Last updated: Sep 27, 2021

Hardback, 530 pages (ISBN-13: 9780521853866; ISBN-10: 0521853869)

Paperback, 512 pages (ISBN-13: 9780521136563)

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 —

- 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 (not as stated previously).
- 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 and 1.3.6, 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 .
- 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, 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 . In the next paragraph, the inclusion should be , and the constraint may be relaxed to .
- 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.
- Page 132: In the statement of Lemma 3.24, one needs to add the hypothesis that all the sums in are distinct.
- 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, 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). [Update: this is slightly inaccurate, see revised erratum below.]
- 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”. In the third display, should be .
- 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. In Exercise 7.1.1, Lemma 7.3 should be Lemma 7.2.
- 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 . In Exercise 8.3.4, should be . In Exercise 8.3.5, should be .
- 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 , 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.6. In Exercises 1.1.6, 1.1.7, should be .
- p. 7: in the display after (1.10), add an additional right parenthesis before the equality sign.
- p. 12. In Example 1.12, “Legendre’s theorem” should be “Lagrange’s theorem”.
- p. 15. In the final display, (or ) should be (this supersedes the previous erratum for this line).
- p. 16. In the final display, should be . On the next line, “Corollary 1.10 (or Corollary 1.8)” should read “Corollary 1.9”.
- p.17. In Exercise 1.3.4, should be . Also add “The variance of a complex variable is defined to be .
- p.18. In Exercise 1.3.10, add “with not both equal to ” before the first comma.
- p. 23: In (1.30), third summation, should be .
- p. 26: In the penultimate paragraph, should be .
- 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 elements of size less than .”
- p. 42: Exercise 1.8.1 contains a number of inaccuracies (namely, “” should be “, and “no two elements” should be “no two
*distinct*elements” (and so should really be 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. 45: After equation (1.45), the assertion should instead be , where is the logarithmic integral.
- p. 48: in the first bound of Exercise 1.10.3, the error term of should be .
- p.54: After the proof of Lemma 2.1, “Corollary 5.13” should be “Lemma 5.13”.
- p. 58: In Exercise 2.2.6, should be . (This error is not present in the first edition.)
- Page 61. In the penultimate bullet point of Proposition 2.7, add the requirement . After Proposition 2.7, “, , or respectively” should be “, , or respectively”.
- p.62: In (2.7), after Definition 2.8, should be .
- p. 63: In the sixth line of the first display, should be .
- p. 66: In Exercise 2.3.2, should be .
- p. 74: In Corollary 2.23, add the hypothesis .
- p. 83: In Exercise 2.5.5, the small K case is somewhat more difficult than the hint suggests, since if for some small then a direct application of Exercise 2.5.4 will give losses of rather than . 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. 88: In the definition of , should be .
- p. 100: In Corollary 2.52, A should be assumed to be a finite subset, rather than a finite subfield.
- p. 104: the last paragraph here (from “Now let …” onwards) can be deleted, and replaced by the much shorter “Since and contains both 0 and 1, we have “.
- p. 116: In the proof of Corollary 3.6, should be , and “for some ” should be “for some .
- p. 118: In 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. 121: After Lemma 3.11, “exponentially in |A|” should be “exponentially in K”.
- p. 128: In the proof of 3.19, the normalisation should not be used; instead, one should normalise (and the comment that it suffices to show should be deleted). The second display should then be restricted to the range (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. 131: In the second paragraph of the proof of Lemma 3.21, “translates of ” should be “translates of “.
- p. 132: In Lemma 3.23, the hypothesis that the cosets for are disjoint should be added.
- p. 133: The Kronecker approximation theorem should more accurately be called a simultaneous Dirichlet approximation theorem.
- p. 134: In the first line, “with” should be “within”.
- p. 135: In the definition of , should be .
- p. 136, in the first line, “directional basis for A” should be “directional basis for B”.
- p. 142: In the proof of Lemma 3.36, and should be and respectively (for consistency).
- p. 143?: Exercise 3.5.3 is
**false**. In its place, one can put the following example (from this reference): Let be the convex hull of the four points for a natural number . Show that has cardinality , but has cardinality at least . Thus one cannot remove the symmetry hypotheses from Lemma 3.21. - p. 154: In display (4.9) there is a full stop missing.
- p. 155: In Exercise 4.1.5, “pairwise independent” should be “pairwise uncorrelated”. In Exercise 4.1.7, should be . In Exercise 4.1.8, should be . In Exercise 4.1.4, “for all ” should be “for all “.
- p. 156: In the second display (defining ), should be .
- p. 158: 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, followed by Minkowski’s inequality to estimate the norm of f.”
- p. 162: In the last line, “sum of two elements” should be “sum of two squares”.
- 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). In Exercise 4.3.2, should be . In the discussion of Lemma 4.16,the term in the approximation for after the display should be deleted. In the statement of Lemma 4.16, should be .
