Last updated: Dec 18, 2019

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 .
- 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. 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. 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.)
- 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. 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. 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 .
- 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. 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 .
- 242: Before (5.19), a right parenthesis should be inserted after $\mathrm{Tor}(Z)$.
- 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 .
- Page 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. 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, Mikhail Gabdullin, Moubariz Garaev, Stephen Ge, Paul Hagelstein, Le Thai Hoang, Tom Koornwinder, Jarek Kuben, 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, 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.

## 203 comments

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28 January, 2008 at 6:58 am

HoangOn p.263 :”Let us call a pair (a,a’) bad if they are not connected by at least paths of length 2″, shouldn’t that number be ?. The same also appears and need to be replaced on p.264

28 January, 2008 at 10:37 am

Terence TaoThanks for the correction!

31 January, 2008 at 3:16 pm

AnonymousWould you mind elaborating a bit on (Open) Question 1.4 in section 1.1 of the text. I’m assuming you are using n here to indicate |A| in the previous theorem. As I read it this result can’t be true as (n/3)+10 can exceed the size of the set for small sets. That aside I’m curious about the constant 10. Bourgan’s paper appears to show that there exists B (in the language of Theorem 1.3) such that |B|>= |A|/3 +2. Has there been further improvement to the constant increasing it to 9? Are counterexamples know for 11? Thanks for the clarification and great blog!!

31 January, 2008 at 4:16 pm

HaoDear Anonymous,

I think you can consider n sufficiently large so that small sets are not important here. Bourgain’s result should be |A|/3+2/3, slightly better than |A|/3+1/3 by the probabilistic method.

1 April, 2008 at 2:25 am

AnonymousDear Professor Tao,

Sorry for bothering you.

I have some questions on the proof of Lemma 4.25, on p. 170.

a. In line 3, when invoking Lemma 4.20, where does the number 5 in the log come from? Why isn’t that a 4?

b. In line 5, where it says that this should hold for all : I cannot really make sense of it. Did you mean ?

c. To be honest to the point of embarrassment: I cannot easily see why the Bohr set is then regular. For the quotients of the densities of the two Bohr sets, I always get a useless estimate: instead of …..

1 April, 2008 at 2:49 pm

Terence TaoDear Anonymous,

You are correct, one can replace with (i.e. with 2) in that bound, and that the condition should be .

For your last question, one has that whenever . This follows from the inequalities whenever , which follow from the fundamental theorem of calculus (since is bounded between 1 and 5 in the interval ).

2 April, 2008 at 12:34 am

Vicky NealeOn page 107 (in the proof of Proposition 2.61) there seems to be a reference to Exercise 1.1.8. I haven’t found Exercise 1.1.8; is there a typo?

Thanks!

2 April, 2008 at 9:59 am

Terence TaoHmm, that’s strange, that exercise managed to get deleted from the print version. It’s now in the errata above.

14 April, 2008 at 10:39 pm

AnonymousDear Prof Tao

I have some questions about the proof of Proposition 10.32.

* On p. 394, 2nd, 3rd & 4th displayed equation, I think some of the s should be replaced by s?

* 4th displayed equation on that page: should that be instead of ?

Thanks!

17 April, 2008 at 10:00 am

Terence TaoDear anonymous: Thanks for the correction!

13 June, 2008 at 7:42 am

Isabel LugoExercise 6.1.4 (“Cauchy-Schwarz for bipartite graphs”) appears to be in error.

You ask for proofs:

(a) that a bipartite graph contains at least paths of length two with both endpoints in , including degenerate paths;

(b) that it contains at least cycles of length 4.

A Cauchy-Schwarz type argument gives the answer for the first part. Another C-S argument, when I attempted it, gave me a bound of for the second part. And this appears to be sharp; for example consider the complete bipartite graph . This graph has edges and a number of 4-cycles which grows like for some positive constant C — the exact count depending on how degeneracies are counted and whether things like (1, 2, 3, 4) and (3, 4, 1, 2) are counted as different cycles. The bound given in the problem is .

13 June, 2008 at 7:46 am

Isabel LugoAlso, although this is irrelevant to this post — my previous comment shows up with the time stamp “13 June, 2008 at 7:42 am”. I made it at 11:42 am US Eastern time (my local time); I assume since you’re in Los Angeles your blog uses US Pacific time (your local time), hence it should show 8:42 am. I suspect for some reason WordPress is not adjusting for daylight savings time.

13 June, 2008 at 2:43 pm

Isabel LugoAnother potential erratum: the second displayed equation on p. 310 (lower bound for the crossing number) is given as

but this should be

.

14 June, 2008 at 10:53 am

Terence TaoDear Isabel: Thanks for the corrections! WordPress.com does not seem to have an easy way to automatically adjust for daylight savings; I could move the blog time zone back and forth an hour every six months, but this seems to be more trouble than it is worth…

7 October, 2008 at 12:00 pm

Finite subsets of groups with no finite models « What’s new[…] larger than A itself if A has some additive structure; see this paper of Ruzsa (or Lemma 5.26 of my book with Van Vu) for a precise statement. This projection trick has a number of important uses in additive […]

12 October, 2008 at 2:00 pm

Paul HagelsteinIn Appendix 1.10 p. 47, the last two equations for the sums over primes less than or equal to n feature an “n” on the LHS but not the RHS. Offhand the only correction I see involves an “integration by parts” argument frequently associated to Abel’s summation technique, but I have the impression that you have a more efficient argument in mind!

12 October, 2008 at 2:49 pm

Terence TaoDear Paul: Thanks for pointing out the issue! I have adjusted the erratum for that page to provide what I think fixes the problem.

6 December, 2008 at 10:43 am

SeanHere are some possible typos:

– p. 1: I think the last equation should end “E(X^2) – E(X)^2” (no absolute values).

– p. 8: should read .

– p. 58, Exercise 2.2.5: There are examples of additive sets A with non-empty subsets B which have doubling constant larger than \sqrt{ \sigma[A]|A|/2 }. Take A = B with |A| \leq 2, or (less trivially), A = {0,1,2,3} and B= {0,1,3}. However, it is easy to show that the doubling constant of B is at most \sqrt{ \sigma[A]|A| }. Is this the correct bound for the exercise, or is the bound meant asymptotically? I’m guessing that changing the bound in Ex 2.2.5 also affects the bound in Ex 2.2.6.

Thanks!

9 December, 2008 at 6:48 am

Terence TaoDear Sean, thanks for the corrections!

4 October, 2012 at 6:32 am

ablmffHi, for 2.2.6, isn’t it easy to prove the bound \sqrt{ 2 \sigma[A] |A| } ? We are confused why 2 should be outside the square root.

[Yes, you’re right; this was actually correct in the first edition, but was mistakenly revised in the second edition. I’ll add a new erratum to reflect this. -T]4 February, 2009 at 4:01 am

Vicky NealeI’m a bit anxious about the spelling of Blichtfeld/Blichfeldt as it appears on page 134 (Lemma 3.27) and in the index. For example, the splendid Mactutor history of maths site has a biography of Blichfeldt (http://www-history.mcs.st-and.ac.uk/Biographies/Blichfeldt.html). I’m certainly not an expert, though!

5 February, 2009 at 9:12 am

Terence TaoOops, you’re right, thanks!

3 March, 2009 at 6:55 pm

JohnCould it be the case that the second exponent in exercise 2.3.13., p.68, is wrong? seems hard to prove at this point of the text (whereas would follow along the same lines of the first inequality). Thanks!

4 March, 2009 at 8:56 am

Terence TaoDear John,

I think it works in both directions (for the second inequality, you have to substitute -A for B).

7 March, 2009 at 7:47 pm

AnonymousDear Professor Tao,

In page 122, after the first math display, should one read “In particular we have “?

8 March, 2009 at 9:55 am

Terence TaoDear anonymous:

Thanks for the correction!

25 April, 2009 at 6:13 am

AnonymousDear Prof Tao,

This comment is somewhat tangential, but concerns the proof of the Squeezing Lemma (page 137):

How does one go about showing the existence of a continuous right inverse of ? I can’t seem to find a rigorous definition of centre of mass.

