Microsoft Office 2010 Pro Plus Additive combinator

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Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao

Last updated: Feb 13, 2011
Additive Combinatorics
Terence Tao, Van Vu
Cambridge University Press
Hardback, 530 pages (ISBN-13: 9780521853866; ISBN-10: )
This book covers the basic tools in additive combinatorics: sum set estimates, inverse theorems, graph theory techniques, crossing numbers, algebraic methods, Szemerédi’s theorem.  A revised edition is forthcoming.
Sample chapters (contents, probabilistic method, sum set estimates, additive geometry)
— Deleted scenes —
Arithmetic Ramsey Theory
Menger’s theorem
Santalo’s inequality
Entropy sumset estimates
— Precursor material —
Gowers’ proof of Szemeredi’s theorem for progressions of length 4
Non-commutative sum set estimates
The Roth-Bourgain theorem
Math 254A (some highlights of arithmetic combinatorics)
Math 262 (Topics in Combinatorics)
— Errata —
p. xi: Reference [116] should be [113].
p. xiv: “Chevalley-Waring” should be “Chevalley-Warning”.
p. 1, last display: should be .
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, should be (two occurrences). In Q. 1.1.5, and should be and respectively.
p. 6: Exercise 1.1.8 is missing. This exercise reads: “Let be an additive set. Show that there exists a subset of cardinality such that . (Hint: translate randomly by independent elements of , and use the first moment method.)”
p. 8: In the fourth display, should be . should be .
p. 9: In Exercise 1.2.4, the inequality should be reversed.
p. 11: In the paragraph after (1.17), should be .
p. 14: In the 9th line from below, should be (two places); similarly before (1.23) on P. 15. In the final display, should be .
p. 15: In the first display, the first sign should be .
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 needs to be odd. In Exercise 1.3.4, the first expectation should be a probability. In Exercises 1.3.5 and1.3.6, needs to be positive.
p.18: In Exercise 1.3.8, the expectation should be and the variance should be . In Exercise 1.3.11, should be .
p. 21: In Definition 1.21,”we say that is a -complementary base of ” should be”we say that is a -complementary base of order of “.
p. 22, before second display: Lemma 1.53 should be Theorem 1.53.
p. 23: Near top: “the event that is good” should be “the event that is good”. In the last line of the proof of Lemma 1.24, should be .
p. 24: In the last display, the third should be .
p. 25: In the first display, should be .
p. 36: In the first line of the proof of Theorem 1.39, add “Without loss of generality we may assume .   In the third display on the last line, should be .
p. 37: In the first display, should be |A|.
p. 38: In the RHS of the display in Corollary 1.42 RHS of the formula, should be .
p. 40: In the RHS of the last display should be .
p. 41: In the first display, should be .
p. 44: One line before the third display, should be .
p. 47: In the penultimate equation in the proof of Proposition 1.51, the summation should be , and one should observe that the integrand vanishes for .  In the last equation, should be .  The last sentence of the proof should then say “Direct computation then shows that and , and the claim follows”.
p. 57: In the bounds for , should be (two occurrences). Similarly, should be (two occurrences).
p. 58: In Exercise 2.2.5, should be ; similarly, for Exercise 2.2.6, should be .
p. 68: In Exercise 2.3.14, should be , and should be . Replace “though the set may be symmetric around a non-zero origin ” with “where by symmetry here we mean that ”. (The point is that 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, should be .
p. 78: In, Exercise 2.4.11, Proposition 2.4.11 should be Proposition 2.27.
p. 79: In (2.20), should be .
p. 80: In the first paragraph, should be . 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 and ” should be the iterated sum and difference sets of and “.
p. 82: In Exercise 2.5.3, add after . In Exercise 2.5.4, “is can be used” should be “can be used”.
p. 85: In, Lemma 2.23, “there exists a set ” should be “there exists a set “.
