Citation Context

...22, 36] for related work on multiple averages and nilmanifolds in ergodic theory. (viii) The analogue of GI(s) in finite fields of large characteristic was established by ergodic-theoretic methods in =-=[4, 34]-=-. (ix) A weaker “local” version of the inverse theorem (in which correlation takes place on a subprogression of [N ] of size ∼ N cs) was established by Gowers [11]. This paper provided a good deal of ...

Citation Context

...polynomial is Ω(1). Lovett et al. [4] and Green and Tao [3] disproved this conjecture as stated, but a modification of the conjecture remains open, and was recently proven in high characteristic [9], =-=[10]-=-. These conjectures and the Gowers norms have been extremely influential. For example, Green and Tao [11] used the Gowers norms over the integers to prove that the primes contain arbitrarily long arit...

Citation Context

...partial higher order analogues, although14 BEN GREEN AND TERENCE TAO the situation here is much less well understood. To illustrate this phenomenon, consider the following result, recently proven in =-=[3, 27]-=-. Here and for the rest of the section we write F := F5 for definiteness, although the same arguments would work for Fp for any fixed prime p � 5. There are definite issues in extremely low characteri...

Citation Context

... will need later on. Defining the decompositions requires us to introduce the Gowers norm first. 2.1 The Gowers Norm We define Gowers norms in the general setting of arbitrary finite Abelian groups. Definition 7 (Gowers norm) Let G be a finite abelian group and f : G → C. For an integer k ≥ 1, the k’th Gowers norm of f , denoted ‖f‖Uk is defined by: ‖f‖2 k Uk = Ex,y1,y2,...,yk∈G ∏ S⊆[k] Ck−|S|f ( x+ ∑ i∈S yi ) where C denotes the complex conjugation operator. Two facts about the Gowers norm will be absolutely crucial in what follows. First is the Gowers Inverse theorem, established by [BTZ10, TZ10]. Throughout, we let e (x) denote the complex number e2πix/p for x ∈ Fp. Theorem 8 (Gowers Inverse Theorem) Given a positive integer d < p, for every δ > 0, there exists ǫ = ǫ8(δ) such that if f : F n p → R satisfies ‖f‖∞ ≤ 1 and ‖f‖Ud+1 ≥ δ, then there exists a polynomial P : Fnp → Fp of degree at most d so that |Ex[f(x) · e (P (x))] |≥ ǫ. The second is a lemma due to Green and Tao [GT10b] based on repeated applications of the Cauchy-Schwarz inequality. Refer to Definition 5 for the term “complexity”. Lemma 9 Let f1, . . . , fm : F n p → [−1, 1]. Let L = {L1, . . . , Lm} be a system of m line...

Citation Context

...degree d is a function P from Fn to Zpk such that the (d + 1)-th order derivative of P is zero. The integer k − 1 is called the “depth” of P . Classical polynomials have depth 0. We use the result of =-=[TZ11]-=- to obtain non-classical polynomials P1, . . . , PC of degree 6 d such that each g(σi) = Γi(P1, . . . , PC) for some function Γi. We return now to the goal of lower-bounding Eq. (2). By a sequence of ...

Citation Context

...s = 1 and 2 (see [11]) as well as the truth of analogues of the conjecture in both ergodic theory [18, 28] and in the “finite field model” in which [N ] is replaced by Fn for some small prime field F =-=[1, 26]-=-. It is also known that this conjecture is necessary, in the following sense. Proposition 1.4 (Necessity of inverse conjecture). Suppose that f : [N ] → C is a 1-bounded function, that (F (g(n)Γ))n∈Z ...