We prove the inverse conjecture for the Gowers U s+1 [N ]-norm for all s 1; this is new for s 4. More precisely, we establish that if f :there is a boundedcomplexity s-step nilsequence F (g(n)Γ) that correlates with f , where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.

Abstract. Let F a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm U k (X) for an ergodic action (T g ) g∈F ω of the infinite abelian group F ω on a probability space X = (X, B, µ) is generated by phase polynomials φ : X → S 1 of degree less than C(k) on X, where C(k) depends only on k. In the case where k char(F) we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for Z by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k char(F), with a partial result in low characteristic.

Let X = ( X 0 , B , μ , T ) X=(X^0,\mathcal {B},\mu ,T) be an ergodic probability measure-preserving system. For a natural number k k we consider the averages (*) 1 N ∑ n = 1 N ∏ j = 1 k f j ( T a j n x ) \begin{equation*} \tag {*} \frac {1}{N}\sum _{n=1}^N \prod _{j=1}^k f_j(T^{a_jn}x) \end{equation*} where f j ∈ L ∞ ( μ ) f_j \in L^{\infty }(\mu ) , and a j a_j are integers. A factor of X X is characteristic for averaging schemes of length k k (or k k -characteristic) if for any nonzero distinct integers a 1 , … , a k a_1,\ldots ,a_k , the limiting L 2 ( μ ) L^2(\mu ) behavior of the averages in (*) is unaltered if we first project the functions f j f_j onto the factor. A factor of X X is a k k -universal characteristic factor ( k k -u.c.f.) if it is a k k -characteristic factor, and a factor of any k k -characteristic factor. We show that there exists a unique k k -u.c.f., and it has the structure of a ( k − 1 ) (k-1) -step nilsystem, more specifically an inverse limit of ( k − 1 ) (k-1) -step nilflows. Using this we show that the averages in (*) converge in L 2 ( μ ) L^2(\mu ) . This provides an alternative proof to the one given by Host and Kra.

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps > 0$, we show that there are infinitely many integers $x,m$ with $1 \leq m \leq x^\eps$ such that $x+P_1(m), ..., x+P_k(m)$ are simultaneously prime. The arguments are based on those in Green and Tao, which treated the linear case $P_i = (i-1)\m$ and $\eps=1$; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.Comment: 82 pages, 1 figure. A minor error in the paper (concerning the definition of the polynomial forms condition, which had too weak of a requirement on the degree of the polynomials involved) has been fixe

Abstract. We establish the inverse conjecture for the Gowers norm over finite fields, which asserts (roughly speaking) that if a bounded function f : V → C on a finitedimensional vector space V over a finite field F has large Gowers uniformity norm f U s+1 (V ) , then there exists a (non-classical) polynomial P : V → T of degree at most s such that f correlates with the phase e(P ) = e 2πiP . This conjecture had already been established in the "high characteristic case", when the characteristic of F is at least as large as s. Our proof relies on the weak form of the inverse conjecture established earlier by the authors and Bergelson [3], together with new results on the structure and equidistribution of non-classical polynomials, in the spirit of the work of Green and the first author [22] and of Kaufman and Lovett [28].