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 I also have a page of unpublished notes which may be of some interest.

41. Approximate groups, II: The solvable linear case
10 pages, submitted Emmanuel Breuillard

40. Approximate groups, I: The torsion-free nilpotent case
23 pages, submitted Emmanuel Breuillard

39. An equivalence between inverse sumset theorems and inverse conjectures for the U³-norm
23 pages, to appear in Math. Proc. Camb. Phil. Soc. Terence Tao

38. A note on Elkin's improvement of Behrend's construction
4 pages, submitted to Mel Nathanson special volume Julia Wolf

37. On a variant of the large sieve
7 pages

36. The Mobius function is strongly orthogonal to nilsequences
20 pages, submitted Terence Tao

35. The distribution of polynomials over finite fields, with applications to the Gowers norms
33 pages, submitted Terence Tao

34. Three topics in additive prime number theory
40 pages, expository article, to appear in Current Developments in Mathematics 2007.

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, submitted. 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
Ann. of Math. (2) 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
83 pages. Annals of Math, to appear.
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, 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.

41. Approximate groups, II: The solvable linear case 10 pages, submitted	Emmanuel Breuillard
40. Approximate groups, I: The torsion-free nilpotent case 23 pages, submitted	Emmanuel Breuillard
39. An equivalence between inverse sumset theorems and inverse conjectures for the U³-norm 23 pages, to appear in Math. Proc. Camb. Phil. Soc.	Terence Tao
38. A note on Elkin's improvement of Behrend's construction 4 pages, submitted to Mel Nathanson special volume	Julia Wolf
37. On a variant of the large sieve 7 pages
36. The Mobius function is strongly orthogonal to nilsequences 20 pages, submitted	Terence Tao
35. The distribution of polynomials over finite fields, with applications to the Gowers norms 33 pages, submitted	Terence Tao
34. Three topics in additive prime number theory 40 pages, expository article, to appear in Current Developments in Mathematics 2007.
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, submitted.	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 Ann. of Math. (2) 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 83 pages. Annals of Math, to appear. 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, 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.