The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches

The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches, Discrete Analysis 2023:11, 48 pp.

Let $G$ be a finite Abelian group and let $f:G\to\mathbb C$. Then for every $u\in G$ we define the function $\partial_uf$ by setting $\partial_uf(x)$ to be $f(x)\overline{f(x-u)}$. For each $k\geq 2$ the $U^k$norm of $f$ is then defined by the formula
$$\|f\|_{U^k}^{2^k}=\mathbb E_{u_1,\dots,u_k,x}\partial_{u_1}\dots\partial_{u_k}f(x)$$
where the $\mathbb E$ notation denotes the average over all $(u_1,\dots,u_k,x)\in G^{k+1}$. This family of norms was introduced by Gowers as an important ingredient of a quantitative proof of Szemerédi’s theorem. Key to the proof was a local inverse theorem for $G=\mathbb Z/n\mathbb Z$, which in the case $k=3$ (which turns out to be the case necessary for analysing arithmetic progressions of length 4) states that if $\|f\|_\infty\leq 1$ and $\|f\|_{U^3}\geq c$, then there is an arithmetic progression $P$ of length $n^{\alpha(c)}$ and a quadratic function $q:G\to G$ such that $|\mathbb E_{x\in P}f(x)\exp(2\pi iq(x)/n)|\geq\beta(c)$.

Later, this result was extended in many ways, in particular to a global inverse theorem – that is, one that shows that $f$ correlates with a highly structured function defined on all of $G$, and not just on a longish arithmetic progression. Green and Tao did this for $\mathbb Z/N\mathbb Z$ in the case $k=3$, also obtaining a similar result when $G=\mathbb F_p^n$ for fixed $p\geq 3$, then Bergelson, Tao and Ziegler proved the $\mathbb F_p^n$ case for all $k$ when $p>k$, and finally Green, Tao and Ziegler proved the statement for $\mathbb Z/N\mathbb Z$ for all $k$, a result that has important applications. The case $k=3$ for $\mathbb F_2^n$ was proved independently by Green and Tao and by Samorodnitsky, while the general low-characteristic case (that is, the $U_k$ norm for $\mathbb F_p^n$ when $k\leq p$) was established by Tao and Ziegler. On the quantitative side, results with explicit bounds have been obtained by Manners for $\mathbb Z/N\mathbb Z$ and by Gowers and Milićević for $\mathbb F_p^n$ in the high-characteristic case. The $U^3$ bounds mentioned above were quantitative: in the low-characteristic case there is also a quantiative result due to Tidor for the $U^4$ norm.

Before this paper, the state of knowledge about the $U^3$ norm was that a quantitative inverse theorem was known for $\mathbb F_2^n$, as already mentioned, and for all finite Abelian groups of odd order – the latter statement due to Green and Tao. One might have thought, given this, that it should be relatively straightforward to obtain a result for all finite Abelian groups, but it turns out that 2-torsion introduces genuine subtleties, related to the fact that polarization identities connecting quadratic forms to symmetric bilinear forms require one to divide by 2, that are not straightforward to deal with. While these difficulties are not completely new, and indeed have to be faced in the low-characteristic results mentioned above, using a notion of “generalized polynomial”, it is not clear how to adapt the ideas from the $\mathbb F_p^n$ setting to that of general finite Abelian groups. Indeed, it was not even obvious how to formulate the statement of the inverse theorem in that context.

The authors give two proofs. One follows the basic contours of the Green-Tao proof for groups of odd order, but at a certain point it adds a crucial extra step that is largely algebraic – passing from an algebraic structure that appears in the Green-Tao argument and obtaining from it in a non-obvious way a different algebraic structure that is more useful. The other proof is ergodic theoretic. It introduces a novel and surprising correspondence principle with an ergodic system $(X,H)$ for which the group $H$ is countably generated, rather than being a more usual finitely generated group. The paper also contains conjectures that point the way forward to a possible generalization of their results to the case of general $U^k$ norms.

Taken together, the results of the paper deal with a long-standing deficiency in the literature in a novel and interesting way.