.3.15, the transformation $M$ needs to be assumed to be symmetric.
p. 166In Lemma 4.20, the additional hypothesis $\rho \leq 1$ should be added.
p. 168-169Before Proposition 4.23, “dispense with” should be “almost dispense with”. In Proposition 4.23, the proper progression $P$ should be the proper coset progression $P+H$, where $H$ is a subgroup of ${\bf Z}_N$ (i.e. the multiples of $M$ for some factor $M$ of $N$). The last sentence of the proposition is redundant and can be deleted. In the proof, one first handles the case when $\alpha$ has order $N$ in ${\bf Z}^d$, in which case the first display of p.169 is an equality, and one obtains a progression of rank $d$. In the general case, when $\alpha$ has order $M$ for some $M$ dividing $N$, one then passes from ${\bf Z}_N$ to ${\bf Z}_M$ by quotienting out the group $H$ generated by $M$, and then applies the previous case.
p. 170In the proof of Lemma 4.25, “find thus find” should be “thus find”.
p. 171In Exercise 4.4.7, “lower” should be “increase”.
p. 174In Remark 4.31, the sentence “The converse statement is true up to logarithmic factors; see exercises.” should be deleted.
p. 176-177The denominator $4+\varepsilon$ in (4.34) can be improved to $2+\varepsilon$, by setting $\sigma$ equal to $\lambda$ instead of $\lambda/2$ in page 177 (changing the fourth and fifth displays appropriately).
p. 179In Exercise 4.5.8, $|Z|^{1/h}$ should be $O(h |Z|^{1/h})$. In Exercise 4.5.7, $\min$ should be $\max$.
p. 180Exercise 4.5.11 does not work as stated and should be deleted.
p. 181In Lemma 4.35, add “$k \geq 0$ is an integer, ” after “where”. In the proof, “incrementing $k+1$” should be “incrementing $k$ to $k+1$“. “all dissociated subsets of $S$” should be “all dissociated subsets of $R$“.
p. 183In the statement of Lemma 4.37, the second “an” should be “a”. In the proof, “convexity” should be “concavity”, and $e(x, \xi+\xi')$ should be $e((\xi+\xi') \cdot x)$.
p. 187In the statement of Theorem 4.41, the upper bound for $\|H\|_u$ should be $p^{-\varepsilon} {\bf P}_F(H)$ rather than just $p^{-\varepsilon}$.
p. 188At the end of the proof of Theorem 4.41, add the following parenthetical remark: “The upper bound on $|A|$ required for Corollary 2.62 is supplied by (4.37) and the lower bound on $|H|$.” In Exercise 4.6.2: $\hbox{Sym}_0$ should be $\hbox{Sym}_1$ (two occurrences).
p. 189In the discussion after Theorem 4.42, the bound $E(A,A) \geq |A|^2$ should be $E(A,A) \geq P(A) |A|^3$.
p. 191In the third equation of the display, the $L^2(Z)$ norms should be $\ell^2(Z)$ (and the final term $\widehat{1_{A_3}}(\xi)$ should just be $\widehat{1_{A_3}}$.
p. 193Lemma 4.46 has a redundant “such that”.
p. 194In Theorem 4.47, the $\frac{(\log \log N)^3}{\log N}$ should be $\frac{(\log \log N)^{3/2}}{\log^{1/2} N}$, and $(\delta \log N)^{1/3}$ should be $(\delta^2 \log N)^{1/3}$.
p. 196A right parenthesis is missing after the reference to Green [149]. In the penultimate line in the proof of Theorem 4.47, as well as in the sentence immediately after this proof, and in the second sentence on page 197, $(\delta \log N)^{1/3}$ should be $(\delta^2 \log N)^{1/3}$.
p. 199In the second sentence in the paragraph before the statement of Lemma 5.3, “Lemma” should be uncapitalised.
p. 201In the second-t0-last displayed equation, $A_{(0)} + H \backslash A_{(0)}$ should be $(A_{(0)} + H) \backslash A_{(0)}$.
p. 208The sixth line of the long display should be $|A| + 3 |\phi_h(A)| |H| - (|H|-2) |\phi_h(A)| - |\phi_h(A)| - 2|H| + 2(|H|-2) - 2k + 1$ (here we use $|\phi_h(A)| \geq 2$). The seventh line should be $|A| + 2 |\phi_h(A)| |H| + |\phi_h(A)| - 2k - 3$. The eighth line should be $|A| + 2 (|A|+k-1) + |\phi_h(A)| - 2k - 3$. The final line remains as $3|A|-3$ (here we again use $|\phi_h(A)| \geq 2$).