p. 86: In the first display of the proof of Lemma 2.34, should be G.
p. 87: In the fourth display, the on the LHS and RHS should be . In the last sentence in the statement of Theorem 2.35, should be .
p. 95: Lemma 2.41, while correct, does not imply Corollary 2.42 as stated (since does not control ). Thanks to Mei-Chu Chang for pointing out the problem. Replace “As , this implies” by “A similar argument (see [Proposition 4.5, 362]) gives”. Rename Corollary 2.42 as Proposition 2.42, replace “” by “” in the first sentence, replace “” by “” in the second sentence, and replace “” by “” in the display.
p. 96: “verstion” should be “version”.
p. 97: In Theorem 2.48 (ii), should be .
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 should be an .
p. 115, fourth line of Lemma 3.4: should be .
p. 116,Microsoft Office Pro Plus, proof of Corollary 3.6: “induce” should be “induct”. In the last two sentences of this proof, should be (three occurrences).
p. 117, proof of Theorem 3.7: “induce” should be “induct”.
p. 118: Exercise 3.1.1 should be p. 120: In the second display, should be .
p. 121, after the proof of Lemma 3.11, “an progression” should be “a progression”.
p. 122, after first display: “for all real ” should be “for all non-negative reals “. In the next sentence, “for any integer ” should be “for any positive integer “, and “” should be ““.
p. 123, proof of Theorem 3.13: should be . On the same line, “whch” should be “which”.
p. 124: should be (two occurrences). In the first display and two lines further down, should be . In the second-to-last display, should be . In the last paragraph, should be .
p. 128: In the proof of Proposition 3.19, should be .
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, should be .
p. 130: In Exercise 3.4.5, “amd” should be “and”.  In Exercise 3.4.6, all occurrences of should be .
p. 131: Delete the sentence beginning with “The lower bound is trivial…” from the proof of Lemma 3.21.
p. 133: On the second display, should be . Just before the third display, should be .  In Corollary 3.25, should be . In the next display, delete .
p. 134: In Lemma 3.27, Blichtfeld should be Blichfeldt.
p. 139: In the last display, |B| should be .
p. 140: In the proof of Theorem 3.34, should be .
p. 141-142: Throughout the statement and proof of Lemma 3.36, should be replaced with , and replaced with . In the last two equations in the display, replace and both by .
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 and is not proper” with “If , is not proper, and is torsion-free”. In the proof, “We induce on ” should be “We induct on ”. On the last line, “unless is torsion-free” should be deleted.
p. 147: In the second paragraph, “” should be ““.
p. 155: In Q 4.1.7, should be .  In Q 4.1.10, the range of should be rather than C.
p. 157: In (4.14), should be .
p. 158: In the proof of Lemma 4.10, should be . Between the second-to-last and third-to-last lines of the long display in that proof, insert the lines “” and “”.
p. 159: In Exercise 4.2.2, “let the dual exponent” should be “let be the dual exponent”.
p. 162: In the proof of Lemma 4.14, should be . In the third line of Corollary 4.15, should belong to rather than , and should be . (Remark: the order estimate only gives the qualitative result for sufficiently large, but the small values of can be done by hand.) In the last word of the last sentence, should be .
p. 164, in Q. 4.3.10, “” should be ““.
p. 165, bottom paragraph: “combinatorialinformation” should be “combinatorial information”.
p. 170, proof of Lemma 4.25: should be 2, and the condition should be .
p. 171, in Q 4.4.6, should be .
p. 173: In the proof of Proposition 4.29, (4.31) should be (4.30).  In the second and last displays, should be .
p. 174: In the third display, should be .
p. 175: In the four summations involving , should range over Z rather than S.  After the second display, and should be and respectively, and similarly on the third display.
p. 176: In the last display, should be .
p. 177: In the first display, should be .   In the third display, should be .  In the penultimate display, should be . “If take” should be “Taking”.  In the last display, should be =.
