Ben's preprints and papers

Here, in reverse chronological order, are all the papers I have finished writing at the moment. I also have a page of unpublished notes which may be of some interest.

57. A nilpotent Freiman dimension lemma
8 pages, submitted to special edition of EJC in honour of Hamidoune. Emmanuel Breuillard
Terence Tao

56. Contractions and expansion
6 pages, submitted to special edition of EJC in honour of Hamidoune. Emmanuel Breuillard

55. The structure of approximate groups
91 pages, submitted. Emmanuel Breuillard
Terence Tao

54. On (not) computing the Mobius functions using bounded depth circuits
10 pages, to appear in Combinatorics, Probability and Computing.

53. A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups
5 pages, submitted. Emmanuel Breuillard
Terence Tao

52. Strongly dense free subgroups of semisimple algebraic groups
27 pages, to appear in Israel J. Math. Emmanuel Breuillard
Bob Guralnick
Terence Tao

51. An inverse theorem for the Gowers U^s+1[N]-norm
116 pages, to appear in Annals of Math. Terence Tao
Tamar Ziegler

50. Approximate groups, III: The unitary case
19 pages, to appear in Turkish J. Math. Emmanuel Breuillard

49. An inverse theorem for the Gowers U^s+1-norm (announcement)
Electron. Res. Annouce. Math. Sci. 18 (2011), 69-90. Terence Tao
Tamar Ziegler

48. Approximate subgroups of linear groups
GAFA 21 (2011), no. 4, 774-819. Emmanuel Breuillard
Terence Tao

47. Suzuki groups as expanders
Groups Geom. Dyn 5 (2011), 281--299. Emmanuel Breuillard
Terence Tao

46. Yet another proof of Szemeredi's theorem
6 pages, in An irregular mind: Szemeredi is 70, Bolyai Society Math. Studies 21 (2010). Terence Tao

45. An arithmetic regularity lemma, associated counting lemma, and applications
58 pages, in An irregular mind: Szemeredi is 70, Bolyai Society Math. Studies 21 (2010). Terence Tao

44. Linear approximate groups (research announcement)
Electron. Res. Announc. Math. Sci 17 (2010), 57-67. Emmanuel Breuillard
Terence Tao

43. An inverse theorem for the Gowers U⁴-norm
Glasgow Math. J. 53 (2011), 1-50. Terence Tao
Tamar Ziegler JV

42. Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak
25 pages, Current Events Bulletin of the AMS, 2010.

41. Approximate groups, II: The solvable linear case
Quart. J. Math. 62 (2011), no. 3, 513-521. Emmanuel Breuillard

40. Approximate groups, I: The torsion-free nilpotent case
J. Inst. Math. Jussieu 10 (2011), no. 1, 37-57. Emmanuel Breuillard JV

39. An equivalence between inverse sumset theorems and inverse conjectures for the U³-norm
Math. Proc. Camb. Phil. Soc. 149 (2010), no. 1, 1-19. Terence Tao

38. A note on Elkin's improvement of Behrend's construction
in Additive Number Theory (Festschrift in honour of Mel Nathanson), Springer 2010, 141-144. Julia Wolf

37. On a variant of the large sieve
7 pages

36. The Mobius function is strongly orthogonal to nilsequences
20 pages, to appear in Annals of Math. Terence Tao

35. The distribution of polynomials over finite fields, with applications to the Gowers norms
33 pages, Contrib. Discrete Math 4 (2009), no. 2, 1-36. Terence Tao

34. Three topics in additive prime number theory
40 pages, Current Developments in Mathematics 2007, 1-41, Int. Press, Somerville MA.

33. On the maximal number of three-term arithmetic progressions in subsets of Z/pZ
Bull. Lond. Math. Soc. 40 (2008), no. 6, 945--955. Olof Sisask

32. The quantitative behaviour of polynomial orbits on nilmanifolds
62 pages, to appear in Annals of Math. Terence Tao

31. Freiman's theorem in finite fields via extremal set theory
Combinatorics, Probability and Computing 18 (2009), no. 3, 335--355. Terence Tao

30. A note on the Freiman and Balog-Szemerédi-Gowers theorems in finite fields
J. Aust. Math. Soc. 86 (2009), no. 1, 61--74. Terence Tao

29. A quantitative version of the idempotent theorem in harmonic analysis
Annals. of Math. 168 (2008), no. 3, 1025--1054. Tom Sanders