- p. 165: In Exercise 4.3.16, the transformation needs to be assumed to be symmetric.
- p. 166: In Lemma 4.20, the additional hypothesis should be added.
- 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. 179: In Exercise 4.5.8, should be . In Exercise 4.5.7, should be .
- p. 180: Exercise 4.5.11 does not work as stated and should be deleted.
- p. 181: In Lemma 4.35, add “ is an integer, ” after “where”. In the proof, “incrementing ” should be “incrementing to “. “all dissociated subsets of ” should be “all dissociated subsets of “.
- 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. 191: In the third equation of the display, the norms should be (and the final term should just be .
- 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. 201: In the second-t0-last displayed equation, should be .
- p. 208: The sixth line of the long display should be (here we use ). The seventh line should be . The eighth line should be . The final line remains as (here we again use ).
- p. 211: In the statement of Lemma 5.13, “let suppose” should be just “suppose”.
- p. 226: In Exercise 5.3.4, the hypothesis “” is missing.
- p. 236: In the proof of Proposition 5.41, all occurrences of should be .
- p. 239: The example in Exercise 5.5.17 is incorrect; the set should be in , where is the standard basis and one adopts the convention ; “quadrilateral” should be “pentagon” and and should be replaced by throughout.
- p. 241: In the last line of the first display in the proof of Lemma 5.45, should be .
- p. 242: Before (5.19), a right parenthesis should be inserted after $\mathrm{Tor}(Z)$.
- p. 248: In the proof of Theorem 6.1, “probablistic” should be “probabilistic”.
- p. 249: In the second paragraph of the proof of Theorem 6.2, all occurrences of should be .
- p. 254: In 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. 260: In Exercise 6.3.5, should be replaced with , with the note “where is the restricted sumset of with itself” added.
- p. 263: In the final display, the right parenthesis after should be deleted. Similarly for the first display on page 264.
- p. 271: In Section 6.5.2, should be (two occurrences).
- p. 272: The definition of needs to be moved from page 273 to just before Lemma 6.34.
- p. 273: In Exercise 6.4.2, “Claim 6.31 and Claim 6.4” should be “equations (6.3) and (6.4)”.
- p. 279: In Proposition 7.7, should be .
- p. 281: In the last paragraph, “ is odd” should be “ is odd”.
- p. 285: Corollary 5.25 should be Lemma 5.25.
- 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, 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. 293: should be elements of rather than . In Example 7.19, “which have volume” should be “which has volume”.
- p. 294: After the first proof of (7.20), “lower dimension its” should be “lower its dimension”.
- p. 299: In the first display, should be .
- p. 313: In Theorem 8.10, “at most lines” should be “at most curves”.
- p. 315: In the proof of Theorem 8.14, should be .
- p. 316: In the third to last line, should be .
- p. 331: In the third and fifth displays, the summation should run up to rather than . In the proof of Theorem 9.2, should be .
- 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. 358: In the second display of the proof of Lemma 9.43, should be .
- 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. 371: In Exercise 10.0.3, add “The upper density of a subset of the integers is defined as .”.
- p. 372: In Exercise 10.0.8, add the hypothesis that no non-zero element of the groups containing or has order less than .
- p. 386: In the first display, should be .
- 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 simultaneous 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. 394: in the second display, should be .
- p. 396: In 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. 397: In Exercise 10.4.2, Lemma 10.15 should be Lemma 10.29.
- p. 402: In (10.21), should be , and similarly in the folowing display.
- p. 405: In Exercise 10.5.1, should be .
- p. 412: In the second case of Corollary 10.50 the word ‘and’ should be inserted.
- p. 413: In Exercise 10.7.2, should be . In the second display, should similarly be (which is permissible since ).
- p. 414: In the first paragraph, “motivation” should be “motivations”.
- p. 430: In the last line, a right parenthesis is missing after $\exp(O(\eta^{-O(1)})$.
- p. 435: In the penultimate display, should be .
- p. 436: In the equation before (11.21), should be , and the right brace should be deleted.
- p. 437: In the definition of , should be .
- p. 439. In proposition 11.12, In the last line, a right parenthesis is missing after $\exp(O(\eta_k^{-O_k(1)})$.
- P. 449: In 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, should be .
- p. 450: In the last line of the proof, “non-zero ” should be “non-zero “.
- p. 452: In Theorem 11.25, should be .
- 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, Ryan Alweiss, Andrea, Louigi Addario-Berry, Khodakhast Bibak, Thomas Bloom, Lin Bo, Richard Brent, Walter Bridges, Adam Brown, Tom Brown, Andres Caicedo, Rafael Tesoro Carretero, Kestutis Cesnavicius, Mei-Chu Chang, Sheng-Fu Chiu, David Colvert, Paul Delhanty, RFZ, Mikhail Gabdullin, Moubariz Garaev, Stephen Ge, Marcel Goh, Paul Hagelstein, Le Thai Hoang, Zach Hunter, Tom Koornwinder, Jarek Kuben, Nitish Kumar, Pham Lam, Choongbum Lee, Mark Lewko, Victor Lie, Chao-Ming Lin, Yang Liu, Isabel Lugo, Freddie Manners, Heiko Mattern, Vicky Neale, Danny Nguyen, Lam Pham, Gyan Prakash, Aditya Guha Roy, Juanjo Rué, Sean, Mark Sellke, Tomer Shalev, Olof Sisask, Justin W. Smith, Sai Teja Somu, Srivastan, Thomas, Matthew Tointon, Sam van Gool, Marco Vitturi, and Mate Weirdl for corrections.