Thanks

27 April, 2009 at 10:28 am

Terence TaoDear anonymous,

The centre of mass of a bounded non-empty open set A of in is defined as the vector . The dominated convergence theorem can be used to establish continuity of the centre of mass with respect to the type of perturbations in A one would encounter in the squeezing lemma.

28 May, 2009 at 2:18 am

Olof SisaskExercise 10.2.1 appears to duplicate exercises 10.0.8 and 10.0.11, and this may not be intentional. (I would say that 10.2.1 is the outlier, rather than 10.0.8 and 10.0.11.)

Thanks.

28 May, 2009 at 9:15 am

Terence TaoHmm, how strange. It’s not exactly an erratum, but I’ll make a note of it nevertheless.

29 May, 2009 at 9:30 am

Olof SisaskA small typo: four lines into the proof of Cauchy-Davenport on p. 200, the |A_e| + |B_e| on the left-hand side of the inequality should be |A_e + B_e|.

[Updated, thanks – T.]25 June, 2009 at 5:13 pm

Sumset and inverse sumset theorems for Shannon entropy « What’s new[…] Probability, and Computing. This paper evolved from a “deleted scene” in my book with Van Vu entitled “Entropy sumset estimates“. In those notes, we developed analogues of the […]

8 July, 2009 at 8:50 pm

Mark LewkoOn page 465, should q and p be the same parameter in the definition of P_{W,b,N}? Also on page 465, I think there might be a typo in the phrase “again needed to order to neglect the r = 0 diagonal term”.

[Corrected, thanks – T.]17 July, 2009 at 10:00 am

Hales-Jewett 定理的归纳证明 « Liu Xiaochuan’s Weblog[…] der Waerden’s theorem都是它的特例。下面的证明是在Terence Tao和Van Vu的书 Additive Combinatorics […]

23 July, 2009 at 2:10 am

VickyA very tiny point: I think that in Exercise 4.3.14 (page 165), Polya has lost his accent. This seems to be the case in the corresponding index entry too.

Thanks.

[Corrected, thanks – T.]5 August, 2009 at 8:55 am

AndreaDear Prof. Tao,

on page 331, in the third display one sum extremum shouldn’t be instead of ? The same in the fifth display.

[Corrected, thanks -T.]20 August, 2009 at 1:16 pm

Mark LewkoHere are a few possible typos I have collected. In the proof of Lemma 4.25 on page 170, the phrase “find thus find” reads like a typo. On page 292, should the comma in exercise 7.3.4 be an apostrophe? Related to this, did the title of reference [367] change? It doesn’t seem to appear on your website (at least under this name). Lastly, I think the page numbers for reference [41] are off (or its a rather lengthy paper!).

20 August, 2009 at 7:53 pm

Terence TaoThanks for the corrections! The paper [367] is still on the backburner, but we’ll get to it at some point…

26 October, 2009 at 5:59 pm

Mark LewkoCould you give me a(n additional) hint on exercise 4.5.11, specifically on how to “reverse the proof of Lemma 4.30”? It appears that you need to be able to show that can’t be large very often, however it isn’t clear to me how to deduce this from the hypothesis.

Also, assuming that you can prove the estimate for characteristic functions (which is what my question refers to), I think you should be able remove the logarithm that arises from the dyadic pigeonhole argument with the tensor product trick.

26 October, 2009 at 7:51 pm

Terence TaoAck, yes, you’re right. The problem still works if the hypothesis is strengthened to , where is the analogue of for , but this is not nearly as interesting. I don’t see an easy way to salvage the original spirit of the exercise and so it will unfortunately have to be deleted.

31 October, 2009 at 5:57 pm

Mark LewkoThe statement of Lemma 4.46 on page 193 contains an unneeded “such that”.

[Corrected, thanks – T.]12 November, 2009 at 10:26 am

An elementary non-commutative Freiman theorem « What’s new[…] established. However, I wanted to record here an elementary argument (based on Exercise 2.6.5 of my book with Van Vu, which in turn is based on this paper of Izabella Laba) that handles the case when is very close […]

16 December, 2009 at 9:02 am

OlofSmall typo: p. 474: In Theorem 12.5, “constants” should be “constant”.

Thanks.

[Corrected, thanks – T.]9 January, 2010 at 10:11 am

ThomasOn page 333, in the proof of Lemma 9.3, the second to last sentence reads “…P vanishes on contains…” where it should presumably read “…P vanishes on a set which contains…”

[Thanks for the correction – T.]12 January, 2010 at 1:53 pm

Justin W Smithpg 315. Proof of Theorem 8.14. Small typo.

Should say “…namely the points $(b + c, ac)$ with $c \in A$.”

[Corrected, thanks – T.]23 February, 2010 at 11:38 pm

RyosukeDear Prof. Tao,

On page 145, is theorem 3.40 true for any addivitve group? I found there is a additional condition “near torsion-free” in reference [365] in remark 3.41 and I don’t know if this condition can be removed or not. (In [365], you say it can be replaced by bound |P|.) Thanks.

24 February, 2010 at 12:07 am

Terence TaoThe torsion-free hypothesis is only needed to ensure that the proper progression Q is symmetric; otherwise, the arguments still work with torsion (though they become a little bit lengthier).

In Corollary 1.18 of our subsequent paper in

http://arxiv.org/abs/math.CO/0701005

Van and I obtained a sharper result (using some ideas communicated to us by Ben Green). The symmetry problem can be avoided by redefining the concept of a generalised arithmetic progression in a manner that allows one to define non-integer dilates P_t of a progression P.

12 March, 2010 at 10:20 pm

RyosukeDear Prof. Tao,

I don’t know how to apply Chernoff’s inequality on Lemma 4.16.. In fact, I don’t know how to define random variables like Chernoff’s inequality on p 11. Could you give me a hint? thanks.

14 March, 2010 at 12:10 am

Terence TaoOne can write the indicator function as , where is the Kronecker delta at , and the are iid Bernoulli random variables.

15 March, 2010 at 9:26 pm

RyosukeDear Prof. Tao,

On p 168, Lemma 4.22, since inequality (4.28) is proved by discrete John’s lemma, I think its power of d’ should be -3d’/2. Isn’t it?

19 March, 2010 at 2:51 pm

Terence TaoThe argument uses the leftmost inclusions of the discrete John theorem, rather than the rightmost inclusions.

20 April, 2010 at 12:41 am

RyosukeDear Prof. Tao,

On p 169, you use 2nd Minkowski theorm to prove lemma 4.23, but it seems that to get lower bound of |P| we need upper bound of mes(R^d/Gamma) in stead lower bound (1/N in p169).

20 April, 2010 at 9:40 am

Terence TaoOops, good point! One has to fix the proposition by replacing P by P+H, where H is a subgroup of (i.e. a cyclic group generated by some factor of ). The point being that the if the order of is a factor of , rather than being equal to , then one has to quotient out by the subgroup first before the top display of p.169 becomes an equality. I’ve added an erratum to this page to reflect the changes.

26 April, 2010 at 1:35 am

RyosukeDear Prof. Tao,

On p 176, I find that right hand side of (4.34) in lemma 4.33 can be written as exp(- (lambda^2)/(2+epsilon)). Since we can simply multiply “i” to get exp(- (lambda^2)/4) in proof.

I want to consult an other question: In proposition 4.29. Is there any inductive formula of alpha(h, |S|)?

Thanks!

26 April, 2010 at 10:50 am

Terence TaoDear Ryosuke,

I don’t see how to use multiplication by i to improve the exponent to ; using the inequality only gives .

In principle can be written down explicitly, but I doubt that there is a formula for it that does not involve at least one summation sign. In the asymptotic limit it becomes .

28 April, 2010 at 5:25 pm

RyosukeDear Prof. Tao,

In your Errata, p177 and taking lambda=sigma, I get the fourth display should be e^(-lambda^2)/2 rather than e^(-lambda^2)/4. So I get 4 rather than 8.

28 April, 2010 at 7:48 pm

Terence TaoAh, fair enough; I’ve added the improvement as an erratum.

4 May, 2010 at 4:29 am

GyanDear Professor Tao,

The proof of Theorem 4.41 as per the edition ISBN-13 978-0-521-85386-6

is a bit unclear to me.

