Back to my preprints page.If you are interested in long arithmetic progressions in the primes, but don’t want to plunge directly into all the details, I can suggest the following surveys (in roughly increasing order of technical level of treatment):
Title | With | Status | Download |
A sum-product estimate for finite fields, and applications | GAFA 14 (2004), 27-57 | ||
The primes contain arbitrarily long arithmetic progressions | Annals of Math. 167 (2008), 481-547 | ||
New bounds for Szemeredi's Theorem, I: Progressions of length 4 in finite field geometries | Proc. Lond. Math. Soc. 98 (2009), 365-392 | ||
Restriction theory of the Selberg Sieve, with applications | Journal de Théorie des Nombres de Bordeaux 18 (2006), 137—172 | ||
A quantitative ergodic theory proof of Szemer\'edi's theorem |
| Electron. J. Combin. 13 (2006). 1 No. 99, 1-49. | |
On random $\pm 1$ matrices: Singularity and Determinant | Random Structures and Algorithms 28 (2006), 1—23. [An extended abstracted is also in: STOC’05: Proceedings of the 37thannual ACM symposium on the theory of computing, 431—440, New York 2005.] | ||
Arithmetic progressions and the primes |
| Collectanea Mathematica (2006), Vol. Extra., 37-88. [Proceedings, 7th International Conference on Harmonic Analysis and Partial Differential Equations] | |
On the singularity probability of random Bernoulli matrices | J. Amer. Math. Soc. 20 (2007), 603-628 | ||
The Gaussian primes contain arbitrarily shaped constellations |
| J. d’Analyse Mathematique 99 (2006), 109-176 | |
An inverse theorem for the Gowers $U^3(G)$ norm | Proc. Edin. Math. Soc. 51 (2008), 73-153 | ||
| J. Combin. Thy. A 113 (2006), 1257--1280 | ||
Szemeredi's regularity lemma revisited |
| Contrib. Discrete Math. 1 (2006), 8-28 | |
Random symmetric matrices are almost surely non-singular | Kevin Costello | Duke Math. J. 135 (2006), 395-413 | |
Obstructions to uniformity, and arithmetic patterns in the primes |
| Quarterly J. Pure Appl. Math. 2 (2006), 199-217 [Special issue in honour of John H. Coates, Vol. 1 of 2] | |
Compressions, convex geometry, and the Freiman-Bilu theorem | Quarterly J. Math. 57 (2006), 495-504 | ||
Inverse Littlewood-Offord theorems and the condition number of random discrete matrices | Annals of Math. 169 (2009), 595-632 | ||
New bounds for Szemeredi's Theorem, II: A new bound for r_4(N) | Analytic number theory: essays in honour of Klaus Roth, W. W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt, R. C. Vaughan, eds, Cambridge University Press, 2009. 180-204. | ||
New bounds for Szemeredi's Theorem, III: A polylog bound for r_4(N) | In preparation |
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Quadratic uniformity of the M\"obius function | Annales de l’Institut Fourier 58 (2009), 1863—1935. | ||
Linear equations in primes | Annals of Math. 171 (2010), 1753-1850 | ||
The dichotomy between structure and randomness, arithmetic progressions, and the primes |
| 2006 ICM proceedings, Vol. I., 581--608 | |
Product set estimates in noncommutative groups |
| Combinatorica 28 (2008), 547-594 | |
| J. d’Analyse Mathematique 103 (2007), 1--45. | ||
The ergodic and combinatorial approaches to Szemer\'edi's theorem |
| Centre de Recerches Math\'ematiques, CRM Proceedings and Lecture Notes Vol. 43 (2007), 145--193 | |
The primes contain arbitrarily long polynomial progressions | Acta Math. 201 (2008), 213—305. | ||
John-type theorems for generalized arithmetic progressions and iterated sumsets | Adv. in Math. 219 (2008), 428—449. | ||
A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields | J. Aust. Math. Soc. 86 (2009), 61-74. | ||
The condition number of a randomly perturbed matrix | Proceedings of the thirty-ninth annual ACM symposium on Theory of computing (STOC) 2007, 248-255 | ||
Freiman's theorem in finite fields via extremal set theory | Combin. Probab. Comput. 18 (2009), no. 3, 335--355 | ||
Szemeredi's theorem | Scholarpedia, p. 15573 | ||
Norm convergence of multiple ergodic averages for commuting transformations | Ergodic Theory and Dynamical Systems 28 (2008), 657-688 | ||
Structure and randomness in combinatorics | Proceedings of the 48th annual symposium on Foundations of Computer Science (FOCS) 2007, 3-18 | ||
Random Matrices: The circular Law | Communications in Contemporary Mathematics, 10 (2008), 261--307 | ||
The quantitative behaviour of polynomial orbits on nilmanifolds | Annals of Math. Volume 175 (2012), Issue 2, 465-540. | ||
The M\"obius function is asymptotically orthogonal to nilsequences | To appear, Annals of Math. | ||
The distribution of polynomials over finite fields, with applications to the Gowers norms | Contrib. Discrete Math. 4 (2009), no. 2, 1--36. | ||
On the testability and repair of hereditary hypergraph properties | Random Structures and Algorithms 36 (2010), 373-463 | ||
A remark on primality testing and decimal expansions | 91 (2011), 405-413 | ||
On the permanent of random Bernoulli matrices | Adv. Math. 220 (2009), 657—669. | ||
Smooth analysis of the condition number and the least singular value | Mathematics of Computation, 79 (2010), 2333-2352 | ||
The sum-product phenomenon in arbitrary rings | Contrib. Discrete Math. 4 (2009), no. 2, 59--82. | ||
Random matrices: Universality of ESDs and the circular law | Manjunath Krishnapur(appendix) | Annals of Probability 38 (2010), no. 5, 2023--2065. | |
