ErgodicPNT

one views the regularity lemma not as a structure theorem for large dense graphs, but rather as a structure theorem for events or random variables in a product probability space.

we deduce from Chen's theorem, Roth's theorem, and a transference principle that there are infinitely many arithmetic progressions p1 < p2 < p3 of primes, such that pi + 2 is either a prime or a product of two primes for each i = 1, 2, 3.

[converted to Unicode] The proof uses a quadratic version of Littlewood-O?ord type results concerning the concentration functions of random variables and can be extended for more general models of random matrices.

DR Heath-Brown - Number theory and algebraic geometry-to Peter Swinnerton-

I've just uploaded to the arXiv my lecture notes "Structure and randomness in combinatorics" for my tutorial at the upcoming FOCS 2007 conference in October. This tutorial covers similar ground as my ICM paper (or slides), or my first two Simons lectures, but focuses more on the "nuts-and-bolts" of how structure theorems actually work to separate objects into structured pieces and pseudorandom pieces, for various definitions of "structured" and "pseudorandom".  Given that the target audience consists of computer scientists, I have focused exclusively here on the combinatorial aspects of this dichotomy (applied for instance to functions on the Hamming cube) rather than, say, the ergodic theory aspects (which are covered in Bryna Kra's lecture notes from Montreal, or my notes from Montreal for that matter).  While most of the known applications of these decompositions are number-theoretic (e.g. my theorem with Ben Green), the number theory aspects are not covered in detail in these notes.  (For that, you can read Bernard Host's Bourbaki article, Ben Green's http

Freiman's theorem in an arbitrary abelian group
Green and Ruzsa J. London Math. Soc..2007; 0: jdl021v1-13

there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes.

Rather than give another exposition of this result, we have chosen to take a broader view, surveying the collection of structural theorems which underlie the proof of such results as Theorem 1.1 and Theorem 1.2.

Published version, can be downloaded freely. PAMQ is a new journal with many beautiful papers.

it is reasonable to conjecture that an analogous result to Theorem 1.3 also holds in higher dimensions.This is however still open even in the linear case, the key difficulty being that the tensor product of pseudorandom measures is not pseudorandom.

The combinatorial and ergodic approaches may seem rather different at first glance, but we will try to emphasise the many similarities between them.

The setting of this paper was deliberately placed at a midpoint between graph theory and ergodic theory, and the author hopes that it illuminates the analogies and interconnections between these two subjects.

Introduction to Additive Combinatorics

Fields Institute thematic program, Spring 2008

Denote the Gowers Inverse conjecture by 'GI(s)' and denote the M¨obius and nilsequences conjecture by 'MN(s)', Our results are therefore unconditional in the case s = 2, and in particular we can obtain the expected asymptotics for the number of 4-term

Define r4(N) to be the largest cardinality of a set A ⊆ {1, . . . ,N} which does not contain four elements in arithmetic progression.

Manfred Einsiedler and Thomas Ward

many good notes