p. 211In the statement of Lemma 5.13, “let suppose” should be just “suppose”.
p. 226In Exercise 5.3.4, the hypothesis “$0 \in A$” is missing. In exercise 5.3.2, $\{0,1,n\}$ should be $\{0,1,n+2\}$.
p. 236In the proof of Proposition 5.41, all occurrences of ${\bf R}$ should be ${\bf Z}$.
p. 239The example in Exercise 5.5.17 is incorrect; the set $A$ should be $\{ 2e_i, e_i + e_{i+1}: i=1,\dots,5\}$ in ${\bf Z}^5$, where $e_1,\dots,e_5$ is the standard basis and one adopts the convention $e_6 =1$; “quadrilateral” should be “pentagon” and ${\bf Z}^4$ and ${\bf Z}^2$ should be replaced by ${\bf Z}^5$ throughout.
p. 241In the last line of the first display in the proof of Lemma 5.45, $(|X|+2n-2)|X|$ should be $(|X|+2n-2)^{|X|}$.
p. 242Before (5.19), a right parenthesis should be inserted after $\mathrm{Tor}(Z)$.
p. 248In the proof of Theorem 6.1, “probablistic” should be “probabilistic”.
p. 249In the second paragraph of the proof of Theorem 6.2, all occurrences of $n-i$ should be $i-1$.
p. 254In exercise 6.2.8, the adjective “triangle-free” should be deleted. In Exercise 6.2.3, the final period should be inside the parenthesis.
p. 260In Exercise 6.3.5, $B+B$ should be replaced with $B \hat + B$, with the note “where $B \hat + B := \{ b_1+b_2: b_1,b_2 \in B, b_1 \neq b_2\}$ is the restricted sumset of $B$ with itself” added.
p. 263In the final display, the right parenthesis after $(a,b) \in E$ should be deleted. Similarly for the first display on page 264.
p. 271In Section 6.5.2, $G_1 \circ \ldots \circ G_k$ should be $G_k \circ \ldots \circ G_1$ (two occurrences).
p. 272The definition of $C_i^\pm$ needs to be moved from page 273 to just before Lemma 6.34.
p. 273In Exercise 6.4.2, “Claim 6.31 and Claim 6.4” should be “equations (6.3) and (6.4)”.
p. 279In Proposition 7.7, $\lfloor (|X|+i)/2\rfloor$ should be $\lfloor |X|/2\rfloor + i$.
p. 281In the last paragraph, “$Z$ is odd” should be “$|Z|$ is odd”.
p. 285Corollary 5.25 should be Lemma 5.25.
p. 286In the second paragraph, the condition $\alpha \geq {\bf E}_{Z_p}(G)$ should be replaced by $\alpha > \min G$, and the phrase “by Markov’s inequality” should be dropped. Similarly, in the third and fifth displays, the first occurrence of ${\bf E}_{Z_p}(G)$ in each should be $\min G$.
p. 291In the definition of $w(\xi)$ after the first display, a $d\xi_1$ is needed at the end of the integral. In the first paragraph, add “but still bounded” after “sufficiently large”.
p. 292In Exercise 7.3.4, the comma should be an apostrophe.
p. 293$v_1,\dots,v_n$ should be elements of $P$ rather than $V$. In Example 7.19, “which have volume” should be “which has volume”.
p. 294After the first proof of (7.20), “lower dimension its” should be “lower its dimension”.
p. 299In the first display, $3 \log_ 2 n$ should be $3n/\log_2 n$.
p. ?In Section 7.7 and in reference [101], the accent in Esseen should be dropped.
p. 313In Theorem 8.10, “at most $\beta$ lines” should be “at most $\beta$ curves”.
p. 315In the proof of Theorem 8.14, $c \in P$ should be $c \in A$.
p. 316In the third to last line, $|A|^{18/3}$ should be $|A|^{8/3}$.
p. 319In the proof of Theorem 8.17, one should be counting the number of narrow isosceles triangles, rather than all isosceles triangles; and in the preceding sentence, “points in $p$” should be “points in $P$“.
p. 331In the third and fifth displays, the summation should run up to $|S_n|-1$ rather than $|S_n|$. In the proof of Theorem 9.2, $t_{n-1}^{d_n-1}$ should be $t_{n-1}^{d_{n-1}}$.
p. 333In the proof of Lemma 9.3, “P vanishes on contains” should be “P vanishes on a set which contains”.
p. 346“Cauchy’s theorem” should be “Lagrange’s theorem”.
p. 358In the second display of the proof of Lemma 9.43, $s^{|G|}-4$ should be $s^{|G|}-r$.