The lower bound obtained for with is fine. Then on page 188 third line, the Reader is

asked to use Corollary 2.62 to obtain a contradiction. However to use

Corollary 2.62 directly, we will need to have some upper bound for the

cardinality of A. For instance will suffice.

I do not know how to obtain such a bound.

The case when can be however easily handled

by appealing to Exercise 4.6.3 to obtain a lower bound for and then applying Corollary 2.62 to obtain a contradiction.

4 May, 2010 at 8:39 am

Terence TaoI believe (4.37) establishes such an upper bound on |A| using the lower bound hypothesis on |H|. I’ll add a note in the errata to clarify this.

4 May, 2010 at 10:33 pm

GyanUsing the assumed lower bound for and (4.37), we will get that |A| is not more than . Therefore we obtain that |A| is at most Since the present statement of Corollary 2.62 gives result only for those A with |A| at most , it is not clear how to use it

directly.

However I believe that the statement of Corollary 2.62 is true even for those A with |A| at most . This follows from the same proof as given in the book. Once one observes this, the problem mentioned above disappears.

5 May, 2010 at 2:45 am

GyanDear Prof. Tao,

Sorry, I missed the trivial fact that implies that . Therefore Corollary 2.62 is trivially applicable for the sets A with This clears all my earlier confusions.

A small typo in the statement of Theorem 4.41.

Shouldn’t “…. ….” be replaced by ?

[Corrected, thanks. – T.]6 May, 2010 at 5:53 am

RyosukeYes, I also faced this problem, and I find the same way as you did. So happy it is right.

6 May, 2010 at 5:59 am

Thomas BloomA couple of minor errors.

On page 346, “Cauchy’s theorem” should be “Lagrange’s theorem” and on page 414, “motivation” should be “motivations” (first paragraph).

[Added, thanks – T.]18 May, 2010 at 4:41 pm

Weiyu Xua few comments and questins on proving Lemma 7.17 (Esseen Concentration Inequality).

Comments:

1. On Page 291, in the definition of w(\xi), I believe we are missing a “d\xi_{1}”

2. Right before the definition of w(\xi), , should the equality be replaced with an inequality?

one question: In Equation (7.4) on page 290, is \Omega(1) dependent on C?

18 May, 2010 at 9:52 pm

Terence TaoThanks for the corrections! For 2, both equality and inequality would be appropriate here (due to the nature of the Omega(1) notation). The constants here do depend on C, but C will be taken to be bounded (this should have been mentioned at the top of page 291).

20 May, 2010 at 9:45 pm

254B, Notes 5: The inverse conjecture for the Gowers norm I. The finite field case « What’s new[…] lemma and Freiman’s theorem for vector spaces; see Section 11.3 of my book with Van for further discussion. The proof is elementary but a little lengthy and would take us too far […]

1 June, 2010 at 5:37 pm

RyosukeDear Prof. Tao,

On p 279, proposition 7.7, I think [(|X|+i)/2] should be replaced by [|X|/2]+i, then it will be the same result in lemma of [206].

I am not sure if you want to state proposition as I thought.

1 June, 2010 at 8:42 pm

Terence TaoI believe the two sums are the same (both are equal to the sum of the k largest binomial coefficients).

1 June, 2010 at 9:17 pm

RyosukeYse, I know that

{sum C(|X|,[(|X|+i)/2] ) from i=1 to k} = {sum C(|X|, [|X|/2]+i) from i = integers between -k/2 to (k-1)/2}

but in your book, the express is {sum C(|X|, [(|X|+i)/2) from i = integers between -k/2 to (k-1)/2}.

1 June, 2010 at 10:57 pm

Terence TaoAh, I see what you are saying now. Thanks for the correction!

7 June, 2010 at 7:24 am

Thomas BloomIn Exercise 6.2.8, G shouldn’t be ‘triangle-free’, as later in the same sentence it has ‘T triangles’.

Also, in Exercise 1.1.7, do you need the 10 in loglog(10+Z)? I did the exercise without needing it, and isn’t loglog(10+Z)=O(loglogZ) (and vice versa) anyway?

Thanks.

7 June, 2010 at 9:28 am

Terence TaoThanks for the correction! The 10 is there purely to prevent the double logarithm from becoming negative.

8 June, 2010 at 5:46 pm

RyosukeDear Prof. Tao,

Can you give me some hint to drive lest display of proof of lemma 7.14 ?

Because everytime I use pointwise bound F < G^theta(u'/u), I get RHS of lest inequality = E(G)^c instead power theta(u'/u).

thank you.

8 June, 2010 at 7:49 pm

Terence TaoAh, the cutoff in here was not optimised correctly. The first case should instead be enlarged to (which does not affect the contribution of this case), so that the second case can be shrunk down to (at which point the stated argument works). I've added an erratum accordingly.

8 June, 2010 at 8:57 pm

RyosukeYes, you’re right! thank you.

20 June, 2010 at 6:10 am

AnonymousDear Prof. Tao,

Can you explain first display of last paragraph on p 295? I cannot figure out how to apply volume-packing argument on it.( I cannot find additive structure of these sets. )

thank you.

21 June, 2010 at 5:29 pm

Terence TaoThis particular display is just a consequence of the monotonicity properties of stated at the beginning of the paragraph. (It is a component of the volume packing argument, rather than a consequence of it.)

22 June, 2010 at 12:06 am

RyosukeBut it seems strange. RHS is a average of (And I think there miss a ‘ for ) with center at . If RHS should be small than LHS, I still cannot understand how to use monotonicity to get this.

22 June, 2010 at 8:59 am

Terence TaoWhen m=0 one can use the fact that this expression is comparable to 1/k for , as mentioned in the preceding paragraph.

22 June, 2010 at 5:50 pm

RyosukeSo let , I know we have: 1. decreasing for . 2. is even. 3. for all and for all . So you mean we can use 1~3 to prove:

?

22 June, 2010 at 6:33 pm

Terence TaoYes, this is correct (together with the fact that f is nonnegative). For m = O(k) one uses 3; for |m| >> k one uses 1 and 2.

23 June, 2010 at 12:19 am

RyosukeOh, I get it! thanks!

20 June, 2010 at 10:33 pm

RyosukeDear Prof. Tao,

On p 299, I cannot derive conclusion form first display. Can you explain how to computate it? Thanks.

21 June, 2010 at 5:35 pm

Terence TaoThe dominant contribution is when is close to . In that regime, the final factor in the first display is already much smaller than (it’s basically . The first two factors are , so together they smaller than the desired bound (with a bit of room to spare).

21 June, 2010 at 5:02 pm

TomFor exercise 1 in chapter 1…what am I missing: Want to show that . So . Or:

Thus: .

Is that correct? Can we use Markov’s Inequality?

21 June, 2010 at 5:39 pm

Terence TaoI’m sorry, your equations are not making sense. (For instance, you are using both as a free variable and as a dummy variable of integration.) Also, you seem to be trying to derive consequences of the desired identity, rather than trying to prove it. I would recommend trying to explicitly compute a concrete example, e.g. let X be a random variable that equals 5 with probability 0.3 and 0 with probability 0.7, as this may help clear up some confusion.

21 June, 2010 at 6:11 pm

TomDear Terry,

I was trying to do the following: (assuming is non-negative).

23 June, 2010 at 8:45 am

Terence TaoTry expressing in terms of the probability density function . (Strictly speaking, this only works for continuous probability distributions, unless one uses measure theory.)

24 June, 2010 at 9:06 pm

Tom.

Is this correct?

25 June, 2010 at 5:27 am

GPNow try Fubini !!!

