An equivalence between inverse sumset theorems and inverse conjectures for the \(U^{3}\) norm. (English) Zbl 1229.11132

Green, Ben; Tao, Terence

Math. Proc. Camb. Philos. Soc. 149, No. 1, 1-19 (2010).

Let \(A\) be a finite subset of some ambient abelian group \(G\). \(A\) is said to be a \(K\)-approximate group if \(A\) is symmetric and if the sumset \(A+A\) is covered by \(K\) translates of \(A\). Let \(A\) and \(B\) be two sets in \(G\), and \(K\geq 1\). \(B\) is said to \(K\)-control \(A\) if \(|B|\leq K|A|\) and if there is some set \(X\) in \(G\) with \(|X|\leq K\) and such that \(A\subseteq B+X\).
The authors first recall some results and conjectures in this area and then prove that some of these conjectures are equivalent.
Conjecture 1.5 (Polynomial Freiman-Ruzsa over \(\mathbb{F}_2^{\infty}\)). Suppose that \(A \subset \mathbb{F}_2^{\infty}\) is a \(K\)-approximate group. Then \(A\) is \(K^C\)-controlled by a finite subgroup. Here \(C\) denotes some positive constant.
Conjecture 1.6 states a “weak” polynomial version over \(\mathbb{Z}\): Suppose \(A \subset \mathbb{Z}\) is a \(K\)-approximate group. Then \(A\) is \(e^{K^{o(1)}}\)-controlled by a symmetric generalized arithmetic progression with dimension \(d\leq K^{o(1)}\).
The authors then turn to a – seemingly – different topic: Let \(f: G \rightarrow \mathbb{C}\). One says \(f\) \(\delta\)-correlates with another function \(F: G \rightarrow \mathbb{C}\) if the inner product \(\langle f,F\rangle := \mathbb{E}_{x \in G} f(x) \overline{F(x)}\) is at least \(\delta\).
Let \(U^3(G)\) denote the Gowers \(U^3\)-norm. A function \(f:G \rightarrow \mathbb{C}\) is called a \(K\)-approximate quadratic if \(\|f\|_{\infty} \leq 1\) and \(\|f\|_{U^3(G)} \geq 1/K\).
Conjecture 1.10 (Polynomial Gowers inverse conjecture over \(\mathbb{F}_2^{n}\)): Suppose \(f: \mathbb{F}_2^{n} \rightarrow \mathbb{C}\) is a \(K\)-approximate quadratic. Then \(f\) \(K^{-C}\) correlates with a quadratic phase \((-1)^{\psi}\).
Conjecture 1.11 is the “weak” version over \(\mathbb{Z}/N \mathbb{Z}\): Suppose that \(f:\mathbb{Z}/N \mathbb{Z} \rightarrow \mathbb{C}\) is a \(K\)-approximate quadratic. Then \(f\) \(\exp(-K^{o(1)})\)-correlates with an elementary 2-step nilsequence \(F(g^n\, x_0)\), where \(F:G/\Gamma \rightarrow \mathbb{C}\) is Lipschitz of order at most \(\exp(K^{o(1)}), g \in G, x_0 \in G/\Gamma\) and \(G/\Gamma\) is an elementary 2-step nilsystem of dimension at most \(K^{o(1)}\).
The authors then prove: Conjectures 1.5 and 1.10 are equivalent. (This was independently proved by Shachar Lovett.) Also, the authors prove that Conjectures 1.6 and 1.11 are equivalent.

Reviewer: Christian Elsholtz (Graz)

MSC:

11P70	Inverse problems of additive number theory, including sumsets
11B30	Arithmetic combinatorics; higher degree uniformity

Keywords:

polynomial Freiman-Ruzsa conjecture; polynomial Gowers inverse conjecture

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