p. 363in the last paragraph “the finite field $F_p$” should be “the ring $F_p[z]$ of formal polynomials with coefficients in the finite field $F_p$“.
p. 370In the last paragraph, the reference “Meshulam [248]” should instead be “Brown and Buhler [BB] (see also Frankl, Graham, and Rodl [FGR]”, where [BB] is “T. C. Brown and J. C. Buhler, Lines imply spaces in density Ramsey theory, J. Combin. Theory Ser. A 36 (1984), 214-220.”, and [FGR] is “P. Frankl, R. Graham, and V. Rodl, On subsets of abelian groups with no 3-term arithmetic progresion, J. Combin. Theory Ser. A 45 (1987), 157-161.”
p. 371In Exercise 10.0.3, add “The upper density of a subset $A$ of the integers is defined as $\lim\sup_{N \to \infty} \frac{|A \cap \{-N,\dots,N\}|}{2N+1}$.”.
p. 372In Exercise 10.0.8, add the hypothesis that no non-zero element of the groups containing $A$ or $B$ has order less than $k$.
p. 386In the first display, $\alpha = o( |Z|^{-1/10})$ should be $\alpha \geq |Z|^{-1/10}$.
p. 388In Lemma 10.25, there should be no absolute value signs around ${\bf E}_{n \in P'} f(n)$. In the last display, $\delta N/4$ should be $\delta/8$, and in the first display on page 389, $\sigma |P_0|/4$ should be $\sigma |P_0|/8$.
p. 388390-391: The Kronecker approximation theorem should more accurately be called a simultaneous Dirichlet approximation theorem.
p. 391In Lemma 10.29, $[1,n]$ should be $[1,N]$, there should be an additional hypothesis of ${\bf E}_{n \in [1,N]} |f(n)| \leq \delta$. The bound $\Omega(\sigma N^{1/(|S|+1)})$ should be $\Omega(N^{1/(|S|+1)})$, and the bound $\Omega(\sigma)$ should be $\Omega(\sigma/\delta)$. In the third display of the proof, the right-hand side of 1 should be $\delta$, and in the fifth display, the right-hand side should be $\Omega(\sigma/\delta)$.
p. 394in the second display, $1_{\mathrm{Bohr}(S,\rho-\rho_1)}(r)$ should be $1_{\mathrm{Bohr}(S,\rho-\rho_1)}(x)$.
p. 396In each of the two displays after “From (10.16), we have”, there is an extra right parenthesis on the right-hand side that should be deleted.
p. 397In Exercise 10.4.2, Lemma 10.15 should be Lemma 10.29.
p. 402In (10.21), $\in a+bi+\alpha \in B$ should be $- a-bi-\alpha \in B)$, and similarly in the folowing display.
p. ?In the statement of Theorem 10.31, the bound for $\Lambda(f,f,f)$ should be $\Omega({\bf E}_Z(f)^{O({\bf E}_Z(f)^{-2})})$ rather than $\Omega({\bf E}_Z(f)^{-O({\bf E}_Z(f)^{2})})$.
p. 405In Exercise 10.5.1, $K^2$ should be $O_{K,\sigma}(1)$.
p. 412In the second case of Corollary 10.50 the word ‘and’ should be inserted.
p. 413In Exercise 10.7.2, $O(k)$ should be $O(k+r)$. In the second display, $O( \frac{N}{d} )$ should similarly be $O( \frac{N}{d} + v_i )$ (which is permissible since $\sum_{i=1}^d v_i = O(N)$).
p. 414In the first paragraph, “motivation” should be “motivations”.
p. 430In the last line, a right parenthesis is missing after $\exp(O(\eta^{-O(1)})$.
p. 435In the penultimate display, $V_2$ should be $V_1$.
p. 436In the equation before (11.21), ${\bf b}(y)$ should be ${\bf b}(x+y+h)$, and the right brace should be deleted.
p. 437In the definition of $W$, $h \in W_1$ should be $x \in W_1$.
p. 439In proposition 11.12, In the last line, a right parenthesis is missing after $\exp(O(\eta_k^{-O_k(1)})$.
p. 449In the sixth line from the bottom, “positive upper progressions” should be “positive upper density”. In the first display in the proof of Theorem 11.23, $((c_j))_{j=1}^\infty$ should be $((c_j)_{j=1}^\infty)$.
p. 450In the last line of the proof, “non-zero $B$” should be “non-zero $n$“.
p. 452In Theorem 11.25, ${\bf Z}^d$ should be ${\bf Z}$.
p. 465In the definition of $P_{W,b,N}$, $Wp+b$ should be $Wq+b$; also, on the last line, “to order to” should be “in order to”.
p. 477In the top line, “an positive” should be “a positive”.
p. 492[98] and [100] refer to the same article; [98] should be deleted and redirected to [100].