25 June, 2010 at 6:44 pm

The uncertainty principle « What’s new[…] under addition a “reasonable fraction of the time”; for a precise definition, see my book with Van Vu). However, there are dyadic models of Euclidean domains, such as the field of formal Laurent […]

29 June, 2010 at 3:17 am

Thomas BloomSome more typos:

p. 131, the second \Gamma in the second paragraph of the proof should be \Gamma’

p. 134, first line, ‘with’ should be ‘within’

p. 135, in the definition of \lambda_j, k should be j.

p. 196, the second line after the proof finishes is missing a closing bracket after \delta

p. 388, last line, \sigma N/4 should be \sigma/4

p. 391, in the statement of Lemma 10.29, [1,n] should be [1,N]

p. 397, in Exercise 10.4.2, Lemma 10.15 should be Lemma 10.29

[Corrected, thanks – T.]30 June, 2010 at 1:50 pm

Yiannishi,

i have a simple question — is there a corresponding development of sumset and inverse sumset theorems (or at least partial results) for continuous subsets of the real line or, more generally, for infinite subsets of arbitrary (abelian, locally compact) additive groups, with respect to Haar measure instead of set cardinality? it seems that some of the basic sumset bounds generalize very easily (at least to the case of the real line) but some other results don’t. (in fact, even some of the simplest bounds fail in this case.)

i understand this has probably been discussed at length somewhere, so i apologize in advance for my ignorance!

thanks!

yiannis

1 July, 2010 at 8:41 am

Terence TaoDear Yiannis,

I discuss this in my paper http://front.math.ucdavis.edu/math.CO/0601431 , particularly the last section.

1 July, 2010 at 10:00 am

Yiannishi — got it, thanks!

yiannis

9 November, 2010 at 7:27 pm

FernandoDear Prof Tao,

In the proof of Theorem 1.39 about B_h[g] set, it says in the end that one can establish E(partial derivative of Y_m)=O(m^{-1/h}). But let’s say h=2, and so

Y_m=t_1*t_{m-1}+t_2*t_{m-2}+…

So the partial derivative of Y_m with respect to t_{m-1} is just t_1, but E(t_1) is a constant, and is not bounded by m^{-1/h} for large m. Is there something wrong with this argument or did I misunderstand the statement?

Thanks,

Fernando

17 November, 2010 at 10:33 am

Terence TaoYes, this is a moderately serious issue. It can be fixed by a “splitting” argument, in which one first only considers representations in which all terms have size at least (say). The remaining terms can be handled by an induction hypothesis, noting from the Borel-Cantelli lemma that we can assume that there is a uniform bound on the number of elements of A in any interval of the form if is small enough. I’ll add some details of the fix as an erratum.

10 November, 2010 at 3:42 am

Olof SisaskHiya, some small typos:

p. 388: There should not be absolute value signs in the conclusion (density increment vs density decrement), and the bound should probably be delta/8 rather than delta/4 from the proof given: in getting the last displayed inequality on the page one presumably divides by 2N for the pigeonholing and then by 2 for the Re(1+e(…)) term. This change propagates to changing the 4 to an 8 in the first display on p. 389.

[Corrected, thanks – T.]Thanks.

18 November, 2010 at 4:26 am

Olof SisaskHiya, I think there might be an issue with the bounds in Theorem 4.47 on p. 194: perhaps the length of the arithmetic progression ought to be (with a rather than ) and the lower bound on ought perhaps to be the square root of what it currently is. It seems that this is needed in getting the bound in the first display on p. 196, given that |S| is of the order . If correct, this change needs to propagate to the third and penultimate lines of the proof of Theorem 4.47, the discussion afterwards, and to the second line of p. 197.

[Added, thanks. I checked the original reference of Bourgain, and of course he writes the exponents correctly; I can’t remember how the error slipped in initially. – T.]Thanks.

1 December, 2010 at 2:11 am

HI think there’s still some problem with Exercise 1.3.1. Take for instance $\lambda=n-1$ for some even $n$. Since $n$ is even, the only way the sum can be $>=\lambda$ is if the sum equals $n$.

1 December, 2010 at 10:42 am

Terence TaoI think it’s fine in this case; both sides are equal to . (The left-hand event occurs if and only the first signs are equal to .)

7 December, 2010 at 10:15 am

HeikoHello,

i was wondering if the case k=1=m should be ruled out, because we could choose B=N and that would be a perfectly regular basis of order 1 (although a rather boring one).

Corrected, thanks – T.]7 December, 2010 at 10:18 am

HeikoIn Exercise 1.3.10, i meant. Sorry

6 January, 2011 at 10:53 am

Máté WierdlTerry, on page 17, Exercise 1.3.4, in the estimate the negative sign is missing from the second exponential in front of lambda.

[Corrected, thanks – T.]1 February, 2011 at 11:53 am

AnonymousEquation (2.7): I believe min(|A||B|) should read min(|A|,|B|) (this is in the softcover edition… I couldn’t find an errata page for that edition)

[Corrected, thanks – T.]7 February, 2011 at 9:35 am

Polymath6: thoughts on empirically testing the finite-field heuristic for very low ‘n’ « Paul Delhanty[…] context is relatively undemanding. For example, consider the concept of a Bohr set. (See also Tao & Vu’s Additive Combinatorics – 4.4 Bohr Sets.) In the context of the cap-set problem, Bohr sets are just subspaces, so […]

13 February, 2011 at 8:10 am

Erik AasOn p.271, not only should G_1 o … o G_k be changed to G_k o … o G_1, but also

||G_1 o … o G_k|| _ {1/k} to ||G_k o … o G_1|| ^{1/k},

I think.

[Corrected, thanks -T.]13 February, 2011 at 8:24 am

Thomas Bloomp. 391: in the statement of Lemma 10.29, the $\Omega(\sigma N^{1/(\lvert S\rvert+1})$ should be $\Omega(N^{1/\lvert S\rvert+1})$, and the final bound of $\Omega(\sigma)$ should be $\Omega(\sigma\delta^{-1})$, where $\delta=\mathbb{E}f$, otherwise one can’t get the Heath-Brown/Szemeredi bound. Also in the proof, the L^1 norm of f should be bounded by $2\delta$ rather than $1$ to enable this improved statement.

[Corrected, thanks -T.]Also, on pages 390, 391, 388, 133, should the references to the Kronecker approximation theorem be instead to a multidimensional Dirichlet approximation theorem, this being the name usually used for this kind of result? (Kronecker applying to linearly independent sets).

[Hmm, tough call. There does not appear to be prior literature referring to a multidimensional Dirichlet approximation theorem. You are right that the Kronecker approximation theorem, as stated in the literature, is slightly different (although it does imply the qualitative version, at least, of the result used here). On balance, though, I guess the former term is more appropriate. – T.]17 June, 2011 at 11:13 pm

Christian Elsholtz>>There does not appear to be prior literature referring to a multidimensional Dirichlet approximation theorem. <<

"Simultaneous approximation" (rather than multidimensional approximation) is a standard term, so that I suggest "simultaneous Dirichlet approximation".

[Adjusted, thanks – T.]9 June, 2011 at 11:38 pm

Smooth geometry vs (nonsmooth calculus and combinatorics) | chorasimilarity[…] Additive combinatorics, see this book by Terence Tao, Van Vu. […]

15 June, 2011 at 5:44 am

DavidJCovert@gmail.comOne small typo: at the bottom of page 316 (in the first edition), $C|A+A|^{8/3}/|A|^{18/3}$ should be $C|A+A|^{8/3}/|A|^{8/3}$.

[Added, thanks – T.]19 July, 2011 at 1:00 am

tomerat p.162, the last line should be <= |F| (rather than = |F|)

[Sorry, I was unable to locate this issue in either the paperback or hardback versions. – T.]26 July, 2011 at 4:39 am

Anonymousat Lemma 4.14 (Gauss sum estimate), the last line of the proof

it should be <= |F| (rather than = |F|)

26 July, 2011 at 4:47 am

Anonymoussorry for the trouble – my mistake

28 July, 2011 at 11:19 pm

tomer shalevPage 191, the third line in the computation segment (of |f(x) – f(0)|), the L-2 norms due to cauchy-schwarz should be l-2 norm (with small L) ?

[Corrected, thanks – T.]30 July, 2011 at 7:03 am

tomer shalevanother error: page 63, the sixth line in the computation ( of E(A,B) ),

a’ – b should be omitted and instead should be y variable.

[Corrected, thanks – T.]8 August, 2011 at 11:36 am

David CovertVery minor typo in revised addition (Additive_sample.pdf), page 87, Lemma 2.13: One instance of $X_{+}$ should be $X_{-}$.

