Here are my publications, grouped by rough category. I also have a page of unpublished notes which may be of some interest. All except my earliest few papers are available on the arxiv, where they are listed in reverse chronological order. In some cases the published version may differ slightly from the version linked here.

**Approximate groups and applications** (for the abelian case see Freiman's Theorem).

- A nilpotent Freiman dimension lemma (with E. Breuillard and T. Tao) European J. Combinatorics
**34**(2013), no. 8, 1287-1292. - Contractions and expansion (with E.Breuillard), European J. Combinatorics
**34**(2013), no. 8, 1293-1296. - The structure of approximate groups (with E. Breuillard and T.Tao) Publ. Math. IHES
**116**(2012), 115-221. - A note on approximate subgroups of GL
_{n}(C) and uniformly nonamenable groups (with E. Breuillard and T. Tao) - Approximate groups, III: The unitary case (with E. Breuillard) Turkish J. Math
**36**(2012), no. 2, 199-215. - Approximate subgroups of linear groups (with E. Breuillard and T. Tao)
*GAFA***21**(2011), no. 4, 774-819. - Linear approximate groups (research announcement) (with E. Breuillard and T. Tao)
*Electron. Res. Announc. Math. Sci***17**(2010), 57-67. - Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak 25 pages, Current Events Bulletin of the AMS, 2010.
- Approximate groups, II: The solvable linear case (with E. Breuillard)
*Quart. J. Math.***62**(2011), no. 3, 513-521. - Approximate groups, I: The torsion-free nilpotent case (with E. Breuillard)
*J. Inst. Math. Jussieu***10**(2011), no. 1, 37-57.

**Miscellaneous (classical topics in analytic number theory, expansion in groups, combinatorics ....)**

- A note on multiplcative functions on progressions to large moduli,
*to appear in Proc Royal Soc. Edinburgh Ser. A*, 17pp. - Invariable generation of the symmetric group (with Sean Eberhard and Kevin Ford), 15pp.
- Permutations fixing a k-set (with Sean Eberhard and Kevin Ford), IMRN.
- Long gaps between primes (with Kevin Ford, Sergei Konyagin, James Maynard and Terence Tao).
- Large gaps between consecutive prime numbers (with Kevin Ford, Sergei Konyagin and Terence Tao), Annals of Math
**183**(2016), 935-974. - Inverse questions for the large sieve (with Adam Harper), GAFA 24 (2014), no. 4, 1167-1203.
- Expansion in finite simple groups of Lie type (with E. Breuillard, R.Guralnick and T. Tao), J. Eur. Math. Soc 17 (2015), no. 6, 1367-1434.
- On sets containing few ordinary lines (with Terry Tao), Discrete Comp. Geom. 50 (2013), no. 2, 409-468.
- On (not) computing the Mobius function using bounded depth circuits Comb. Prob. Computing 21 (2012), no. 6, 942-951.
- Strongly dense free subgroups of semisimple algebraic groups (with E. Breuillard, R.Guralnick and T. Tao) Israel J. Math 192 (2012), no. 1, 347-379.
- Suzuki groups as expanders (with E. Breuillard and T. Tao)
*Groups Geom. Dyn***5**(2011), 281--299. - On a variant of the large sieve, manuscript, 7pp

** Gowers norms, nilsequences, applications to primes**

- An inverse theorem for the Gowers U
^{s+1}[N]-norm (with T.Tao and T. Ziegler) Annals of Math**176**(2012), no. 2, 1231-1372. - An inverse theorem for the Gowers U
^{s+1}-norm (announcement) (with T. Tao and T. Ziegler)*Electron. Res. Annouce. Math. Sci.***18**(2011), 69-90. - An inverse theorem for the Gowers U
^{4}-norm (with Terry Tao and Tamar Ziegler)*Glasgow Math. J.***53**(2011), 1-50. - The Mobius function is strongly orthogonal to nilsequences (with Terence Tao)
*Annals of Math.***175**(2012), no. 2, 541-566. - The distribution of polynomials over finite fields, with applications to the Gowers norms (with Terence Tao)
*Contrib. Discrete Math***4**(2009), no. 2, 1-36. - The quantitative behaviour of polynomial orbits on nilmanifolds (with Terry Tao)
*Annals of Math.***175**(2012), no. 2, 465-540. Erratum - Linear equations in primes (with Terry Tao)
*Annals of Math.***171**(2010), no. 3, 1753-1850. ICM slides ICM talk notes - Quadratic uniformity of the Möbius function (with Terry Tao)
*Annales de l'Institut Fourier (Grenoble)***58**(2008), no. 6, 1863-1935. - Montréal lecture notes on quadratic Fourier analysis in
*Additive Combinatorics (Montréal 2006, ed. Granville et al.)*, CRM Proceedings vol.**43**, 69-102, AMS 2007. - An inverse theorem for the Gowers U
^{3}-norm, with applications (with Terry Tao)*Proc. Edinburgh Math. Soc.***51**(2008), no. 1, 73-153. - The primes contain arbitrarily long arithmetic
progressions (with Terry Tao)
*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) - Restriction theory of the Selberg sieve, with
applications (with Terry Tao)
*Jour. Th. Nombres Bordeaux***18**(2006), 147--182. - 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.