Show errata for 1 older edition (216 corrections)
First edition
216 corrections
p. xiReference [116] should be [113].
p. xiv“Chevalley-Waring” should be “Chevalley-Warning”.
p. 1last display: $E(|X|^2)-E(|X|)^2$ should be $E(X^2)-E(X)^2$.
p. 3after Lemma 1.2: “with probability n” should be “with probability 1”.
p. 4after Question 1.4, “Kleiman” should be “Kleitman”.
p. 5In 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. 6Exercise 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. 8In the fourth display, $\ln$ should be $\log$. $|B|-10 \leq \nu(x) \leq |B|$ should be $|B| \leq \nu(x) \leq |B|+10$.
p. 9In Exercise 1.2.4, the inequality should be reversed.
p. 11In the paragraph after (1.17), $\ln$ should be $\log$.
p. 14In 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 $16$ (not $\kappa/16$ as stated previously).
p. 15In the first display, the first $\leq$ sign should be $\geq$.
p. 17In 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 and 1.3.6, $\lambda$ needs to be positive.
p. 18In 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. 21In 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. 22before second display: Lemma 1.53 should be Theorem 1.53.
p. 23Near 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. 24In the last display, the third $\overline{A_i}$ should be $\overline{A_j}$.
p. 25In the first display, $\geq$ should be $\leq$.
p. 36In 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. 37In the first display, $F_h(g,N)$ should be |A|.
p. 38In the RHS of the display in Corollary 1.42 RHS of the formula, $D_{k,n}$ should be $(D_{k,n}-1)^k$.
p. 40In the RHS of the last display $D_n$ should be $(D_n-1)^k$.
p. 41In the first display, $D_n + 1$ should be $(D_n + 1)^k$.
p. 44One line before the third display, $m=[n^{1/k}]$ should be $m=[n^{1/r}]$.
p. 47In the penultimate equation in the proof of Proposition 1.51, the summation $\sum_{p\leq t}$ should be $\sum_{p \leq \min(n,t)}$, and one should observe that the integrand vanishes for $t <2$. In the last equation, $\int_1^\infty (\log t + O(1)) \ldots$ should be $\int_2^\infty (\log \min(n,t) + O(1)) \ldots$. The last sentence of the proof should then say “Direct computation then shows that $\int_2^\infty \log \min(n,t) \frac{dt}{t \log^2 t} = \log \log n + O(1)$ and $\int_2^\infty \frac{dt}{t \log^2 t} = O(1)$, and the claim follows”.
p. 57In 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. 58In Exercise 2.2.5, $\sqrt{\sigma(A) |A|/2}$ should be $\sqrt{\sigma(A) |A|}$.
p. 68In 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. 69In Exercise 2.3.20, a square root is missing from the left-hand side.
p. 76In Exercise 2.4.4, Corollary 2.19 should be Corollary 2.20.
p. 77In Exercise 2.4.7, $\frac{1}{(KK')^3}$ should be $(KK')^3$.
p. 78In, Exercise 2.4.11, Proposition 2.4.11 should be Proposition 2.27.
p. 79In (2.20), $K^4$ should be $K^5$.
p. 80In 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. 82In 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. 85In, Lemma 2.23, “there exists a set $G \subset |S|^2$” should be “there exists a set $G \subset S^2$“.
p. 86In the first display of the proof of Lemma 2.34, $\tilde G$ should be G.
p. 87In 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. 95Lemma 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. 97In Theorem 2.48 (ii), $G \subset A \cdot B$ should be $G \subset A \times B$.
p. 107before Corollary 2.62, “Corollary 2.60” should be “Theorem 2.60”.
p. 112bottom: “in Section 4.4” should be “to Section 4.4”.
p. 114line 3: the last $Z$ should be an $X$.
p. 115fourth line of Lemma 3.4: ${\Bbb R}^d$ should be ${\Bbb R}^k$.
p. 116proof of Corollary 3.6: “induce” should be “induct”. In the last two sentences of this proof, $w$ should be $v$ (three occurrences).
p. 117proof of Theorem 3.7: “induce” should be “induct”.
p. 118Exercise 3.1.1 $\phi (Z')$ should be $\phi (Z)$
p. 120In the second display, $(m,m,\ldots, m)$ should be $(m-1, m-1, \ldots, m-1)$.
p. 121after the proof of Lemma 3.11, “an progression” should be “a progression”.
p. 122after 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 positive integer $n$“, and “$|n| \cdot A$” should be “$n \cdot A$“.
