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However, it turns out that if one of the sets, say A, is sufficiently “uniform” or “pseudorandom”, then one can always solve this Goldbach-type problem, regardless of what the other two sets are. This type of fact is often established by Fourier-analytic means (or by closely related techniques, such as spectral theory), but let me give a heuristic combinatorial argument to indicate why one would expect this type of phenomenon to occur.

quares Primes Lagrange’s four square theorem: For every positive integer N, there exists a pattern in of the form . Vinogradov’s theorem: For every sufficiently large integer N, there exists a pattern in of the form . Fermat’s two square theorem: For every prime number , there exists a pattern in of the form . Even Goldbach conjecture: For every even number , there exists a pattern in of the form . Fermat’s four square theorem: There does not exist any pattern in of the form with . Green-Tao theorem: For any , there exist infinitely many patterns in of the form with . Pell’s equation: There are infinitely many patterns in of the form . Sophie Germain conjecture: There are infinitely many patterns in of the form .

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.