**Freiman's theorem and related topics**

- An equivalence between inverse sumset theorems and inverse conjectures for the U
^{3}-norm (with Terence Tao)*Math. Proc. Camb. Phil. Soc.***149**(2010), no. 1, 1-19. - Freiman's theorem in finite fields via extremal set theory (with Terry Tao)
*Combinatorics, Probability and Computing***18**(2009), no. 3, 335--355. - A note on the Freiman and Balog-Szemerédi-Gowers theorems in finite fields (with Terry Tao)
*J. Aust. Math. Soc.***86**(2009), no. 1, 61--74. - Compressions, convex geometry and the Freiman-Bilu
theorem (with Terry Tao)
*Quart. Jour. Math.***57**(2006), no. 4, 495-504. - Freiman's theorem in an arbitrary abelian group (with Imre Ruzsa)
*Jour. London Math. Soc.***75**(2007), no. 1, 163-175. - Sets with small sumset and rectification (with Imre Ruzsa)
*Bull. London Math. Soc.***38**(2006), no. 1, 43-52.

**Szemeredi's Theorem and arithmetic progressions**

- Sárkozy's theorem in function fields 7pp.
- New bounds for Szemerédi's theorem, II: A new bound for r
_{4}(N) (with Terry Tao)*Roth Feschrift* - New bounds for Szemerédi's theorem, Ia: Progressions of length 4 in finite field geometries revisited (with Terry Tao), 16pp (replaces this earlier paper from 2005)
- Yet another proof of Szemerédi's theorem (with Terry Tao)
*An irregular mind, 335--342, Bolyai Soc. Math. Stud., 21, Janos Bolyai Math. Soc., Budapest*, 2010. - An arithmetic regularity lemma, associated counting lemma, and applications (with Terry Tao)
*An irregular mind,*261-334, Bolyai Soc. Math. Stud.,**21**, Janos Bolyai Math. Soc., Budapest, 2010. - A note on Elkin's improvement of Behrend's construction (with Julia Wolf)
*in Additive Number Theory (Festschrift in honour of Mel Nathanson)*, Springer 2010, 141-144. - On the maximal number of three-term arithmetic progressions in subsets of Z/pZ (with Olof Sisask)
*Bull. Lond. Math. Soc.***40**(2008), no. 6, 945-955. - A Szemerédi-type regularity lemma in abelian groups
*GAFA***15**(2005), no. 2, 340-376. - On arithmetic structures in dense sets of integers
*Duke Math. Jour.***114**(2002) no. 2, 215-238.

**Harmonic analysis**

- A quantitative version of the idempotent theorem in harmonic analysis (with Tom Sanders)
*Annals. of Math.***168**(2008), no. 3, 1025-1054. - Boolean functions with small spectral norm (with Tom Sanders)
*GAFA***18**(2008), 144-162. - On the Littlewood problem modulo a prime (with Sergei Konyagin)
*Canad. J. Math.***61**(2009), no. 1, 141-164. - On the Hardy-Littlewood majorant problem (with Imre Ruzsa)
*Math. Proc. Camb. Phil. Soc.***137**(2004), no. 3, 511-517. - Spectral structure of sets of integers
*in Fourier analysis and convexity, 83-96, Appl. Numer. Harmon. Anal., Birkhauser Boston, Boston, MA, 2004.* - Some constructions in the inverse spectral theory of cyclic
groups
*Comb. Prob. Comp.***12**(2003) no. 2, 127-138.

**Classical topics in combinatorial number theory (sum-free sets, sumsets, Sidon sets...) **

- Monochromatic solutions to x + y = z
^{2}(with Sofia Lindqvist), 33pp. - Monochromatic sums and products (with Tom Sanders), Discrete Analysis 2016:
**5**, 43pp. - On the chromatic number of random Cayley graphs, 26pp, to appear in Combinatorics, Probability and Computing.
- Counting sets with small sumset and applications (with Rob Morris), Combinatorica
**36**(2016), no. 2, 129-159. - Sets of integers with no large sum-free subset (with Sean Eberhard and Freddie Manners), Annals of Math
**180**(2014), no. 2, 621-652. - Sum-free sets in abelian groups (with Imre Ruzsa)
*Israel J. Math***147**(2005), 157-189. - The Cameron-Erdõs Conjecture
*Bull. London Math. Soc.***36**(2004), no. 6, 769-778. - Counting sets with small sumset, and the clique number of random
Cayley graphs
*Combinatorica***25**(3) (2005), 307-326. - Counting sumsets and sum-free sets modulo a prime (with Imre Ruzsa)
*Studia Sci. Math. Hungarica***41**(2004), no.3, 285-293. - Arithmetic progressions in sumsets
*GAFA***12**(2002) no. 3, - The number of squares and B
_{h}[g] sets*Acta Arithmetica***100**(2001) no. 4, 365-390.

**Survey or expositional articles**

- Approximate algebraic structure (for ICM 2014), 29pp.
- Sets of small doubling in groups (survey article, with E. Breuillard and T. Tao), 23pp.
- Three topics in additive prime number theory 40 pages, Current Developments in Mathematics 2007, 1-41, Int. Press, Somerville MA.
- 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. - Generalising the Hardy-Littlewood method for primes
*International Congress of Mathematicians*. Vol. II, 373-399, Eur. Math. Soc., Zurich, 2006. - 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.