p. 123proof 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. 128In the proof of Proposition 3.19, $\sup_x f(x) = \sup_y f(y) = 1$ should be $\int_{\Bbb R} f = \int_{\Bbb R} g = 1$. In the next paragraph, the inclusion $\supseteq$ should be $\subseteq$, and the constraint $1 > \lambda > 0$ may be relaxed to $1 > \lambda \geq 0$.
p. 129In 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. 130In Exercise 3.4.5, “amd” should be “and”. In Exercise 3.4.6, all occurrences of $n$ should be $d$.
p. 131Delete the sentence beginning with “The lower bound is trivial…” from the proof of Lemma 3.21.
p. 132In the statement of Lemma 3.24, one needs to add the hypothesis that all the sums in $\Gamma+P$ are distinct.
p. 133On the second display, $(x-V) \cap (\Gamma+P-P)$ should be $(x-V) \cap (\Gamma+P)$. 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. 134In Lemma 3.27, Blichtfeld should be Blichfeldt.
p. 139In the last display, |B| should be $|B \cap \Gamma|$.
p. 140In the proof of Theorem 3.34, $\Gamma \cap V_{i-1} +W$ should be $\Gamma \cap V_i$.
p. 141-142Throughout 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. 144In the last pragraph of the proof of Theorem 3.38, the superfluous ) should be deleted.
p. 145In 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. 147In the second paragraph, “$M_1, \ldots, M_{d-1}$” should be “$M := (M_1,\ldots , M_{d-1})$“.
p. 155In 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. 157In (4.14), $E(1_A,1_B)$ should be $E_Z (1_A*1_B)^2$.
p. 158In 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. 159In Exercise 4.2.2, “let $p'$ the dual exponent” should be “let $p'$ be the dual exponent”.
p. 162In 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. 164in Q. 4.3.10, “$A\cdot 2A = A$” should be “$A \cdot 2A=2A$“.
p. 165bottom paragraph: “combinatorialinformation” should be “combinatorial information”.
p. 170proof 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. 171in Q 4.4.6, $\sqrt{2}$ should be $1/4$.
p. 173In 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. 174In the third display, $\|c_\xi\|_H^4$ should be $|c_\xi|^4$.
p. 175In 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. 176In the last display, $e(x,\xi)$ should be $e(\xi \cdot x)$.
p. 177In the first display, $e(x,\xi)$ should be $e(\xi \cdot x)$. In the third display, $e^{\sigma^2/\lambda}$ should be $e^{\sigma^2/2}$. 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. 179In Exercise 4.5.4, $e(x,\xi)$ should be $e(\xi \cdot x)$.
p. 181In the proof of Lemma 4.35, “incrementing k+1” should be “incrementing k”.
p. 182middle: “$S' \in S_\theta$” 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. 183In 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. 184In the fourth and fifth displays, ${\bf P}_Z(A)$ should appear on the right-hand side. (Also, to avoid ambiguity, one may wish to rewrite $\alpha^2/2|S|^2$ in the fifth display as $\frac{1}{2} \alpha^2 |S|^2$.)
p. 185In the proof of Proposition 4.39, $e(x,\xi)$ should be $e(x\cdot \xi)$.
p. 186In Proposition 4.40, $\sqrt{8 \varepsilon K}$ can be improved to $\sqrt{\varepsilon K/2}$ (though the existing bound is certainly correct). In the line before the last display, $|A| e(\xi \cdot(x-y))$ should be $|A|( e(\xi \cdot (x-y)) - 1 )$.
p. 187In Theorem 4.41, ${\Bbb Z}_p \backslash 0$ should be ${\Bbb Z}_p \backslash \{0\}$.
p. 190In the first display, 1/2 should be 1/6.
p. 191The second equality in the third line should be an inequality $\leq$.
p. 193In the third line of the first display, $\sum_{h \in H} \|T^h f\|_{L^p(Z)}^{1/p}$ should be $(\sum_{h \in H} \|T^h f\|_{L^p(Z)}^p){1/p}$. In the second and final displays (and in the first display of p194 and the second, third, and fourth displays in p195), $j \in J$ should be $j \leq J$ (seven occurrences).
p. 194In the first display, $e(x+jr,\xi)$ should be $e(\xi \cdot (x+jr))$. After the second display, ${\Bbb E}_Z f$ should equal $\delta^2$ rather than $\delta$.
p. 195first line: $S_{err}$ should be $\Gamma_{err}$. In the first display, $10 \log \frac{1}{\delta}$ should be M.
p. 196After 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). In the fourth line of the proof, $|A_{(e)}|+|B_{(e)}|$ should be $|A_{(e)}+B_{(e)}|$.
