Preprints in additive combinatorics and number theory
Math 254A home page - Arithmetic combinatorics (2003)
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):
- Terry Tao, “Long arithmetic progressions in the primes”- slides, aimed at undergraduate audience
- Terry Tao, “Long
arithmetic progressions in the primes”- slightly more detailed
version of previous
- Terry Tao, “Multiscale analysis of the primes” - slides
- Bryna Kra, “The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view”
- Bernard Host, “Progressions arithmétiques dans les nombres premiers, d'après B. Green et T. Tao”. Seminaire Bourbaki,Mars 2005, 57eme annee, 2004-2005, no. 944.
- Terry Tao, “The dichotomy between structure and randomness, arithmetic progressions, and the primes” (ICM lecture)
- Terry Tao, “Obstructions to uniformity, and arithmetic patterns in the primes”
- Ben Green, “Long arithmetic progressions of primes”
- Terry Tao, “Arithmetic progressions and the primes - El Escorial lectures” (A harmonic analysis perspective)
Papers, and projects close to completion
|
Title |
With |
Status |
Download |
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A sum-product estimate for finite fields, and applications |
GAFA 14 (2004), 27-57 |
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The primes contain arbitrarily long arithmetic progressions |
Annals
of Math. 167 (2008), 481-547 |
math.NT/0404188 |
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New bounds for Szemeredi's Theorem, I: Progressions of length 4 in finite field geometries |
To
appear, Proc. Lond. Math. Soc. |
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Restriction theory of the Selberg Sieve, with applications |
Journal
de Théorie des Nombres de Bordeaux 18 (2006), 137—172 |
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A quantitative ergodic theory proof of Szemer\'edi's theorem |
|
Electron.
J. Combin. 13 (2006). 1 No. 99, 1-49. |
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On random $\pm 1$ matrices: Singularity and Determinant |
Random
Structures and Algorithms 28 (2006), 1—23. |
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Arithmetic progressions and the primes |
|
Collectanea
Mathematica (2006), Vol. Extra.,
37-88. |
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On the singularity probability of random Bernoulli matrices |
J. Amer. Math.
Soc. 20 (2007), 603-628 |
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The Gaussian primes contain arbitrarily
shaped constellations |
|
J.
d’Analyse Mathematique 99 (2006), 109-176 |
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An inverse theorem for the Gowers $U^3(G)$ norm |
To
appear, Proc.
Edin. Math. Soc. |
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|
J.
Combin. Thy. A 113 (2006), 1257--1280 |
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Szemer\’edi’s regularity lemma revisited |
|
Contrib.
Discrete Math. 1 (2006), 8-28 |
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Random symmetric matrices are almost surely non-singular |
Kevin Costello |
Duke Math.
J. 135 (2006), 395-413 |
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Obstructions to uniformity, and arithmetic patterns in the primes |
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Quarterly
J. Pure Appl. Math. 2 (2006), 199-217 [Special issue in honour of
John H. Coates, Vol. 1 of 2] |
|
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Compressions, convex geometry, and the Freiman-Bilu theorem |
Quarterly
J. Math. 57 (2006), 495-504 |
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Inverse Littlewood-Offord theorems and the condition number of random discrete matrices |
To
appear, Annals
of Math. |
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New bounds for Szemeredi's Theorem, II: A new bound for r_4(N) |
To
appear, a volume in honour of Roth's 80th
birthday |
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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 |
To
appear, Annales de
l’Institut
Fourier |
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Linear equations in primes |
To
appear, Annals
of Math. |
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The dichotomy between structure and randomness, arithmetic progressions, and the primes |
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2006
ICM proceedings, Vol. I., 581--608 |
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Product set estimates in noncommutative groups |
|
To
appear, Combinatorica |
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|
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J.
d’Analyse
Mathematique 103 (2007), 1--45. |
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The ergodic and combinatorial approaches to Szemer\'edi's theorem |
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Centre de Recerches Math\'ematiques, CRM Proceedings and Lecture Notes Vol. 43 (2007), 145--193 |
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The primes contain
arbitrarily long polynomial progressions |
To appear, Acta Math. |
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John-type theorems
for generalized arithmetic progressions and iterated sumsets |
To
appear, Adv.
in Math. |
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A note on the
Freiman and Balog-Szemeredi-Gowers theorems in finite fields |
To
appear, J.