[Thanks! Actually this was already corrected in the published version – T.]25 August, 2011 at 9:15 am

The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 « What’s new[…] by a standard Fourier-analytic computation involving Riesz products (see Chapter 7 of my book with Van Vu, or this recent preprint of Maples). Using this Fourier representation, the fact that this […]

27 August, 2011 at 11:35 am

254A, Notes 0 – Hilbert’s fifth problem and related topics « What’s new[…] of additive or multiplicative structure are “essentially” equivalent; see Chapter 2 of my book with Van Vu (or this paper of mine in the non-commutative case) for more […]

12 November, 2011 at 7:18 am

Tomer Shalevpage 16, theorem 1.16. last line of the page regarding the inequality of the expectation of the representing function.. should 8(logn) be 32(logn)?

otherwise I can’t get the O(1/n^2) that should follow from corollary 1.9.

another point: it is said in the theorem that corollary 1.10 is applicable as well, but then you have the set P of primes which is infinte and a finite random set B then one can’t use corollary 1.10 due to the infiniteness of P.

I will be happy to figure out where i did not follow or understand.

page 181, lemma 4.35 (cube covering lemma): it should be noted in the theorem that the union of D_1,…., D_k may be empty as well.

the reading of the theorem made me believe at first that these sets exist with the fixed (d + 1) cardinality, which is of course not the case when one reads the proof.

[These errata have been added, thanks – T.]1 December, 2011 at 11:35 pm

Tomer Shalevpage 81, Lemma 2.30 . I believe that the definition of S should be

with a power of 2 in the left hand side

i.e S:= {x\in A+B… |A \cap (x- B)|^2 >= …}

otherwise I can’t follow the deduction of the inequality computation in the third line. If so true, then all of the proof should be adjusted and rewritten from there on, possibly changing the theorem slightly.

4 December, 2011 at 5:06 pm

Terence TaoActually, I think the definition of S is correct as it stands. The point is that one should only estimate one of the two factors of , thus

.

4 December, 2011 at 10:19 pm

Tomer Shalevofcourse, thanks for making this point clearer

6 December, 2011 at 4:26 am

Win-Vector Blog » What to do when you run out of memory[…] Historically computer scientists have concentrated on streaming or online algorithms (that is algorithms that work with the data in the order it is available and use limited memory). For many problems we have found this an insufficient model and it is much better to assume you can re-order and replicate data (such as scattering data to many processors and re-collecting it to sort). The scatter/gather paradigm is ubiquitous and is the underpinning of large scale sorting, databases and Map Reduce. So in one sense databases and Map Reduce different APIs on top of very related technologies (journaling, splitting and merging). Replicating data (or even delaying duplicate elimination) that is already “too large to handle” may seem counterintuitive; but it is exploiting the primary property of secondary storage: that secondary storage tends to be much larger than primary storage (typically by 2 orders of magnitude, compare a 2 terabyte drive to an 8 gigabyte memory stick).In our web age, the typical big data problems are inverting indices (for fast search lookup) and computing term frequencies (for TF/IDF scoring or for things like Naive Bayes classifiers). Since these are over-worked examples we will use a mathematical problem from “Additive Combinatorics”, Terence Tao, Van Vu, (ISBN-13: 9780521853866; ISBN-10: 0521853… […]

22 December, 2011 at 9:21 am

Xi WuHi,

For Q 4.1.7, I think \E_{H^{\perp}} is correct, you need this to give a factor |H^{\perp}| and then apply (4.6), i.e. |Z| = |H||H^{\perp}|.

For Q 4.1.8. in the last line of the furthermore part, I think ” f((\phi^{\dag})^{-1}(x)) ” should be ” \hat(f)((\phi^{\dag})^{-1}(x)) ”

[Added, thanks – T.]12 March, 2012 at 11:33 pm

Tomer ShalevHi Prof. Tao.

can you please elaborate more on the solution

of exercise 2.4.10 in page 78. the hint in the exercise unfortunately

didn’t make any sense to me

12 March, 2012 at 11:35 pm

Tomer Shalevit’s just that approximate groups work well for doubled sets.

13 March, 2012 at 11:50 am

Terence TaoDear Tomer,

After placing A in a translate x+H of an approximate group, and B in a translate y+K of another approximate group, it is easy to see that H+K is also an approximate group. The main difficulty is to show that A+B is a large subset of x+H+y+K, which is where the hint comes in.

13 March, 2012 at 11:18 pm

Tomer ShalevHi Prof Tao.

this was my strategy before I wrote here. but I am

still missing something.

This is what it looks like:

essentialy H = A-A, K = B-B . H + K = A – A + B -B

A+B is not a large subset of H+K at first impression, unless H+K has

a small doubling.

i can get the following:

|A+B| >= |H+K| / ( (K_1)^5 * (K_2)^5 * |A-A()B-B| )

My problem is ofcourse showing that |A-A()B-B| is comparable to

the doubling constant of A and B.

BTW, all of my playing around with the representing function, Approximate groups and the doubling constant makes me believe that the

doubling constant is not the most natural measure. No one ever justified

the doubling constant in terms of covering A-A with O(K) translates of A,

where K is the doubling constant. it actually happens when A is a sidon set or arithmatic progression of dimension 1. while this seems nowhere justifiable, strange properties seem to happen like sub multiplicitivity under sum set operation.

It is easy to see that another measuring can strengthen the doubling measure. for example: suppose the doubling constant was the cardinality

of the maximal sub collection of {-A + a : a \in A} obeying the property

{-A + a()-A + b} = {0}, in special cases like sidon and arithmatic progression these measure is equivilent to the doublng constant.

14 March, 2012 at 7:45 am

Terence TaoHere is a sketch of proof. To simplify the notation let us suppress the dependence on K_1, K_2. We place A in a coset x+H of an approximate group, and B in a coset y+K of another approximate group, with and . Since H and K are approximate groups, H+K is also. (indeed, if 2H can be covered by a bounded number of translates of H, and 2K can be covered by a bounded number of translates of K, then 2(H+K)=2H+2K can be covered by a bounded number of translates of H+K). So

To finish the job we need to show that

From the inequality in the hint one has

.

On the other hand, from the Ruzsa triangle inequality one has

and the claim follows.

I agree with you that the doubling constant is not the most tractable quantity to measure additive structure; its presence in the subject is primarily for historical reasons. Fortunately, the basic additive combinatorics theory allows one to efficiently convert control of doubling constants to more useful control, such as control of the approximate group parameter.

14 March, 2012 at 9:48 am

Tomer ShalevThanks prof. Tao

I believe that the last computation should be:

since and

where

then by Ruzsa:

My problem was justifying :

14 March, 2012 at 9:52 am

Tomer Shalevlatex cut the first line after “ruzsa”. if you hit the reply button you will see the rest. I now see that you corrected the first sketch with which I had problems at the end.

thank you

14 March, 2012 at 2:56 am

Vorlesung Kombinatorische Zahlentheorie » notebook[…] – Terence Tao and Van Vu, Additive Combinatorics, Cambridge 2006. University Press; see also https://terrytao.wordpress.com/books/additive-combinatorics/ – Alfred Geroldinger and Imre Ruzsa, Combinatorial number theory and additive group theory, […]

22 May, 2012 at 3:08 am

AnonymousHi Prof. Tao,

Reporting a typo in the old edition of the book, in p. 116 in the proof of Corollary 3.6. (1) In line no. 9 of the proof, $w \in \mathbf Z^d$ should be $\mathbf Z^n$. (2) The symbol “n” in two different ways, leading to some curious phrases like $n \in \mathbf Z^n$; e.g., see line no. 3 of the proof. In case it helps, I checked the previous pages, and found that you had avoided using “n” for the dimension prior to this paragraph.

Also, I would like a clarification reg. the proof of Corollary 3.9. In the very first line, you define the torsion subgroup of G, and invoke Corollary 3.8 for this subgroup to conclude that it is a direct sum of finite cyclic groups. But then, should one not first prove that the torsion subgroup is finite? This would of course follow if one shows that (a) any subgroup of a f.g. abelian group is f.g., and (b) if a f.g. abelian group is also a torsion group, then it is finite. I browsed through the earlier theorems to see whether either of these statements have been proved/mentioned, but I did not find anything relevant. Am I missing something here? Thank you.

26 May, 2012 at 4:15 pm

Terence TaoThanks for the corrections!

Regarding the facts that (a) any subgroup of a f.g. abelian group is f.g., and (b) any f.g. torsion abelian group is finite, claim (a) follows for lattices from Lemma 3.4, and then for arbitrary f.g. groups by viewing such groups as homomorphic images of lattices. For (b), observe that each generator of a f.g. torsion group has finite order, and so there are only finitely any distinct additive combinations of such generators.