p. 203the subscript ${0}$ should be $(0)$ (three occurrences).
p. 208In the sixth line of the last display, $2k$ should be replaced by $(|H|-2)$ (here we use the bound $|\phi_h(A)| \geq 2$), and similarly for the seventh through ninth lines (and the tenth line should be deleted). [Update: this is slightly inaccurate, see revised erratum below.]
p. 209In Exercise 5.1.2, Lemma 5.1 should be Lemma 5.2. In Exercise 5.1.6, “Schirelmann” should be “Schnirelmann”.
p. 210In Exercise 5.1.9, $A_1+\ldots + A_N$ should equal ${\Bbb Z}_p$ rather than p.
p. 211In the proof of Lemma 5.13, “induce” should be “induct” (two occurrences). “Frieman” should be “Freiman”.
p. 212In the proof of Lemma 5.14, $(A \cup B) \backslash 0$ should be $(A \cup B) \backslash \{0\}$.
p. 213In 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. 214In the proof of Corollary 5.16, “induce” should be “induct”.
p. 215middle, “closed unit interval” should be “closed unit ball”. In the proof of Lemma 5.18, “induce” should be “induct”.
p. 216middle: “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. 217In 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. 218fourth line: $\varepsilon/K|A|$ should be $\frac{\varepsilon}{K} |A|$. Similarly on the sixth line. Also, on that line, $|A''| \leq \varepsilon/K|A|$ should be $|A''| \geq \frac{\varepsilon}{K}|A|$.
p. 219Before the third display, (5.14) should be (5.13).
p. 221In 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. 225In 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. 226In Exercise 5.3.7, the range of $\phi$ should be $A' \uplus B'$ rather than $A' \uplus B$.
p. 229In the proof of Theorem 5.30, Corollary 2.23 should be Lemma 5.26. In the last line of this proof, P should be Frieman isomorphic to Q, rather than to P. In Lemma 5.31, $2K(1+\log_2(KK'))$ should be $4K(1+\log_2(KK'))$.
p. 230In the first display, $K' K^{n+1}$ should be $K' K^{n+1} |B|$. In the second display, $A_{N_1}$ should be $A_{N-1}$.
p. 231In the fifth line, $w'_d$ should be $w_{d'}$. In Theorem 5.33, the hypothesis $|A| \geq 100 K^2$ should be included, and the first paragraph of the proof of Theorem 5.33 should be deleted. Throughout this proof, $\exp(O(K^{O(1)})))$ should be $\exp(O(K^{O(1)}))$.
p. 236In Proposition 5.41, “Frieman” should be “Freiman”. In Corollary 5.42, K should be d.
p. 237In Exercise 5.5.1, Hom should be $\hbox{Hom}_k$.
p. 240In Theorem 5.44, “Then there a” should be “Then there is a”
p. 241The end of proof box on the fourth line should be moved to the end of the first paragraph in page 244.
p. 243In the last line, $B$ should be $B \cap {\Bbb Z}^d$.
p. 247In Exercise 6.1.2, “symmetruc” should be “symmetric”, and one should assume $0 \not \in A$.
p. 248In Exercise 6.1.4, the denominator $|V_1| |V_2|$ should be $|V_1|^2 |V_2|^2$.
p. 249In the second line from the bottom, Ruzsa[297] should be Ruzsa[303].
p. 250In the first display, $r^2$ should be r (and in the previous line, $D_r$ should have radius $\sqrt{r}$ rather than r).
p. 253In Remark 6.8, “We say that S” should be “We say that S is”.
p. 255In 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. 256In 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. 257In 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. 258In 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. 259first paragraph: “the bound…which follow…are” should be “the bound…which follows…is”. “being of growing” should be “growing”.
p. 260in Exercise 6.3.3, $\{x+y,xy\}$ should be $\{x,y,x+y,xy\}$.
p. 263In 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. 264In 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. 265In Theorem 6.21, the first “then” should be an “and”. In the third display, $(a',b') \in A \times B$ should be $(a,b) \in A \times B$.
p. 268In the first line, “smallest” should be “largest”. In the line before Example 6.26, the right parenthesis should be deleted. In the fifth paragraph, $\|H \circ G\| \leq \|G\| \|H \|$ should be $\|H \circ G\| \geq \|G\| \|H\|$.
p. 271In the last paragraph of 6.5.2, $G_1 \circ \ldots \circ G_k$ should be $G_k \circ \ldots \circ G_1$ (two occurrences).
p. 272first display: $\frac{|X|}{|C_k|}$ should be $\frac{|C_k|}{|X|}$.