p. 179: In Exercise 4.5.4, should be .
p. 181: In the proof of Lemma 4.35, “incrementing k+1″ should be “incrementing k”.
p. 182, middle: “” should be ““. After the fourth display, should be .  In the last display, the upper limit of the integral should be infinite.
p. 183: In the last four displayed equations, all signs should be . In (4.38), is missing from the RHS.  In the sixth display, should be .
p. 184: In the fourth and fifth displays, should appear on the right-hand side.  (Also, to avoid ambiguity, one may wish to rewrite in the fifth display as .)
p. 185: In the proof of Proposition 4.39, should be .
p. 186: In Proposition 4.40, can be improved to (though the existing bound is certainly correct).  In the line before the last display, should be .
p. 187: In Theorem 4.41, should be .
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 .
p. 193: In the third line of the first display,Microsoft Office 2007 Standard, should be .  In the second and final displays (and in the first display of p194 and the second, third, and fourth displays in p195), should be (seven occurrences).
p. 194: In the first display, should be .  After the second display, should equal latex \delta^2$ rather than .
p. 195, first line: should be .  In the first display, should be M.
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, should be .
p. 200: “induce” should be “induct” (four occurrences). In the fourth line of the proof, should be .
p. 203: the subscript should be (three occurrences).
p. 208: In the sixth line of the last display, should be replaced by (here we use the bound ), and similarly for the seventh through ninth lines (and the tenth line should be deleted).
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, should equal 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, should be .
p. 213: In the proof of Proposition 5.15, should be , should be , and should be .
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 ” should be “we see that contains a set symmetric around some origin “.
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: should be . Similarly on the sixth line.  Also, on that line, should be .
p. 219: Before the third display, (5.14) should be (5.13).
p. 221: In the first example, should be .  In the fourth line of the fifth example, should be .
p. 225: In the first line, should be .  A period is missing before “Fortunately” in the sentence after the first display.
p. 226: In Exercise 5.3.7, the range of should be rather than .
p. 229: In 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, should be .
p. 230: In the first display, should be .  In the second display, should be .
p. 231: In the fifth line, should be .  In Theorem 5.33, the hypothesis should be included, and the first paragraph of the proof of Theorem 5.33 should be deleted.  Throughout this proof, should be .
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 .
p. 240: In Theorem 5.44, “Then there a” should be “Then there is a”
p. 241: The end of proof box on the fourth line should be moved to the end of the first paragraph in page 244.
p. 243: In the last line, should be .
p. 247: In Exercise 6.1.2, “symmetruc” should be “symmetric”, and one should assume .
p. 248: In Exercise 6.1.4, the denominator should be .
p. 249: In the second line from the bottom, Ruzsa[297] should be Ruzsa[303].
p. 250: In the first display, should be r (and in the previous line, should have radius rather than r).
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 )” should be deleted. “” should be ““. Shortly afterwards, “latter” should be “former”.
p. 256: In the proof of Corollary 6.12, “induce” should be “induct”, and “ contains a” should be “ 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, “ monochromatic” should be “-monochromatic”. In Theorem 6.15, should be . In Proposition 6.16, should be , “exists distinct classes” should be “exist distinct classes”, and “” should be “non-zero “. 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, should be .
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, should be .
p. 263: In the statement of Corollary 6.20, the in the denominator in the last line should be . In the second paragraph of the proof. the denominator should be .
p. 264: In the second paragraph, the denominator should be . In the penultimate display, should be . In the proof of Corollary 6.20, should be (two occurrences), and should be (one occurrence). In the proof of Theorem 2.29, should be (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.  In the fifth paragraph, should be .
p. 271: In the last paragraph of 6.5.2, should be (two occurrences).
p. 272, first display: should be .