28. New bounds for Szemerédi's theorem, II: a new bound for r₄(N).
Analytic Number Theory (special volume in honour of Klaus Roth, ed Chen et al) 180--204. Terence Tao

27. Linear equations in primes
Annals of Math. 171 (2010), no. 3, 1753-1850.
The slides and accompanying notes from my ICM talk give an overview of this paper. Terence Tao

26. Quadratic uniformity of the Möbius function
Annales de l'Institut Fourier (Grenoble) 58 (2008), no. 6, 1863-1935. Terence Tao

25. Boolean functions with small spectral norm
GAFA 18 (2008), 144-162. Tom Sanders

24. Montréal lecture notes on quadratic Fourier analysis
in Additive Combinatorics (Montréal 2006, ed. Granville et al.), CRM Proceedings vol. 43, 69-102, AMS 2007.

23. On the Littlewood problem modulo a prime
Canad. J. Math. 61 (2009), no. 1, 141--164. Sergei Konyagin

22. New bounds for Szemerédi's theorem, I: Progressions of length 4 in finite field geometries
Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 365--392. Terence Tao

21. Long arithmetic progression of primes
in Analytic Number Theory: a tribute to Gauss and Dirichlet (ed. Duke, Tschinkel),
Clay Mathematics Proceedings vol. 7 (2007), 149-168.

20. An inverse theorem for the Gowers U³-norm, with applications
Proc. Edinburgh Math. Soc. 51 (2008), no. 1, 73-153. Terence Tao

19. Generalising the Hardy-Littlewood method for primes
International Congress of Mathematicians. Vol. II, 373-399, Eur. Math. Soc., Zurich, 2006.
Slides and accompanying notes on the slides are available.

18. Compressions, convex geometry and the Freiman-Bilu theorem
Quart. Jour. Math. 57 (2006), no. 4, 495-504. Terence Tao

17. Freiman's theorem in an arbitrary abelian group
Jour. London Math. Soc. 75 (2007), no. 1, 163-175. Imre Ruzsa

16. The primes contain arbitrarily long arithmetic progressions
Annals of Math. 167 (2008), 481-547.
Terry's back of an envelope calculation, which tells us that there are k primes in AP,
all less than 2^2^2^2^2^2^2^(100k) Terence Tao

15. Restriction theory of the Selberg sieve, with applications
Jour. Th. Nombres Bordeaux 18 (2006), 147--182. Terence Tao

14. Sets with small sumset and rectification
Bull. London Math. Soc. 38 (2006), no. 1, 43-52. Imre Ruzsa

13. Finite field models in additive combinatorics
Surveys in Combinatorics 2005, London Math. Soc. Lecture Notes 327, 1-27.
Supplement 1: The polynomial Freiman-Ruzsa conjecture.
Supplement 2: An argument of Shkredov in the finite field setting.

12. Roth's Theorem in the primes
Annals of Math. 161 (2005), no. 3, 1609-1636.
Here are some minor arcs estimates required in this paper.

11. Sum-free sets in abelian groups
Israel J. Math 147 (2005), 157-189. Imre Ruzsa

10. A Szemerédi-type regularity lemma in abelian groups
GAFA 15 (2005), no. 2, 340-376.

9. Counting sets with small sumset, and the clique number of random Cayley graphs
Combinatorica 25 (3) (2005), 307-326.

8. On the Hardy-Littlewood majorant problem
Math. Proc. Camb. Phil. Soc 137 (2004), no. 3, 511-517. Imre Ruzsa

7. Spectral structure of sets of integers
in Fourier analysis and convexity, 83-96, Appl. Numer. Harmon. Anal., Birkhauser Boston, Boston, MA, 2004.

