Asgar Jamneshan and myself have just uploaded to the arXiv our preprint “The inverse theorem for the ${U^3}$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches“. This paper, which is a companion to another recent paper of ourselves and Or Shalom, studies the inverse theory for the third Gowers uniformity norm

$\displaystyle \| f \|_{U^3(G)}^8 = {\bf E}_{h_1,h_2,h_3,x \in G} \Delta_{h_1} \Delta_{h_2} \Delta_{h_3} f(x)$

on an arbitrary finite abelian group ${G}$, where ${\Delta_h f(x) := f(x+h) \overline{f(x)}}$ is the multiplicative derivative. Our main result is as follows:

Theorem 1 (Inverse theorem for ${U^3(G)}$) Let ${G}$ be a finite abelian group, and let ${f: G \rightarrow {\bf C}}$ be a ${1}$-bounded function with ${\|f\|_{U^3(G)} \geq \eta}$ for some ${0 < \eta \leq 1/2}$. Then:
• (i) (Correlation with locally quadratic phase) There exists a regular Bohr set ${B(S,\rho) \subset G}$ with ${|S| \ll \eta^{-O(1)}}$ and ${\exp(-\eta^{-O(1)}) \ll \rho \leq 1/2}$, a locally quadratic function ${\phi: B(S,\rho) \rightarrow {\bf R}/{\bf Z}}$, and a function ${\xi: G \rightarrow \hat G}$ such that

$\displaystyle {\bf E}_{x \in G} |{\bf E}_{h \in B(S,\rho)} f(x+h) e(-\phi(h)-\xi(x) \cdot h)| \gg \eta^{O(1)}.$

• (ii) (Correlation with nilsequence) There exists an explicit degree two filtered nilmanifold ${H/\Lambda}$ of dimension ${O(\eta^{-O(1)})}$, a polynomial map ${g: G \rightarrow H/\Lambda}$, and a Lipschitz function ${F: H/\Lambda \rightarrow {\bf C}}$ of constant ${O(\exp(\eta^{-O(1)}))}$ such that

$\displaystyle |{\bf E}_{x \in G} f(x) \overline{F}(g(x))| \gg \exp(-\eta^{-O(1)}).$

Such a theorem was proven by Ben Green and myself in the case when ${|G|}$ was odd, and by Samorodnitsky in the ${2}$-torsion case ${G = {\bf F}_2^n}$. In all cases one uses the “higher order Fourier analysis” techniques introduced by Gowers. After some now-standard manipulations (using for instance what is now known as the Balog-Szemerédi-Gowers lemma), one arrives (for arbitrary ${G}$) at an estimate that is roughly of the form

$\displaystyle |{\bf E}_{x \in G} {\bf E}_{h,k \in B(S,\rho)} f(x+h+k) b(x,k) b(x,h) e(-B(h,k))| \gg \eta^{O(1)}$

where ${b}$ denotes various ${1}$-bounded functions whose exact values are not too important, and ${B: B(S,\rho) \times B(S,\rho) \rightarrow {\bf R}/{\bf Z}}$ is a symmetric locally bilinear form. The idea is then to “integrate” this form by expressing it in the form

$\displaystyle B(h,k) = \phi(h+k) - \phi(h) - \phi(k) \ \ \ \ \ (1)$

for some locally quadratic ${\phi: B(S,\rho) \rightarrow {\bf C}}$; this then allows us to write the above correlation as

$\displaystyle |{\bf E}_{x \in G} {\bf E}_{h,k \in B(S,\rho)} f(x+h+k) e(-\phi(h+k)) b(x,k) b(x,h)| \gg \eta^{O(1)}$

(after adjusting the ${b}$ functions suitably), and one can now conclude part (i) of the above theorem using some linear Fourier analysis. Part (ii) follows by encoding locally quadratic phase functions as nilsequences; for this we adapt an algebraic construction of Manners.

So the key step is to obtain a representation of the form (1), possibly after shrinking the Bohr set ${B(S,\rho)}$ a little if needed. This has been done in the literature in two ways:

• When ${|G|}$ is odd, one has the ability to divide by ${2}$, and on the set ${2 \cdot B(S,\frac{\rho}{10}) = \{ 2x: x \in B(S,\frac{\rho}{10})\}}$ one can establish (1) with ${\phi(h) := B(\frac{1}{2} h, h)}$. (This is similar to how in single variable calculus the function ${x \mapsto \frac{1}{2} x^2}$ is a function whose second derivative is equal to ${1}$.)
• When ${G = {\bf F}_2^n}$, then after a change of basis one can take the Bohr set ${B(S,\rho)}$ to be ${{\bf F}_2^m}$ for some ${m}$, and the bilinear form can be written in coordinates as