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, should be . In the proof of Lemma 7.6, “induce” should be “induct”.
p. 279: In the third paragraph, Theorem 6.31 should be Theorem 6.32.
p. 280: In Exercise 7.1.4, Proposition 7.9 should be Lemma 7.9.
p. 286: In the first display, there should be no m on the LHS.
p. 289: In the third line, should be .
p. 290: In the display before (7.4), should be .
p. 292: In Exercise 7.3.2, should be .
p. 294: In statement of Proposition 7.21, add “Furthermore, each of the lies in the set “.
p. 295: After the third display, the s should be s.
p. 298: In the last paragraph, should be .
p. 300: In the fourth line of the proof of Corollary 7.28, should be (two occurrences).
p. 308: In the third paragraph, Andrew’s should be Andrews’.
p. 309: In 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. 310: In the second display, should be .
p. 312: In the dual formulation of Corollary 8.5, there is a “:” missing in the set notation.
p. 314: In 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. 316: In Theorem 8.15, should be (two occurrences).
p. 318: On the sixth line, a right parenthesis is missing in .
p. 322: In (8.6) should be .
p. 324: In Exercise 8.4.3, should be .
p. 325: Exercise 8.4.7 is heuristically correct, but the rigorous implementation of this exercise is somewhat subtle; see this post for details.
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 in should be . In Exercise 9.1.3, the Hilbert nullstellensatz of course only applies when F is algebraically closed. In Exercise 9.1.4, should be .
p. 333: In Lemma 9.3, should be .
p. 338: In the proof of Theorem 9.11, “induce” should be “induct”.
p. 341: In Question 9.2.6, should be .
p. 342: In Theorem 9.17, should be , and should be . In the last sentence of the proof of Theorem 9.16, “” should be ““.
p. 343: In the first display, the right-hand side should just be . In the next paragraph, should be ,Microsoft Office 2010 Pro Plus, and in the paragraph after that, should be . In the first line of the last display, should be . In Lemma 9.18, the second sentence should start “Then” rather than “The”.  In the 5th line (the line before 2nd display)  should be .
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 ” should be “be a power of “.
p. 345: In Exercise 9.3.3, “Let ” should be “Set “, and should be .
p. 348: In the proof of Theorem 9.24, should be .
p. 350: In Lemma 9.27, should be .
p. 352-353: In the proof of Lemma 9.31: all occurrences of should be , etc. Also, should be .  In Theorem 9.32, [266] should be [267].  In Exercise 9.5.3, should be .
p. 354: Right before Theorem 9.36, “made a significant progress” should be “made significant progress”.  In the third line of Exercise 9.5.4, should be .  In the RHS of (9.14), should be .  Right after Theorem 9.36, should be .
p. 355: In Remark 9.38, “combining” should be “combined”.  In the first display, should be .  In the last sentence of the proof of Theorem 9.36, should be .
p. 356: In Exercise 9.6.3, should be , and “induce” should be “induct”.
p. 358: In the 3rd display should be .
p. 362, bottom line: “as follows” should be “are as follows”.
p. 363, middle: delete “and using Exercise 9.4.1″. Somewhat further down, should just be .
p. 364, in the proof of Lemma 9.48, “least common multiple” should be “greatest common divisor”.
p. 365: In Theorem 9.5.2, add the condition “ is nonzero”.
p. 366: In 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 fixed” should be “keeping fixed”.
p. 367: In Exercise 9.8.3 should be (two occurrences).
p. 368: In Exercise 9.8.8, delete “, and let be distinct integers in “.  In the second display of Exercise 9.8.11, the sup should be an inf.
p. 370: In Examples 10.3, should be . In Theorem 10.5, the phrase “… with coprime to ” 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, should be . 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, should be . In the third line of the proof of Theorem 10.12 (which should be Proposition 10.12), should be . 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 “” rather than ““.
p. 381: In the second line, should be . (In the next two displays, the 4 is correct.) In the third line, add “and in ” just before “, such that”. In the third and fourth displays, should be .
p. 382: in the paragraph after statement of Theorem 10.20: replace” and ” with just ““.
p. 383: in the third display, the first = sign should be , while should be |.
p. 384: in Lemma 10.22, needs to be between 0 and 1.
p. 385: in third display, should be , and similarly on the next two lines. In the next display, the sum should be enclosed in parentheses.
p. 386: Exercise 10.2.1 is redundant (being essentially the union of 10.0.8 and 10.0.11) and should be deleted.