Aust. Math. Soc. |
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On the condition
number of a randomly perturbed matrix |
Proceedings of
the thirty-ninth annual ACM symposium on Theory of computing
(STOC) 2007, 248-255 |
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| Freiman's theorem in finite fields via extremal set theory | Submitted, Combinatorics,
Probability, and Computing |
math.CO/0703668 discussion |
|
| Szemerédi's theorem |
Scholarpedia,
p. 15573 |
Scholarpedia
article discussion |
|
| Norm convergence of multiple
ergodic averages for commuting transformations |
|
Ergodic
Theory and Dynamical Systems 28 (2008), 657-688 |
arXiv:0707.1117 discussion |
| Structure and randomness in
combinatorics |
Proceedings of the 48th
annual symposium on Foundations of Computer Science (FOCS) 2007, 3-18 |
arXiv:0707.4269 discussion slides discussion of slides |
|
| Random Matrices: The circular
Law |
Van Vu | Communications in Contemporary Mathematics, 10 (2008), 261--307 | arXiv:0708.2895 discussion |
| The quantitative behaviour of
polynomial orbits on nilmanifolds |
Ben Green | Submitted, | arXiv:0709.3562 discussion van der Corput lemma |
| The M\"obius and nilsequences
conjectures |
Ben Green | In preparation | |
| The distribution of
polynomials over finite fields, with applications to the Gowers norms |
Ben Green | Submitted, Contributions to Discrete
Mathematics |
announcement arXiv:0711.3191 discussion |
| On the testability and repair
of
hereditary hypergraph properties |
Tim Austin |
Submitted, Random
Structures and Algorithms |
talk arXiv:0801.2179 discussion |
| A remark on primality testing
and decimal expansions |
Submitted, J.
Aust. Math. Soc. |
arXiv:0802.3361 discussion |
|
| On the permanent of random
Bernoulli matrices |
Van Vu | Submitted, Adv.
Math. |
arXiv:0804.2632 discussion early version |
| Random matrices: A general
approach for the least singular value problem |
Van Vu | Submitted, Israel J. Math. |
arXiv:0805.3167 discussion |
| The sum-product phenomenon in
arbitrary rings |
Submitted, Contributions to Discrete Mathematics | arXiv:0806.2497 discussion |
Short stories
These are generally very short, toy versions of real results due to other people, and are not publication-quality. Caveat emptor. All files other than figures are in dvi format. Unlike the preprints, these articles are fluid and subject to new developments. Please let me know if you have any comments, references, etc. on any of them.
Disclaimer: Many of the notes here are based on papers
written by
other people. My intention here is not to try to "beat" these
authors' work in any way, but rather to isolate the main ingredients of
the
argument, which are often very beautiful, and try to present them in as
simple
and brief a context as possible (often sacrificing generality, rigour,
and/or
details in order to do this). Certainly I do not view these notes
as
worthy of publication in a refereed journal, and are definitely
inferior to the
original article in every single aspect, with the possible exception of
brevity.
Open questions
- Best
bounds for cap sets
- Noncommutative
Freiman theorem
- Triangle
and diamond densities in large dense graphs
- Effective
Skolem-Mahler-Lech theorem
- The parity problem in sieve theory
Miscellaneous
- My book with Van Vu, titled “Additive combinatorics”, is currently in print; see this page for more details.
- Structure and randomness in the prime numbers (UCLA Science Faculty Research Colloquium, Jan 17 2007)
Back to my preprints
page.