From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices | Bull. Amer. Math. Soc. 46 (2009), 377-396 | ||
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle | Analysis & PDE 3 (2010), 1-20 | ||
An inverse theorem for the uniformity seminorms associated with the action of $F^\omega$ | Geom. Funct. Anal. 19 (2010), no. 6, 1539--1596. | ||
A sharp inverse Littlewood-Offord theorem | To appear, Random Structures and Algorithms | ||
Random matrices: the distribution of smallest singular values | GAFA, 20 (2010), 260-297 | ||
Random matrices: universality of local eigenvalue statistics | Acta Math 206 (2011), 127-204 | ||
An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm | Math. Proc. Camb. Phil. Soc. 149 (2010), 1-19 | ||
Freiman’s theorem for solvable groups |
| Contributions to Discrete Mathematics 5 (2010), no. 2, 137–184, | |
Sumset and inverse sumset theorems for Shannon entropy |
| Combinatorics, Probability, and Computing 19 (2010), 603-639 | |
Bulk universality for Wigner hermitian matrices with subexponential decay | Jose Ramírez | Math. Res. Lett. 17 (2010), 793-794 | |
Random matrices: universality of local eigenvalue statistics up to the edge | Communications in Mathematical Physics, 298 (2010), 549-572 | ||
A remark on partial sums involving the Mobius function |
| Bull. Aust. Math. Soc. 81 (2010), 343-349 | |
A new proof of the density Hales-Jewett theorem | Annals of Math. 175 (2012), 1283-1327. | ||
A finitary version of Gromov's polynomial growth theorem | Yehuda Shalom | GAFA 20 (2010), no. 6, 1502–1547. | |
An inverse theorem for the Gowers U^4 norm | |||
Random covariance matrices: Universality of local statistics of eigenvalues | Annals of Probability 40 (2012), 1283--1315. | ||
Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems | Pacific Journal of Mathematics 250-1 (2011), 1--60. DOI 10.2140/pjm.2011.250.1 | ||
Linear approximate groups | Electronic research announcements 17 (2010), 57-67 | ||
An arithmetic regularity lemma, an associated counting lemma, and applications | An Irregular Mind: Szemeredi is 70, Bolyai Society Mathematical Studies, 261-334 | ||
Yet another proof of Szemeredi's theorem | An Irregular Mind: Szemeredi is 70, Bolyai Society Mathematical Studies, 335-342 | ||
The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and F\"uredi | Combinatorica, 2012 | ||
Suzuki groups as expanders | Groups, Geometry, and Dynamics 5 (2011), no. 2, 281–-299. | ||
Approximate subgroups of linear groups | To appear, GAFA | ||
Strongly dense free subgroups of semisimple algebraic groups | To appear, Israel J. Math. | ||
Expansion in simple groups of Lie type | In preparation | ||
An inverse theorem for the Gowers U^{s+1}[N] norm | To appear, Annals of Math. (Announcement: Submitted, Electronic Research Announcements) | ||
Random matrices: Localization of the eigenvalues and the necessity of four moments | Acta Mathematica Vietnamica 36 (2011), 431--449 | ||
Deterministic methods to find primes | Ernie Croot Harald Helfgott | Mathematics of Computation 81 (2012), 1233-1246 | |
Large values of the Gowers-Host-Kra seminorms | To appear, J. d’Analyse Mathematique | ||
Outliers in the spectrum of iid matrices with bounded rank permutations | To appear, Probability theory and related fields | ||
The inverse conjecture for the Gowers norm over finite fields in low characteristic | To appear, Annals of Combinatorics | ||
A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups | Submitted, | ||
The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices | Electronic Journal of Probability 16 (2011), 2104-2121 | ||
Random matrices: Universal properties of Eigenvectors | To appear, Random matrices: Theory and Applications | ||
An incidence theorem in higher dimensions | To appear, Disc. Comp. Geom. | ||
Noncommutative sets of small doubling | To appear, European Journal of Combinatorics | ||
Counting the number of solutions to the Erdös-Straus equation on unit fractions | To appear, J. Aust. Math. Soc. | ||
The structure of approximate groups | To appear, Pub. IHES | ||
A central limit theorem for the determinant of a Wigner matrix | Adv. Math. 231 (2012), 74-101 | ||
Random matrices: The Four Moment Theorem for Wigner matrices | Submitted, MSRI Book series | ||
A nilpotent Freiman dimension lemma | To appear, European Journal of Combinatorics | ||
Random matrices: Sharp concentration of eigenvalues | Submitted, Electronic Journal of Probability | ||
Every odd number greater than 1 is the sum of at most five primes | To appear, Mathematics of Computation | ||
Random matrices: The Universality phenomenon for Wigner ensembles | Submitted, AMS Book series | ||
The asymptotic distribution of a single eigenvalue gap of a Wigner matrix | To appear, Probability Theory and Related Fields | ||
E pluribus unum: from complexity, universality | Daedalus 141 (3) (Summer 2012) | Web version | |
New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited | Ben Green | Submitted, Proc. Lond. Math. Soc. | arXiv:1205.1330 |
Random matrices: Universality of local spectral statistics of non-Hermitian matrices | Submitted, Duke Math. J. | arXiv:1206:1893 | |
On sets defining few ordinary lines | Ben Green | Submitted, Disc. Comp. Geom. | arXiv:1208.4714 |
Back to my preprints page.