6. The Cameron-Erdõs Conjecture
Bull. London Math. Soc. 36 (2004), no. 6, 769-778

5. Counting sumsets and sum-free sets modulo a prime
Studia Sci. Math. Hungarica 41 (2004), no.3, 285-293. Imre Ruzsa

4. Some constructions in the inverse spectral theory of cyclic groups
Comb. Prob. Comp. 12 (2003) no. 2, 127-138.

3. Arithmetic progressions in sumsets
GAFA 12 (2002) no. 3, 584-597.

2. On arithmetic structures in dense sets of integers
Duke Math. Jour. 114 (2002) no. 2, 215-238.

1. The number of squares and B_h[g] sets
Acta Arithmetica 100 (2001) no. 4, 365-390.

57. A nilpotent Freiman dimension lemma 8 pages, submitted to special edition of EJC in honour of Hamidoune.	Emmanuel Breuillard Terence Tao
56. Contractions and expansion 6 pages, submitted to special edition of EJC in honour of Hamidoune.	Emmanuel Breuillard
55. The structure of approximate groups 91 pages, submitted.	Emmanuel Breuillard Terence Tao
54. On (not) computing the Mobius functions using bounded depth circuits 10 pages, to appear in Combinatorics, Probability and Computing.
53. A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups 5 pages, submitted.	Emmanuel Breuillard Terence Tao
52. Strongly dense free subgroups of semisimple algebraic groups 27 pages, to appear in Israel J. Math.	Emmanuel Breuillard Bob Guralnick Terence Tao
51. An inverse theorem for the Gowers U^s+1[N]-norm 116 pages, to appear in Annals of Math.	Terence Tao Tamar Ziegler
50. Approximate groups, III: The unitary case 19 pages, to appear in Turkish J. Math.	Emmanuel Breuillard
49. An inverse theorem for the Gowers U^s+1-norm (announcement) Electron. Res. Annouce. Math. Sci. 18 (2011), 69-90.	Terence Tao Tamar Ziegler
48. Approximate subgroups of linear groups GAFA 21 (2011), no. 4, 774-819.	Emmanuel Breuillard Terence Tao
47. Suzuki groups as expanders Groups Geom. Dyn 5 (2011), 281--299.	Emmanuel Breuillard Terence Tao
46. Yet another proof of Szemeredi's theorem 6 pages, in An irregular mind: Szemeredi is 70, Bolyai Society Math. Studies 21 (2010).	Terence Tao
45. An arithmetic regularity lemma, associated counting lemma, and applications 58 pages, in An irregular mind: Szemeredi is 70, Bolyai Society Math. Studies 21 (2010).	Terence Tao
44. Linear approximate groups (research announcement) Electron. Res. Announc. Math. Sci 17 (2010), 57-67.	Emmanuel Breuillard Terence Tao
43. An inverse theorem for the Gowers U⁴-norm Glasgow Math. J. 53 (2011), 1-50.	Terence Tao Tamar Ziegler	JV
42. Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak 25 pages, Current Events Bulletin of the AMS, 2010.
41. Approximate groups, II: The solvable linear case Quart. J. Math. 62 (2011), no. 3, 513-521.	Emmanuel Breuillard
40. Approximate groups, I: The torsion-free nilpotent case J. Inst. Math. Jussieu 10 (2011), no. 1, 37-57.	Emmanuel Breuillard	JV
39. An equivalence between inverse sumset theorems and inverse conjectures for the U³-norm Math. Proc. Camb. Phil. Soc. 149 (2010), no. 1, 1-19.	Terence Tao
38. A note on Elkin's improvement of Behrend's construction in Additive Number Theory (Festschrift in honour of Mel Nathanson), Springer 2010, 141-144.	Julia Wolf
37. On a variant of the large sieve 7 pages
36. The Mobius function is strongly orthogonal to nilsequences 20 pages, to appear in Annals of Math.	Terence Tao
35. The distribution of polynomials over finite fields, with applications to the Gowers norms 33 pages, Contrib. Discrete Math 4 (2009), no. 2, 1-36.	Terence Tao
34. Three topics in additive prime number theory 40 pages, Current Developments in Mathematics 2007, 1-41, Int. Press, Somerville MA.
33. On the maximal number of three-term arithmetic progressions in subsets of Z/pZ Bull. Lond. Math. Soc. 40 (2008), no. 6, 945--955.	Olof Sisask
32. The quantitative behaviour of polynomial orbits on nilmanifolds 62 pages, to appear in Annals of Math.	Terence Tao
31. Freiman's theorem in finite fields via extremal set theory Combinatorics, Probability and Computing 18 (2009), no. 3, 335--355.	Terence Tao