## 243 comments

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19 January, 2021 at 1:04 am

adityaguharoyOn page 62 line 1, equation number 2.7 the right hand side should be The comma inside the minimum is missing.

19 January, 2021 at 1:15 am

N is a numberCan you please give an idea motivating the name “additive energy” in page 61 ?

[See https://mathoverflow.net/questions/223954/where-did-the-term-additive-energy-originate -T.]21 January, 2021 at 9:41 am

Aditya Guha RoyRelated to an already mentioned errata on page 162, there is a typo in the last line. On the last two lines of that page, I think it should say that 0 can only be represented as a sum of two squares in only one way. The squares is mentioned as elements in the book.

[Correction added, thanks – T.]22 January, 2021 at 6:27 am

Aditya Guha RoyJust some nitpicking : in the statement of lemma 4.20 (size bounds) on page 166, one should have in order to ensure that .

[This correction has already been noted – T.]25 January, 2021 at 9:07 pm

adityaguharoyOn page 61 in the 5th point of the statement of Proposition 2.7 (vanishing of the Ruzsa distance), I think we should have for all non-negative integers with because if both are equal to zero, then whence .

The condition is missing in the text.

[Erratum added, thanks – T.]25 January, 2021 at 9:13 pm

adityaguharoyThe result for follows from by an application of the exact sum set theorem (Proposition 2.2 in the book).

27 January, 2021 at 4:24 am

adityaguharoyOn line 8 of page 63 the sum should be .

[A version of this correction is already implemented – T.]1 February, 2021 at 9:24 pm

Aditya Guha RoyIn Corollary 2.23 shouldn’t we need the condition that because if each then the LHS of the display on the last line of page 73 will be (since those sets contain the identity element ) while the RHS will be .

Another thing:

On the display on the last line of page 73 shouldn’t the right hand side be because when we apply equation 2.17 with the mentioned trivial estimates we would get where the numerator comes from equation 2.17 and the denominator comes from the trivial estimate for any integer non-zero integer $k.$

But somehow I see that if the latter is also true then we need the former condition to be strengthened to because having will give the same problem.

[Erratum added. The small optimisation of the $-1$ factor has been omitted from this preliminary estimate to reduce clutter. -T]16 March, 2021 at 11:36 am

Zach HunterOn the erratum for page 15, I am confused why you say should be replaced by instead of 16. I have not worked out the constants for , so maybe also works, but I would think that 16 is the simpler constant…

[Corrected, thanks – T.]8 April, 2021 at 11:01 pm

Marcel GohHi Prof. Tao, I am wondering if the measure constructed at the top of page 450 is measure-preserving. It needs to be to apply Theorem 11.23, but it is not immediately obvious to me why it is. I can sort of see how one might argue that shifting the index by 1 inside the formula wouldn’t change the or of the sequence of averages that we apply to. But since is only guaranteed to be sandwiched between the two limits, if the inner sequence is not convergent, are we always sure that applied to the sequence is shift-invariant? Would you be able to spell this out a bit more? Thanks in advance!

18 April, 2021 at 9:39 pm

Terence TaoFor any set of integers, and differ by , and hence their generalised limits will agree.

25 May, 2021 at 5:03 am

Zach HunterOn page 51, when describing Goldbach’s conjecture, you write $2\textbf{P}$ rather than $2P$.

25 May, 2021 at 5:59 am

Aditya Guha RoyI think is used to denote the set of all prime numbers, thus the notation.

25 May, 2021 at 6:14 am

Aditya Guha RoyPlease ignore my comment above, because on page 12 (line number 6 from below), it is mentioned that the set of primes will be denoted by and in a previous section it is stated that shall be used to denote the probability of an event.

Apologies for any inconvenience caused due to my comment above.