27 May, 2012 at 9:29 pm

SrivatsanHi, I was the one who posted the previous post. Thank you for your clarification.

I fished out a couple more typos, again in the old edition. (1.) p. 121: In the paragraph following Lemma 3.11, the notation in “exponentially in ” is not defined. It should probably be . (2.) p. 141: The proof of Lemma 3.36 (Discrete John’s theorem) uses for the dimension of the progression, which the statement of the theorem denotes by . This is of course not really a bug, but there does not seem to be any need for the switch either.

Note: The blog post says that it was last updated on May 26, but the errata is not updated. Is this some issue with wordpress?

[Corrections added, thanks – T.]21 August, 2012 at 11:24 am

self studyrevised edition, page 18, exercise 1.3.6: in equation 1.24, can:

$\leq e^{-\lambda^2/4}$ actually be tightened to $\leq e^{-\lambda^2/2}$ via a straight forward application of http://en.wikipedia.org/wiki/Azuma_inequality with $c_i = 2$ ??

27 August, 2012 at 2:25 pm

self studyPage 23, Equation 1.30

Is the “v” in $$\prod_{v \in V} )1-P(A_i))$$ supposed to be an “i”, i.e. $$\prod_{i in V}) (1- P(A_i))$$ ?

Thanks!

[Yes, this has been added to the errata now, thanks – T.]10 September, 2012 at 3:20 pm

adPage 42, Exercise 1.8.1:

Should $$A \in Z^+$$ be $$A \subseteq Z^+$$ ?

10 September, 2012 at 9:37 pm

Terence TaoThanks for the correction! Actually, there is a more serious issue with Ex 1.8.1, which is that the two elements of B should be distinct (i.e. 2B should be ).

10 September, 2012 at 3:21 pm

ad(If the above is true, can you also provide a hint for this problem? :-) )

10 September, 2012 at 3:34 pm

adErr, sorry about spamming your blog, no hint required. Just solved 1.8.1

11 September, 2012 at 12:04 pm

adSo it turns out my proof from yesterday doesn’t quite work.

(1)

The first thing I tried is P(x \in B) = \frac{\log n}{n}, but this breaks down when we have:

A = (N + [1, …, k]) \cup (2N + [1, …, k]) \cup (3N + [1, .., 2k]).

(2)

From that, the intuition is “okay, so maybe I need to somehow bias things toward the larger elements.” Divide the numbers into segments of [1,2), [2, 4), [4,8), … . Then look at the largest non empty segment. If there are $\log n$ elements, we’re done (since the sum of two elements within a segment can’t be in the segment.)

So, if the # of the lements in the segments are s_0, s_1, …, s_m [where s_0 is the largest non-empty segment], we end up getting some equation of the form:

force \sum_i s_{i-1}/s_i * (k_i/s_i)^2 < 1

when \sum_i k_i = \log n

Unfortunately, the case where all the s_i = 1 breaks the above (since the first sum ends up being \log n rather than 1).

I was wondering if you could provide some intuition on how to pick b.

Thanks!

11 September, 2012 at 7:20 pm

Terence TaoHmm, now that I look more closely, it seems that this exercise is misplaced; it is basically a duplicate of Theorem 6.2. I am not sure why it ended up in Section 1.8; it is not particularly easy to prove using the methods from that section…

11 September, 2012 at 11:31 pm

adI was going to stop reading the book after chapter4, but now it looks like I’ll have to get through section 6.2 at the least. :-)

9 October, 2012 at 7:35 am

ablmffIn 2.3.23, we are asked to give an example with certain properties, but do we need to give an example which works for all possible N, or can we choose a particular N?

9 October, 2012 at 6:53 pm

Terence TaoOne should be able to construct an example for all sufficiently large N (of course, the claim is trivial for very small N, e.g. N=1).

14 October, 2012 at 12:58 pm

ablmffI think there might be a bit ambiguity in exercise 2.5.7. It says:

G \subset A \times A is such that every element of A is connected via G to at least K^{-1}|A| elements of |A|.

Here, does the A in “every element of A” refer to the first A in “A \times A”, or to both of them?

[To both; I’ll add an erratum to clarify this. -T]5 November, 2012 at 6:06 am

ablmffI think in Ex 4.3.2, \|T^h A\| should be \|h A\|.

5 November, 2012 at 6:07 am

ablmffSorry, should be \|h + A\|

[Erratum added, thanks – T.]14 November, 2012 at 4:45 pm

Yogesh Anbalaganp. 315 In the proof of theorem 8.14 … the points (b+c, ac) with C ___ … C should be element of A ? Is that a typo?

[Yes; see the errata list for the revised edition. -T.]18 December, 2012 at 2:59 am

Rafael Tesoro CarreteroI found the erratas list very useful, thanks for this!

It seems to me another minor one may be the following. In the middle of page 264 there should be E instead of G, and the first pair should be (a,b) instead of (a’,b’):

|\{ (a,b) \in A \times B : (a,b’),(a’,b’),(a’,b) \in E \}| \ge \frac{|A||B|}{2^{12}K^5}.

[Corrected, thanks – T.]1 February, 2013 at 10:19 am

Small doubling in groups « What’s new[…] much detail the proof techniques used in these results (although the abelian case is discussed in this book of mine with Vu, and the nonabelian case discussed in this more recent book of mine), but instead focuses on the […]

16 March, 2013 at 5:28 pm

Jarek KubenIn the proof of Theorem 9.2, page 331, 15th line, should be t_{n-1}^{d_{n-1}} instead of t_{n-1}^{d_{n}-1}.

[Added, thanks – T.]7 June, 2013 at 2:49 am

Tomer Shalevpage 163: after “(for instance)” if we set A = Z … ||z||_u should be zero, but instead, what is written is ||B||_u = tao*Z + O(…) , (tao*Z should be canceled I believe).

[Corrected, thanks – T.]17 July, 2013 at 6:32 am

RexOn page 4: “Theorem 1.3 was proved by Erdos in 1965 [86]. Several years later, Bourgain [37] used harmonic analysis arguments to improve the bound slightly.”

It seems that a better description might be: “Three decades later, Bourgain used….”

18 July, 2013 at 9:51 am

RexJust so you know, the .dvi file on entropy sumset methods has some formulas that run completely off the page (so they are not visible)

19 July, 2013 at 2:10 am

Rexp. 14: In the last display, the first sign should be . (just as with pg. 15)

19 July, 2013 at 2:12 am

RexOh, nevermind. My mistake

4 November, 2013 at 1:48 pm

Real stable polynomials and the Kadison-Singer problem | What's new[…] as the second moment method, exponential moment method, and zeroth moment method, see Chapter 1 of my book with Van Vu. For a general discussion of the probabilistic method, see the book by Alon and Spencer of the same […]

4 February, 2014 at 7:01 pm

Szemerédi’s Theorem Part I – Equivalent formulations | I Can't Believe It's Not Random![…] Szemerédi used a complicated combinatorial argument. Two years later Furstenberg provided a different proof of this result, using ergodic theory and a correspondence principle which allows one to derive conclusions about sets of integers by studying dynamics on certain measure spaces. It should also be mentioned that a third proof of the Szemerédi’s Theorem was discovered by Gowers in 2001, generalizing Roth’s harmonic analysis proof of the existence of progressions of length in a set of positive density, to progressions of arbitrary length and having the advantage of providing concrete bounds for the finitistic version. More recently a number of different proofs appeared. A good discussion on the different proofs and similarities between them can be found on chapters 10 and 11 of the Tao and Vu’s book. […]

19 March, 2014 at 5:30 pm

Metric entropy analogues of sum set theory | What's new[…] estimates (which are covered, incidentally, in Section 2 of my book with Van Vu) are particularly useful for controlling finite sets of small doubling, in the sense that for […]

23 March, 2014 at 6:55 am

Anonymouscan any one please explain me the concept of “uniformly chosen random subset of a unit interval of cardinality N”.??