p. 276In the first bullet point, “can occur” should be “is”. In the second sentence after these bullet points, “has” should be “have”.
p. 278In 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. 279In the third paragraph, Theorem 6.31 should be Theorem 6.32.
p. 280In Exercise 7.1.4, Proposition 7.9 should be Lemma 7.9. In Exercise 7.1.1, Lemma 7.3 should be Lemma 7.2.
p. 286In the first display, there should be no m on the LHS. In the proof of Lemma 7.14, $\max$ should be $\min$ when applying Cauchy-Davenport.
p. 289In the third line, ${\Bbb P}(X^{(\mu/d)}_{w_1^k w_2^k \ldots w_d^k})$ should be ${\Bbb P}(X^{(\mu/d)}_{w_1^k w_2^k \ldots w_d^k} = 0)$.
p. 290In the display before (7.4), $e^{-\pi |X|^2/2}$ should be $e^{-\pi|X|^2/C}$.
p. 292In Exercise 7.3.2, ${\Bbb P}$ should be ${\Bbb E}$.
p. 294In statement of Proposition 7.21, add “Furthermore, each of the $w_i$ lies in the set $\{v_1,\ldots, v_n\}$“.
p. 295After the third display, the $\eta$s should be $\epsilon$s.
p. 298In the last paragraph, ${\Bbb P}(E_k)$ should be ${\Bbb P}(E_k \backslash E_{k-1})$.
p. 300In the fourth line of the proof of Corollary 7.28, $X_{i-1}$ should be $X_i$ (two occurrences).
p. 308In the third paragraph, Andrew’s should be Andrews’.
p. 309In the second paragraph, the accent on Szemerédi is misplaced. The proof of Theorem 8.1 (as well as the explicit choice of constants) is due to Chazelle, Sharir, and Welzl (see M. Aigner and G. Ziegler, Proofs from the Book, Springer-Verlag, Heidelberg, (2004), viii+239 pp.)
p. 310In the second display, $p^{-2}$ should be $p^{-3}$.
p. 312In the dual formulation of Corollary 8.5, there is a “:” missing in the set notation.
p. 314In Ex 8.2.6, “exactly two points” should be “at least two points”. (It is however an interesting question as to whether the exercise is correct as stated.) In Exercise 8.2.10, a-a should be a-a’.
p. 316In Theorem 8.15, $\Theta$ should be $\Omega$ (two occurrences).
p. 318On the sixth line, a right parenthesis is missing in $\Omega ( \frac{|A|^6}{|A \cdot A|} )$. In Exercise 8.3.4, $|A|^{1/2} |B|^{1/2} |C|^{3/2}$ should be $|A|^{1/2} |B|^{3/2} |C|^{1/2}$. In Exercise 8.3.5, $\Theta(|A|^{5/2})$ should be $\Omega(|A|^{5/2})$.
p. 322In (8.6) $|P|^{3-\frac{1}{d}}r^{-d}$ should be $O_d(|P|^{3-\frac{1}{d}}r^{-d})$.
p. 324In Exercise 8.4.3, $|L \cap P|$ should be $|l \cap P|$.
p. 325Exercise 8.4.7 is heuristically correct, but the rigorous implementation of this exercise is somewhat subtle; see this post for details.
p. 330In the proof of Theorem 9.2, “induce” should be “induct”.
p. 331After the end of the proof, “Combinatorial Nullstelensatz” should be “combinatorial Nullstellensatz”.
p. 332In 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. In Exercise 9.1.4, $P(x_1,\ldots,x_n)=0$ should be $P(x_1,\ldots,x_n) \neq 0$.
p. 333In Lemma 9.3, $F_p$ should be $F$.
p. 338In the proof of Theorem 9.11, “induce” should be “induct”.
p. 341In Question 9.2.6, $V$ should be $\Delta_n$.
p. 342In 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. 343In 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”. In the 5th line (the line before 2nd display) $\prod_{1 \leq i \neq j \leq k} \prod_{r \in R_i}$ should be $\prod_{1 \leq i \leq k}$.
p. 344In 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. 345In Exercise 9.3.3, “Let $P_{\pi}$” should be “Set $P_{\pi}$“, and $P(\pi)$ should be $P_\pi$.
p. 348In the proof of Theorem 9.24, $\hbox{deg}(F)$ should be $\hbox{deg}(P_i)$.
p. 350In Lemma 9.27, $s(|Z|)$ should be $s(Z)$.
p. 352-353In 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 $a_{m+1},\ldots,a_n$. In Theorem 9.32, [266] should be [267]. In Exercise 9.5.3, $2n-1$ should be $2n^2 -1$.