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, “ is bounded” should be “ is bounded by 1″. In third line, should be . In Exercise 10.3.1, add the hint “(You may want to first establish the weaker but easier bound .)”
p. 390: In Proposition 10.28, one should have rather than .
p. 391: The conclusion of Lemma 10.29 should be rather than (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 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. 394: In the second, third, and fourth display, the indicator functions should be applied to x instead of r. Also, in the fourth equation, should be .
p. 398: About half-way down on the page, “to be introduce” should be “to be introduced”.
p. 402: in the last two displays, should be .
p. 407: In Remark 10.44, “Gowers shown” should be “Gowers has shown”.
p. 418: In “choices of “, the parenthesis should be deleted.
p. 439: In the second line, should be . In the fourth line, should be . In the proof of Proposition 11.12, “induce” should be “induct”.
p. 454: In the line before Theorem 11.27, the brackets \{ \} around are missing.
p. 462: In the last line, “in” should be .
p. 465: “and some k-pseudorandom measure , then” should be “for some k-pseudorandom measure . Then”.
p. 474: In the proof of Lemma 12.6, “induce” should be “induct”.   In Theorem 12.5, “constants” should be “constant”.
p. 500: In [288], “arithemtic” should be “arithmetic”.
p. 503: Reference [352] should be redirected to [380].
p. 505: should be , and O(f(n)) needs to be in math mode.
p. 506: “Chevalley-Waring” should be “Chevalley-Warning”, and Blichtfeld should be Blichfeldt.
– Errata to the revised edition –
p.17.  In Exercise 1.3.4, should be .
p.18.  In Exercise 1.3.10, add “with not both equal to ” before the first comma.
p. 30.  In the third display, should be (say), and the third equality should be a sign.
p. 35, first line: a right parenthesis ) missing before .
p. 36-37: The 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 has at most a bounded number of representations as the sum of elements of between and ; the treatment of the remaining sums in which at least one term is less than is left as an exercise.”  Then, in the definition of and also in the computation of , insert the lower bound in the sum.  On the first line of page 37, and in Exercise 1.7.2, replace with for some .  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 for some sufficiently small , by using an induction hypothesis to bound the number of representations by sums of fewer than $k$ elements of size less than .”
p. 45: After equation (1.45), the assertion should instead be , where is the logarithmic integral.
p.54:  After the proof of Lemma 2.1, “Corollary 5.13″ should be “Lemma 5.13″.
p.???: In (2.7), after Definition 2.8, should be .
p. 131: In the second paragraph of the proof of Lemma 3.21, “translates of ” should be “translates of “.
p. 133: The Kronecker approximation theorem should more accurately be called a multidimensional Dirichlet approximation theorem.
p. 134: In the first line, “with” should be “within”.
p. 135: In the definition of , should be .
p. 154: In display (4.9) there is a full stop missing.
p. ???: In the proof of Lemma 4.10, in the final display, the last equals sign should be a sign.  An additional sentence of explanation should be added: “The penultimate inequality follows from Plancherel’s theorem,microsoft Office 2010 Serial, followed by Minkowski’s inequality to estimate the norm of f.”
p. 163: In 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).
p. 168-169: Before Proposition 4.23, “dispense with” should be “almost dispense with”.  In Proposition 4.23, the proper progression should be the proper coset progression , where is a subgroup of (i.e. the multiples of for some factor of ). The last sentence of the proposition is redundant and can be deleted. In the proof, one first handles the case when has order in , in which case the first display of p.169 is an equality, and one obtains a progression of rank .  In the general case, when has order for some dividing , one then passes from to by quotienting out the group generated by , and then applies the previous case.
