Results 1 
5 of
5
The dichotomy between structure and randomness, arithmetic progressions, and the primes
, 2005
"... A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which i ..."
Abstract

Cited by 26 (1 self)
 Add to MetaCart
A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (lowcomplexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the GreenTao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.
Arithmetic progressions in sets of fractional dimension
 Geom. Funct. Anal
"... Let E ⊂ R be a closed set of Hausdorff dimension α. We prove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains nontrivial 3term arithmetic progressions. ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
(Show Context)
Let E ⊂ R be a closed set of Hausdorff dimension α. We prove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains nontrivial 3term arithmetic progressions.
Quadratic uniformity of the Möbius function
, 2005
"... Abstract. This paper is a part of our programme to generalise the HardyLittlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear equations in primes [14]. In particular, the results of this paper may be used, together with the machinery of [14 ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
Abstract. This paper is a part of our programme to generalise the HardyLittlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear equations in primes [14]. In particular, the results of this paper may be used, together with the machinery of [14], to establish an asymptotic for the number of fourterm progressions p1 < p2 < p3 < p4 � N of primes, and more generally any problem counting prime points inside a “nondegenerate ” affine lattice of codimension at most 2. The main result of this paper is a proof of the Möbius and Nilsequences Conjecture for 1 and 2step nilsequences. This conjecture is introduced in [14] and amounts to showing that if G/Γ is an sstep nilmanifold, s � 2, if F: G/Γ → [−1, 1] is a Lipschitz function, and if Tg: G/Γ → G/Γ is the action of g ∈ G on G/Γ, then
A hypergraph regularity method for generalised Turán problems
, 2008
"... We describe a method that we believe may be foundational for a comprehensive theory of generalised Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular config ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
We describe a method that we believe may be foundational for a comprehensive theory of generalised Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that any 3graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2 − q + 1 and n/2, but for PG2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2 − ǫ)n for some small ǫ> 0. We conclude by setting out a program for future developments of this method to tackle other problems.