30. A note on the Freiman and Balog-Szemerédi-Gowers theorems in finite fields J. Aust. Math. Soc. 86 (2009), no. 1, 61--74.	Terence Tao
29. A quantitative version of the idempotent theorem in harmonic analysis Annals. of Math. 168 (2008), no. 3, 1025--1054.	Tom Sanders
28. New bounds for Szemerédi's theorem, II: a new bound for r₄(N). Analytic Number Theory (special volume in honour of Klaus Roth, ed Chen et al) 180--204.	Terence Tao
27. Linear equations in primes Annals of Math. 171 (2010), no. 3, 1753-1850. The slides and accompanying notes from my ICM talk give an overview of this paper.	Terence Tao
26. Quadratic uniformity of the Möbius function Annales de l'Institut Fourier (Grenoble) 58 (2008), no. 6, 1863-1935.	Terence Tao
25. Boolean functions with small spectral norm GAFA 18 (2008), 144-162.	Tom Sanders
24. Montréal lecture notes on quadratic Fourier analysis in Additive Combinatorics (Montréal 2006, ed. Granville et al.), CRM Proceedings vol. 43, 69-102, AMS 2007.
23. On the Littlewood problem modulo a prime Canad. J. Math. 61 (2009), no. 1, 141--164.	Sergei Konyagin
22. New bounds for Szemerédi's theorem, I: Progressions of length 4 in finite field geometries Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 365--392.	Terence Tao
21. Long arithmetic progression of primes in Analytic Number Theory: a tribute to Gauss and Dirichlet (ed. Duke, Tschinkel), Clay Mathematics Proceedings vol. 7 (2007), 149-168.
20. An inverse theorem for the Gowers U³-norm, with applications Proc. Edinburgh Math. Soc. 51 (2008), no. 1, 73-153.	Terence Tao
19. Generalising the Hardy-Littlewood method for primes International Congress of Mathematicians. Vol. II, 373-399, Eur. Math. Soc., Zurich, 2006. Slides and accompanying notes on the slides are available.
18. Compressions, convex geometry and the Freiman-Bilu theorem Quart. Jour. Math. 57 (2006), no. 4, 495-504.	Terence Tao
17. Freiman's theorem in an arbitrary abelian group Jour. London Math. Soc. 75 (2007), no. 1, 163-175.	Imre Ruzsa
16. The primes contain arbitrarily long arithmetic progressions Annals of Math. 167 (2008), 481-547. Terry's back of an envelope calculation, which tells us that there are k primes in AP, all less than 2^2^2^2^2^2^2^(100k)	Terence Tao
15. Restriction theory of the Selberg sieve, with applications Jour. Th. Nombres Bordeaux 18 (2006), 147--182.	Terence Tao
14. Sets with small sumset and rectification Bull. London Math. Soc. 38 (2006), no. 1, 43-52.	Imre Ruzsa
13. Finite field models in additive combinatorics Surveys in Combinatorics 2005, London Math. Soc. Lecture Notes 327, 1-27. Supplement 1: The polynomial Freiman-Ruzsa conjecture. Supplement 2: An argument of Shkredov in the finite field setting.
12. Roth's Theorem in the primes Annals of Math. 161 (2005), no. 3, 1609-1636. Here are some minor arcs estimates required in this paper.
11. Sum-free sets in abelian groups Israel J. Math 147 (2005), 157-189.	Imre Ruzsa
10. A Szemerédi-type regularity lemma in abelian groups GAFA 15 (2005), no. 2, 340-376.
9. Counting sets with small sumset, and the clique number of random Cayley graphs Combinatorica 25 (3) (2005), 307-326.
8. On the Hardy-Littlewood majorant problem Math. Proc. Camb. Phil. Soc 137 (2004), no. 3, 511-517.	Imre Ruzsa
7. Spectral structure of sets of integers in Fourier analysis and convexity, 83-96, Appl. Numer. Harmon. Anal., Birkhauser Boston, Boston, MA, 2004.
6. The Cameron-Erdõs Conjecture Bull. London Math. Soc. 36 (2004), no. 6, 769-778
5. Counting sumsets and sum-free sets modulo a prime Studia Sci. Math. Hungarica 41 (2004), no.3, 285-293.	Imre Ruzsa
4. Some constructions in the inverse spectral theory of cyclic groups Comb. Prob. Comp. 12 (2003) no. 2, 127-138.
3. Arithmetic progressions in sumsets GAFA 12 (2002) no. 3, 584-597.
2. On arithmetic structures in dense sets of integers Duke Math. Jour. 114 (2002) no. 2, 215-238.
1. The number of squares and B_h[g] sets Acta Arithmetica 100 (2001) no. 4, 365-390.