[One can for instance choose a random set where each of the are chosen independently and uniformly from (one can enforce the constraint if one wishes, but it is not necessary as one ends up with the same distribution of a random variable in either case. -T.]27 April, 2014 at 12:45 pm

GeorgeOn page 128 Proposition 3.19, one should have

(as opposed to the containment going the other way). Also, just above this, it may be more clear to write let be arbitary as opposed to let be arbitrary, to emphasize that one may take , but must be avoided to assure that the sets we wish to apply Lemma 3.18 to are nonempty.

[Corrected, thanks – T.]14 August, 2014 at 9:04 am

Marco VitturiDear Prof. Tao,

in the final lines of the proof of Freiman’s 3k-3 Theorem (page 208) it appears that the inequality is used reversed, and I can’t see how the relative errata above can fix this. Am I wrong?

[Gah, the previous erratum was not applied correctly. I’ve now supplied a revised erratum in the second errata list. -T.]27 October, 2014 at 7:28 pm

LamExample 1.12 p.12, “the squares are known to be a basis of order 4 (Legendre’s Theorem)” shouldn’t it be “Lagrange’s Theorem”? (4-square theorem)

[Correction added, thanks – T.]12 November, 2014 at 12:41 pm

LamDear Terry,

On page 294, after the very first proof (of 7.20), it says “however one can lower dimension its by increasing”, probably instead of “lower its dimension by increasing”.

[Correction added, thanks – T.]18 November, 2014 at 2:09 pm

LamDear Terry,

on page 281, last paragraph it says “we can reduce further to the case that $Z$ is odd”, which should probably read “$|Z|$ is odd” or “the order of $Z$ is odd”?

PS: I do not know if it is actually useful to report such minor typos?

[Correction added, thanks -T.]19 November, 2014 at 1:50 pm

Lamp. 293: $v_1,\hdots,v_n$ should probably be elements of $P$ instead of $V$ (volume).

p. 285: Corollary 5.25 should be Lemma 5.25.

[Correction added, thanks – T.]19 November, 2014 at 1:52 pm

LamExample 7.19 p. 293: takes values in the generalized arithmetic progression $nP$ which have volume $n^dV$ (“which has”)

[Correction added, thanks – T.]19 November, 2014 at 2:33 pm

LamIn the proof of Lemma 4.35 p.181, it says “we are left with a remainder $R$ where all dissociated subsets of $S$ have cardinality $d$ or less”. Should $S$ be replaced with $R$?

[Correction added, thanks – T.]10 December, 2014 at 5:58 am

Freddie MannersDear Prof Tao,

On p241, proof of Lemma 5.45, last line of the first displayed equation: I think

should read

as an estimate on the binomial coefficient.

[Correction added, thanks – T.]10 December, 2014 at 6:25 am

Freddie MannersDear Prof Tao,

On p239, exercise 5.5.17: I don’t think this example works. The problem is that the midpoints of the quadrilateral , , , themselves form a parallelogram, and this rules out any non-standard Freiman homomorphisms to .

Replacing the generic quadrilateral with a generic pentagon in should fix it, I think, giving a universal ambient group of .

(Also, I think there is a typo

on the penultimate line: the first “2” should be a “4”.)

[Correction added, thanks – T.]25 February, 2015 at 5:19 am

QuoraAre there any fields of mathematics that incorporate both Combinatorics and Number Theory?Yes. The definition of mathematical fields is very fuzzy, but there are many questions which seem to belong to both “Number Theory” and “Combinatorics”. For example: in how many ways can a positive integer be represented as a sum of four integer sq…

16 June, 2015 at 1:11 pm

tomer shalevHello Professor Tao.

I am getting back to my thesis,

and was wondering about the solution of

exercise 2.1.1.

– how is it possible to come up with a uniform probability

function over the an infinite set of real numbers??

– what am I missing?

all the best

Tomer Shalev

[The exercise is referring to Lebesgue measure on the unit interval [0,1], which is a probability measure. -T.]17 June, 2015 at 6:15 am

Tomer shalevCan you please lay a sketch?

17 June, 2015 at 6:25 am

Tomer shalevThe probability to choose a number is zero according to the Lebesgue measure. And the number of representation of a number is maximized when the two sets differences only share zero element. Since differences have mostly non zero measure, but still u have non countable collection of pair wise numbers that have the same difference x = a-b.

[Uncountable sets can still have zero measure. For instance, the set has zero two-dimensional Lebesgue measure for any given number . -T.]3 July, 2015 at 2:38 pm

tomer shalevHi Professor Tao. Regarding your comment at page 161, before the proof of lemma 4.13:

– you wrote that that P(A_1)***P(A_n) is the quantity one would

expect if random selection is involved conditioning on x =a_1 + .. + a_n,

restricted to these random sets.

– essentially meaning, that the representation function at a fixed point

is expected to be P(A_1)***P(A_n)*(|Z|^(n-1)) restricted to these sets.

the first part of P(A_1)***P(A_n) is understood, but the last part (|Z|^(n-1))

would describe the representation function at a fixes point where solutions

are unconditioned. I find it hard to believe and follow this claim.

please clarify, it is important for me to understand.

T

4 July, 2015 at 7:56 am

Terence TaoFor any fixed , the number of solutions to with is , because can be arbitrary elements of , and then is forced to be .

19 July, 2015 at 8:18 am

tomer shalevHi Prof Tao.

I have a question:

– page 163, exercise 4.3.4 : I understand how to prove it if the group is cyclic.

– does the fact, that the group has bilinear form that indues a isomorpism means, that this is always the case ( because the circle group is cyclic)?

all the best

Tomer

20 July, 2015 at 8:22 pm

A nonstandard analysis proof of Szemeredi’s theorem | What's new[…] is basically thanks to an averaging argument of Varnavides; see for instance Chapter 11 of my book with Van Vu or this previous blog post for a discussion. We have removed the cases as they are trivial and […]

26 August, 2015 at 2:44 pm

A timeline of the polynomial method up-to combinatorial nullstellensatz | Anurag's Math Blog[…] Terence Tao and Van Vu, Chapter 9 in Additive Combinatorics. […]

31 August, 2015 at 2:27 pm

LamDear Terry,

I think there is a minor correction to add p.249 in the proof of Theorem 6.2. It says that “there are only elements of larger than ” but with the ordering, I think it should be . Then for all , and the following sum. Of course the result is the same though.

[Correction added, thanks – T.]22 January, 2016 at 12:28 pm

Incidences Outside of Discrete Geometry – Some Plane Truths[…] led to a variety of results in additive combinatorics that rely on incidences. Tao and Vu’s additive combinatorics book contains a full chapter that is dedicated to incidences. Konyagin and Shkredov’s recent […]

31 March, 2016 at 9:38 am

18.099 Transcript: Chang’s Theorem | Power Overwhelming[…] of the 18.099 discrete analysis reading group at MIT, I presented section 4.7 of Tao-Vu’s Additive Combinatorics textbook. Here were the notes I used for the first part of my […]

4 April, 2016 at 9:42 am

18.099 Transcript: Bourgain’s Theorem | Power Overwhelming[…] of the 18.099 Discrete Analysis reading group at MIT, I presented section 4.7 of Tao-Vu’s Additive Combinatorics textbook. Here were the notes I used for the second half of my […]

12 August, 2016 at 11:30 pm

TomThis is probably my fault, but I am confused by exercise 1.10.3 of pag 48. It is asked to run carefully the central binomial reasoning to get $\sum_{p<N}log(p) \leq Nlog4+O(\sqrt(N))$. But I get something a lot smaller than square root: if K is even then I take care of primes between K and K/2 with the central binomial getting for them a factor Klog2, if K is odd I take care of the primes between K and K/2+1/2 with the biggest binomial getting a factor Klog2. So starting from N the worst thing that can happen is that I have to be always in the second case so that I get the upper bound Nlog2+(Nlog2/2+log2/2)+(Nlog2/4+log2/4+log2/2)+(..) < Nlog4+O(log N) since the other terms are all less then log2 and the number of them is less then $log_2(N)+1$. Where is the mistake? If there isn't why did you put such a bigger function as sqr(N)? Sorry and thanks in advance

[Ah, you’re right, the error does not need to be so large for the upper bound, only for the lower bound. An erratum has been added – T.]4 September, 2016 at 7:16 am

Fat BirdDear Professor Tao,

Hello. I’m reading your book and find chapter 4 an extremely fascinating introduction to Fourier analytic methods. Yet there are two points on which I wonder if you could kindly elaborate a little.

