Linear correlations amongst numbers represented by positive definite binary quadratic forms

Given a positive definite binary quadratic form f, let r(n) = |{(x,y): f(x,y)=n}| denote its representation function. In this paper we study linear correlations of these functions. For example, if r_1, ..., r_k are representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d) ... r_k(n+ (k-1)d).Comment: 60 pages. Small correction

An inverse theorem for the Gowers U^4 norm

We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f ||_{U^4} >= \delta then there is a bounded complexity 3-step nilsequence F(g(n)\Gamma) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s >= 4 as well, and a longer paper will follow concerning this. By combining this with several previous papers of the first two authors one obtains the generalised Hardy-Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p_1 < p_2 < p_3 < p_4 < p_5 <= N of primes.Comment: 49 pages, to appear in Glasgow J. Math. Fixed a problem with the file (the paper appeared in duplicate

Sarnak's Conjecture for nilsequences on arbitrary number fields and applications

We formulate the generalized Sarnak's M\"obius disjointness conjecture for an arbitrary number field $K$, and prove a quantitative disjointness result between polynomial nilsequences $(\Phi(g(n)\Gamma))_{n\in\mathbb{Z}^{D}}$ and aperiodic multiplicative functions on $\mathcal{O}_{K}$, the ring of integers of $K$. Here $D=[K\colon\mathbb{Q}]$, $X=G/\Gamma$ is a nilmanifold, $g\colon\mathbb{Z}^{D}\to G$ is a polynomial sequence, and $\Phi\colon X\to \mathbb{C}$ is a Lipschitz function. The proof uses tools from multi-dimensional higher order Fourier analysis, multi-linear analysis, orbit properties on nilmanifold, and an orthogonality criterion of K\'atai in $\mathcal{O}_{K}$. We also use variations of this result to derive applications in number theory and combinatorics: (1) we prove a structure theorem for multiplicative functions on $K$, saying that every bounded multiplicative function can be decomposed into the sum of an almost periodic function (the structural part) and a function with small Gowers uniformity norm of any degree (the uniform part); (2) we give a necessary and sufficient condition for the Gowers norms of a bounded multiplicative function in $\mathcal{O}_{K}$ to be zero; (3) we provide partition regularity results over $K$ for a large class of homogeneous equations in three variables. For example, for $a,b\in\mathbb{Z}\backslash\{0\}$, we show that for every partition of $\mathcal{O}_{K}$ into finitely many cells, where $K=\mathbb{Q}(\sqrt{a},\sqrt{b},\sqrt{a+b})$, there exist distinct and non-zero $x,y$ belonging to the same cell and $z\in\mathcal{O}_{K}$ such that $ax^{2}+by^{2}=z^{2}$.Comment: 65 page

Correlations of the divisor function

In this paper we study linear correlations of the divisor function tau(n) = sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d).Comment: 33 pages. Corrections and journal referenc

Recurrence and non-uniformity of bracket polynomials

A bracket polynomial on the integers is a function formed using the operations of addition, multiplication and taking fractional parts. For a fairly large class of bracket polynomials we show that if p is a bracket polynomial of degree k-1 on [N] then the function f defined by f(n) = e(p(n)) has Gowers U^k[N]-norm bounded away from zero, uniformly in N. We establish this result by first reducing it to a certain recurrence property of sets of bracket polynomials. Specifically, for a fairly large class of bracket polynomials we show that if p_1, ..., p_r are bracket polynomials then their values, modulo 1, are all close to zero on at least some constant proportion of the points 1, ..., N. The proofs rely on two deep results from the literature. The first is work of V. Bergelson and A. Leibman showing that an arbitrary bracket polynomial can be expressed in terms of a polynomial sequence on a nilmanifold. The second is a theorem of B. Green and T. Tao describing the quantitative distribution properties of such polynomial sequences. We give elementary alternative proofs of the first result, without reference to nilmanifolds, in certain "low-complexity" special cases.Comment: V1 28 pages. V2 31 pages. Section 5 from V1 has been split into two separate sections in V2, and Appendix C has been added. V2 has a different title to V