p. 354Right before Theorem 9.36, “made a significant progress” should be “made significant progress”. In the third line of Exercise 9.5.4, $h$ should be $k$. In the RHS of (9.14), $s(n,d)$ should be $s(m,d)$. Right after Theorem 9.36, $\frac{41}{11}$ should be $\frac{41}{10}$.
p. 355In Remark 9.38, “combining” should be “combined”. In the first display, $(1 - \sigma a_it_i)^{p-1} ((1 - \sigma b_i t_i)^{p-1} -1)$ should be $(1- (\sigma a_i t_i)^{p-1} )(1- (\sigma b_i t_i)^{p-1} )$. In the last sentence of the proof of Theorem 9.36, $x_1 \cdots x_m$ should be $t_1 \cdots t_m$.
p. 356In Exercise 9.6.3, $\frac{41}{11}n$ should be $\frac{41}{10}n$, and “induce” should be “induct”.
p. 358In the 3rd display $s^{|G|} - 4$ should be $s^{|G|} - r$.
p. 362bottom line: “as follows” should be “are as follows”.
p. 363middle: delete “and using Exercise 9.4.1”. Somewhat further down, $f, g \in {\Bbb Z}[z]$ should just be ${\Bbb Z}[z]$.
p. 364in the proof of Lemma 9.48, “least common multiple” should be “greatest common divisor”.
p. 365In Theorem 9.5.2, add the condition “$f$ is nonzero”.
p. 366In the last line of the proof of Theorem 9.53, max should be min. In the first paragraph of the proof of Corollary 9.54, “keeping $\underline{x}$ fixed” should be “keeping $\underline{\xi}$ fixed”.
p. 367In Exercise 9.8.3 $\phi_p(t-1)$ should be $\phi_p(t+1)$ (two occurrences).
p. 368In Exercise 9.8.8, delete “, and let $n_1,\ldots,n_k$ be distinct integers in $[0,p)$“. In the second display of Exercise 9.8.11, the sup should be an inf.
p. 370In 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. 371In 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. 375In the second and third displays, $\Lambda$ should be $\Lambda_3$. In the line afterwards, (4.3) should be (4.2).
p. 376In the second line before the exercises, one of the “that”s should be removed.
p. 377In Exercise 10.1.6, “aproper” should be “a proper”.
p. 378Corollary 10.10 should be Proposition 10.10, and similarly in page 380.
p. 379In 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. 380In 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. 381In 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. 382in the paragraph after statement of Theorem 10.20: replace”$\delta$ and $M$” with just “$M$“.
p. 383in 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. 384in Lemma 10.22, $\eta$ needs to be between 0 and 1.
p. 385in 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. 386Exercise 10.2.1 is redundant (being essentially the union of 10.0.8 and 10.0.11) and should be deleted.
p. 387In the first line, Theorem 10.12 should be Proposition 10.12; in the fourth line, Corollary 10.10 should be Proposition 10.10.
p. 388fifth 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. 389in 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. 390In Proposition 10.28, one should have $|S| = O( \delta^{-5} )$ rather than $|S| = O( \delta^{-3} )$.
p. 391The conclusion of Lemma 10.29 should be ${\Bbb E}_{n \in P'} f(n) = \Omega(\sigma)$ rather than $|{\Bbb E}_{n \in P'} f(n)| =\Omega(\sigma)$ (one wants a density increment, not a density decrement). To get this, one has to add “also, observe that the expression inside the norm has mean zero” just before the final display, replace the $L^\infty$ norm on the final display with a supremum (with no absolute values), and remove the absolute values on the first display of the next page.
p. 394In 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. 398About half-way down on the page, “to be introduce” should be “to be introduced”.
p. 402in the last two displays, $\in a+bi+\alpha \in B$ should be $\in a+bi+\alpha+B)$.
p. 407In Remark 10.44, “Gowers shown” should be “Gowers has shown”.
p. 418In “choices of $f_0,\ldots,f_{k-1})$“, the parenthesis should be deleted.
p. 439In 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. 454In the line before Theorem 11.27, the brackets \{ \} around $v_{k+1}$ are missing.
p. 462In 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. 474In the proof of Lemma 12.6, “induce” should be “induct”. In Theorem 12.5, “constants” should be “constant”.
p. 500In [288], “arithemtic” should be “arithmetic”.
p. 503Reference [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”, and Blichtfeld should be Blichfeldt.
Contributors
Thanks to all those who have contributed corrections. Corrections received on or before 2026-07-09 were reported by the readers listed below over the years; individual per-erratum attributions for these legacy entries were not preserved in migration.