25 May, 2021 at 6:05 am

N is a numberProf. Tao, in exercise 1.1.4 on page 5 of the book you mention the term “popularity principle” for the mentioned inequality . Can you suggest a reason why this nomenclature is assigned to the inequality?

[The term arose from Gowers’ use of “popular differences” in his proof of Szemeredi’s theorem, see e.g., p. 504 of https://www.cs.umd.edu/~gasarch/TOPICS/vdw/sz-thm-gowers-proof.pdf -T]1 June, 2021 at 10:24 am

Zach HunterOn page 26, you write $x’ \in S$ rather than $x’ + S$.

[Erratum added, thanks – T.]23 July, 2021 at 5:38 am

Zach HunterOn page 88, when defining and , I think that should be . Otherwise, the claim that would be false, as the LHS does not depend on , however the RHS says it should get arbitrarily large as approaches 1.

[Erratum added, thanks – T.]13 September, 2021 at 9:17 pm

s1mpleIn the proof of Proposition 10.24, why was it necessary to choose $p$ large enough ($2N < p N?

[If is too small, then there can be arithmetic progressions in that do not arise from arithmetic progressions in (viewed as a susbet of the integers), due to wraparound effects. -T]13 September, 2021 at 9:33 pm

s1mpleIn lemma 10.29, how does the last equation on page 391, follow from holder’s inequality, can you please explain? I am getting $\Omega(\sigma^{1/2}).

Even if we consider the display after ” In particular, from (4.2),(4.9)…” then if we take the maximum out on the LHS and then try to see, then also the last display comes out to be $\Omega(\sigma^{1/2}).

[Here we are using . -T]14 September, 2021 at 1:43 am

s1mpleDear Tao,

Page 394, on the RHS (inequality) of display 2, it should be instead of

[Corrected, thanks -T]21 September, 2021 at 7:28 am

Aditya Guha RoyI wanted to mention a nice expository application of the techniques in the first section of the chapter on Fourier analysis, which I learned from my friend Drago Grozev, in solving a problem from the IMO Shortlist of 1999 (the problem was the 6th combinatorics problem, better known as C6).

I am sharing the details below, because I feel that there exist some more people who will like it, just like how I did.

Let be two odd integers with . We color the integers using 4 colors. Show that there exists two integers of the same color which satisfy .

The Fourier analytic solution to this problem proceeds as follows:

First of all notice that one can assume without losing generality that are positive.

Now let us fix one of the 4 colors, say it is green and we define a function where if and only if is colored green.

Now for sake of contradiction, assume that the to-be-proved claim is false.

Then for each positive integer one has

.

Next, notice that the function must be periodic. Let be a period of , and let be the Fourier transform of .

Now, from the properties of deduced above it follows that and also

Now, using the conditions on and applying the inverse Fourier transform yields

and

where

thus leading to which is not possible, since is integer-valued, a contradiction is obtained as desired.

21 September, 2021 at 8:15 am

N is a numberProf. Tao,

on page 157 in equation 4.14 you have written ; however the additive energy was defined for sets, and here we have their characteristic functions.

Apologies in advance if I missed taking a note of this notation defined somewhere earlier in the text.

[See the existing erratum for this equation – T.]22 September, 2021 at 10:58 pm

N is a numberA small not-so-serious typo: on page 156 and in the 7th line after the section heading “L^p theory” I think you would like to put a colon (:) before the equality sign when you define since this expression defines the quantity on the left.

[Erratum added, thanks – T.]13 November, 2021 at 5:47 pm

RFZDear Prof. Tao,

on page 61 (Proposition 2.7) the fifth condition should be $|nA-mA|=|A|$ for all pairs of non-negative integers $n,m$ with $n+m\geq 1$. Because if $(n,m)=(0,0)$, then $|0A-0A|=|A|$ but $0A$ is defined to be $\{0\}$.

[This errata was already added – T.]14 November, 2021 at 9:03 am

RFZDear Prof. Tao,

on page 61 after proposition 2.7 the sentence “Later on in this chapter we shall generalize this proposition to the case when the Russia distance, difference constant, or doubling constant are little larger than 0,0, or 1, respectively…” to “Later on in this chapter we shall generalize this proposition to the case when the Russia distance, difference constant, or doubling constant are little larger than 0,1, or 1, respectively…”.

[Correction added, thanks – T.]24 November, 2021 at 11:49 am

RFZDear Prof. Tao,

I was wondering is the inequality in problem 2.2.6 on page 58 correct? I guess it should be ” at most “. Since is a Sidon set, then . Also then by problem 2.2.5 we have . Combining them we obtain the following: where is a Sidon set in . Please correct me if I’m wrong. Thanks a lot!

24 November, 2021 at 12:48 pm

RFZSorry I was wrong. I’ve figured out how to solve. Easy to see that but since is a Sidon set in , then . Hence and taking square root we obtain the desired inequality.