p. 170: In the proof of Lemma 4.25, “find thus find” should be “thus find”.
p. 171: In Exercise 4.4.7, “lower” should be “increase”.
p. 174: In Remark 4.31, the sentence “The converse statement is true up to logarithmic factors; see exercises.” should be deleted.
p. 176-177: The denominator in (4.34) can be improved to , by setting equal to instead of in page 177 (changing the fourth and fifth displays appropriately).
p. 180: Exercise 4.5.11 does not work as stated and should be deleted.
p. 183: In the statement of Lemma 4.37, the second “an” should be “a”.
p. 187: In the statement of Theorem 4.41, the upper bound for should be rather than just .
p. 188: At the end of the proof of Theorem 4.41, add the following parenthetical remark: “The upper bound on required for Corollary 2.62 is supplied by (4.37) and the lower bound on .”  In Exercise 4.6.2: should be (two occurrences).
p. 189: In the discussion after Theorem 4.42, the bound should be $latex E(A,A) >= P(A) |A|^3$.
p. 193: Lemma 4.46 has a redundant “such that”.
p. 194: In Theorem 4.47, the should be , and should be .
p.196: A 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, should be .
p. 199: In the second sentence in the paragraph before the statement of Lemma 5.3, “Lemma” should be uncapitalised.
p. 211: In the statement of Lemma 5.13, “let suppose” should be just “suppose”.
p. 254: In exercise 6.2.8, the adjective “triangle-free” should be deleted.
p. 271: In Section 6.5.2, should be (two occurrences).
p. 279: In Proposition 7.7, should be .
p. 286: In the second paragraph, the condition should be replaced by , and the phrase “by Markov’s inequality” should be dropped.  Similarly, in the third and fifth displays, the first occurrence of in each should be .
p. 291: In the definition of after the first display,Office Professional, a is needed at the end of the integral.  In the first paragraph, add “but still bounded” after “sufficiently large”.
p. 292: In Exercise 7.3.4, the comma should be an apostrophe.
p. 299: In the first display, should be .
p. 315: In the proof of Theorem 8.14, should be .
p. 331: In the third and fifth displays, the summation should run up to rather than .
p. 333: In 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. 370: In 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. 388: In Lemma 10.25, there should be no absolute value signs around .  In the last display, should be , and in the first display on page 389, should be .
p. 388, 390-391: The Kronecker approximation theorem should more accurately be called a multidimensional Dirichlet approximation theorem.
p. 391: In Lemma 10.29, should be , there should be an additional hypothesis of .  The bound should be , and the bound should be .  In the third display of the proof, the right-hand side of 1 should be , and in the fifth display, the right-hand side should be .
p. 397: In Exercise 10.4.2, Lemma 10.15 should be Lemma 10.29.
p. 412: In the second case of Corollary 10.50 the word  ’and’ should be inserted.
p. 414: In the first paragraph, “motivation” should be “motivations”.
p. 465: In the definition of , should be ; also, on the last line, “to order to” should be “in order to”.
p. 477: In 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].
Thanks to Eric Aas, Andrea, Louigi Addario-Berry, Khodakhast Bibak, Thomas Bloom, Richard Brent, Tom Brown, Andres Caicedo, Kestutis Cesnavicius, Mei-Chu Chang, Sheng-Fu Chiu, Paul Delhanty, Moubariz Garaev, Paul Hagelstein, Le Thai Hoang, Tom Koornwinder, Choongbum Lee, Mark Lewko, Victor Lie, Isabel Lugo, Heiko Mattern, Vicky Neale, Gyan Prakash, Sean, Olof Sisask, Justin W. Smith, Thomas, Sam van Gool, and Mate Weirdl for corrections.