The first is exercise 4.2.3 (p. 159), about “dyadic decomposition” just below the display. I suppose, following the previous hints, one should split the function into a sum, with each summand ranging within a dilation of a dyadic interval. In this regard, the strategy is unable to cover all scenarios, for example when, arranging values in ascending order, we have the subsequent one is always more than twice (or N times) the preceding. In this case, each dyadic (or N-adic) interval can only contain one value, and the logarithm estimate degrades.

The second has to do with the “Fourier concentration lemma”, in p. 182. In the proof, the sentence before the second display confuses me. To be precise, if one directly uses lemma 4.35, then he merely can find a union of dissociated sets plus a cube, instead of the promised cube alone. If one incorporated the dissociated sets into the cube, then I truly wonder how he could deal with the dimension. In addition, I have taken a glance at [48], but only found the proof of the latter part.

Sincerely I look forward to your reply.

4 September, 2016 at 11:00 am

Terence TaoTo avoid the problem of too many dyadic scales in Exercise 4.2.3, one should first remove from the portion coming from points where is very small, e.g. less than , as the contribution of this case can be treated by crude estimates (e.g. the triangle inequality).

For Lemma 4.36, one applies Lemma 4.35 with d taken to be slightly larger than the upper bound in the second display, then no dissociated sets will occur.

4 September, 2016 at 8:04 pm

Fat BirdI get it. Thanks for your attention!

13 December, 2016 at 7:25 am

AnonymousWhen is the revised addition forthcoming?

20 December, 2016 at 7:12 am

jura05Dear Professor Tao!

On p.163, in Lemma 4.6:

In max(…) there should be instead of . We should apply Chernoff’s bound not for , but for .

Otherwise the bound is too weak for , .

[Correction added, thanks – T.]20 December, 2016 at 1:40 pm

jura05But I think that one should keep — with bound becomes OK.

[Corrected, thanks – T.]8 April, 2017 at 7:55 pm

chaominglinDear Prof. Tao,

Here are a possible typo. On page 318, Ex: 3.3.4, is that |A|^{1/2} |B|^{3/2} |C|^{1/2} rather than |A|^{1/2} |B|^{1/2} |C|^{3/2}.

[Corrected, thanks – T.]9 April, 2017 at 9:09 pm

chaominglinDear Prof. Tao,

Sorry, I made a typo too, that is Ex: 8.3.4, not Ex: 3.3.4.

Here are another possible typo. On page 318, Ex: 8.3.5, is that \Omega(|A|^{5/2}) instead of \Theta(|A|^{5/2}). I’ve checked the reference [79] and found out that the result is not that strong.

[Corrected, thanks – T.]12 April, 2017 at 5:29 am

Sai Teja SomuPage 166: I think the condition $\rho\leq 1$ should be added to Lemma 4.20 as (4.25) is false for $\rho >1$.

[Correction added, thanks – T.]25 May, 2017 at 6:21 am

jura05Dear Prof. Tao!

On p.179, ex.4.5.7, there should be instead of , because inequality is obvious.

[Correction added, thanks – T.]7 July, 2018 at 1:45 pm

1% quasimorphisms and group cohomology | What's new[…] one has a bounded derivative, as one can use the Bogulybov lemma (see e.g, Proposition 4.39 of my book with Van Vu) to find a large Bohr set inside . In applications, the set need not have bounded size, or even […]

24 February, 2019 at 5:13 pm

Walter Bridgesp. 5 and 6, Exc. 1.1.5 and 1.1.6 should be $\{0,1\}^d$, not $[0,1]^d$. (Of course, $\frac{1}{n}v_i$ isn’t necessarily defined for $v_i$ in an arbitrary additive group, but I still found it a bit confusing.

[Correction added, thanks – T.]21 March, 2019 at 12:24 pm

AnonymousI think the errata for p.1 is incorrect. I think the fix should have been \(E(|X|^2)-|E(X)|^2\), since that is what equals \(E(|X-E(X)|^2\) and since exercise 1.3.4 references complex-valued variables.

[Errata added for Ex 1.3.4. (The p1 definition is specifically for real random variables.)10 September, 2019 at 8:06 am

Adam BrownExercise 6.3.5 is false as stated but the spirit of the problem remains intact. Let $A$ be Sidon and choose $C$ to be the dilation $C = 2 \cdot A$.

[Erratum added, thanks – T.]16 December, 2019 at 12:34 am

anonymousUp to page 233, better just to add the latest, only reason I pause is to keep the mind fresh, like multitasking in a factory. Will resume later.

—————————————————————————————————-

P83 Ex 2.5.6 Hint ends with small type, hence needs a large bracket, but due to small type has ended up with a small type bracket to be more clear than in my previous post.…

P109 ex2.8.1 usual, hint ends with small typo bracket yet started with regular bracket.

P178 ex 4.5.3 Usual bug, the hint starts with a normal parenthesis ends with a small type parenthesis, funny , how it just pops up occasionally. Actually puts a bit of pressure on to make sure they are all eliminated.

P179 Usual, ex 4.5.11 hint is closed with a small type bracket…

P196 near the top of page beneath “the claim follows” in the next sentence a parenthetical expression starting with (and …. Has no closing parenthesis, no idea where to end it although I am confident it occurs within the same sentence.

P206: usual, except for once it happened in a Lemma instead of an exercise the first sentence of the proof of Lemma5.10 a clause starting with using Lemma 5.3… ends with a small type parenthesis instead of a correct regular parenthesis.

—————————————————————————————————–

17 December, 2019 at 7:08 am

anonymousAdditive Combinatorics paperback most recent edition sourced from kindle;finalized errata based on a full peruse (PP1-512); minus self promoting vocab; please however delete the above four posts time permitting/when able to please. Won’t at all mind if the entire experience of ‘editing’ proves useless, if the errata proves useful, then a bonus. Good to have perused Additive Combinatorics.

——————————————–

P7 : equation 1.10 it’s a horrible expression to parenthesize perfectly, the covariance in between the two equals signs, if you just look at it twice you’ll see it needs just one more bracket in the first main bracketed expression E((Xi – E(Xi)) should be E((Xi – E(Xi))) ….

P34: below theorem 1.36 there is a paragraph starts with informally ends with a Y = equation…with high probability, …///well if you check that equation again the bracket either goes after E(Y)) hence closing the full side of that equation which seems pointless, or, around 1 + Ok,… I’m sure you’ll work it out, anyhow it’s missing a parenthesis…

P196: near the top of page beneath “the claim follows” in the next sentenced a parenthetical expression starting with (and …. Has no closing parenthesis, no idea where to end it although confident it occurs within the same sentence.

P242 : above equation 5.19 is says “Restricting this to elements of A”…I think the parenthesis needs closed after Tor(z) , so it looks like …”Tor(z)), we obtain”

P253 Ex 6.2.3 period outside parenthesized sentence.,

P263: not fully sure, could be deliberate but there is this “”(a,b) is a member of E)”” (is a member of is in mathematical notation, I just can’t write it in here, not in LaTex anyway, meaning to describe that curvy E symbol,) , that is stated a few times in consecutive equations, might be worth double checking can’t see where the opening parenthesis is from nor why it needs to be parenthesized, probably not due to interval/set notation/parenthesizing.

P355: Sentence mid page starts with “””Finally, when…””,….well, after ….“and hence”” is a summation with “”aixi)”, seems the parenthesis is closing something but hasn’t been opened, suspecting that based on other examples that it might need to be “(aixi)””…

P396: Near top of page it states an equation below “””from (10.16) we have””,….well in that equation it closes with a parenthesis that doesn’t seem to have an opening parenthesis, and same for the next equation below.

P402: middle page just below “”An application of the first moment method easily shows that””….is an equation with an unclosed parenthesis on the left side of the equation.

P430: near bottom of page is an exp() expression which needs just one more bracket, occurs right before,….. “”,then there exists a proper arithmetic progression P in Z of””….,…..

P439: In the second sentence of Proposition 11.12 the exp() expression after “”If”” has an unclosed opening parenthesis.

—————————————–

[Corrections added, thanks – T.]