Quadratic Uniformity of the Mobius Function
Authors:
Ben Green,
Terence Tao
(Submitted on 4 Jun 2006 (v1), last revised 22 Sep 2007 (this version, v2))
Abstract: This paper is a part of our programme to generalise the Hardy-Littlewood
method to handle systems of linear questions in primes. This programme is laid
out in our paper Linear Equations in Primes [LEP], which accompanies this
submission. In particular, the results of this paper may be used, together with
the machinery of [LEP], to establish an asymptotic for the number of four-term
progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally any
problem counting prime points inside a ``non-degenerate'' affine lattice of
codimension at most 2.
The main result of this paper is a proof of the Mobius and Nilsequences
Conjecture for 1 and 2-step nilsequences. This conjecture is introduced in
[LEP] and amounts to showing that if G/\Gamma is an s-step nilmanifold, s <= 2,
if F : G/\Gamma -> [-1,1] is a Lipschitz function, and if T_g : G/\Gamma ->
G/\Gamma is the action of g \in G on G/\Gamma, then the Mobius function \mu(n)
is orthogonal to the sequence F(T_g^n x) in a fairly strong sense, uniformly in
g and x in G/\Gamma. This can be viewed as a ``quadratic'' generalisation of an
exponential sum estimate of Davenport, and is proven by the following the
methods of Vinogradov and Vaughan.
arithwsun arithwsun on 01 Aug 07I'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