$\displaystyle B(h,k) = \sum_{1 \leq i,j \leq m} a_{ij} h_i k_j / 2 \hbox{ mod } 1$

for some ${a_{ij} \in {\bf F}_2}$ with ${a_{ij}=a_{ji}}$. The diagonal terms ${a_{ii}}$ cause a problem, but by subtracting off the rank one form ${(\sum_{i=1}^m a_{ii} h_i) ((\sum_{i=1}^m a_{ii} k_i) / 2}$ one can write

$\displaystyle B(h,k) = \sum_{1 \leq i,j \leq m} b_{ij} h_i k_j / 2 \hbox{ mod } 1$

on the orthogonal complement of ${(a_{11},\dots,a_{mm})}$ for some coefficients ${b_{ij}=b_{ji}}$ which now vanish on the diagonal: ${b_{ii}=0}$. One can now obtain (1) on this complement by taking

$\displaystyle \phi(h) := \sum_{1 \leq i < j \leq m} b_{ij} h_i h_k / 2 \hbox{ mod } 1.$

In our paper we can now treat the case of arbitrary finite abelian groups ${G}$, by means of the following two new ingredients:

• (i) Using some geometry of numbers, we can lift the group ${G}$ to a larger (possibly infinite, but still finitely generated) abelian group ${G_S}$ with a projection map ${\pi: G_S \rightarrow G}$, and find a globally bilinear map ${\tilde B: G_S \times G_S \rightarrow {\bf R}/{\bf Z}}$ on the latter group, such that one has a representation

$\displaystyle B(\pi(x), \pi(y)) = \tilde B(x,y) \ \ \ \ \ (2)$

of the locally bilinear form ${B}$ by the globally bilinear form ${\tilde B}$ when ${x,y}$ are close enough to the origin.
• (ii) Using an explicit construction, one can show that every globally bilinear map ${\tilde B: G_S \times G_S \rightarrow {\bf R}/{\bf Z}}$ has a representation of the form (1) for some globally quadratic function ${\tilde \phi: G_S \rightarrow {\bf R}/{\bf Z}}$.

To illustrate (i), consider the Bohr set ${B(S,1/10) = \{ x \in {\bf Z}/N{\bf Z}: \|x/N\|_{{\bf R}/{\bf Z}} < 1/10\}}$ in ${G = {\bf Z}/N{\bf Z}}$ (where ${\|\|_{{\bf R}/{\bf Z}}}$ denotes the distance to the nearest integer), and consider a locally bilinear form ${B: B(S,1/10) \times B(S,1/10) \rightarrow {\bf R}/{\bf Z}}$ of the form ${B(x,y) = \alpha x y \hbox{ mod } 1}$ for some real number ${\alpha}$ and all integers ${x,y \in (-N/10,N/10)}$ (which we identify with elements of ${G}$. For generic ${\alpha}$, this form cannot be extended to a globally bilinear form on ${G}$; however if one lifts ${G}$ to the finitely generated abelian group

$\displaystyle G_S := \{ (x,\theta) \in {\bf Z}/N{\bf Z} \times {\bf R}: \theta = x/N \hbox{ mod } 1 \}$

(with projection map ${\pi: (x,\theta) \mapsto x}$) and introduces the globally bilinear form ${\tilde B: G_S \times G_S \rightarrow {\bf R}/{\bf Z}}$ by the formula

$\displaystyle \tilde B((x,\theta),(y,\sigma)) = N^2 \alpha \theta \sigma \hbox{ mod } 1$

then one has (2) when ${\theta,\sigma}$ lie in the interval ${(-1/10,1/10)}$. A similar construction works for higher rank Bohr sets.

To illustrate (ii), the key case turns out to be when ${G_S}$ is a cyclic group ${{\bf Z}/N{\bf Z}}$, in which case ${\tilde B}$ will take the form

$\displaystyle \tilde B(x,y) = \frac{axy}{N} \hbox{ mod } 1$

for some integer ${a}$. One can then check by direct construction that (1) will be obeyed with

$\displaystyle \tilde \phi(x) = \frac{a \binom{x}{2}}{N} - \frac{a x \binom{N}{2}}{N^2} \hbox{ mod } 1$

regardless of whether ${N}$ is even or odd. A variant of this construction also works for ${{\bf Z}}$, and the general case follows from a short calculation verifying that the claim (ii) for any two groups ${G_S, G'_S}$ implies the corresponding claim (ii) for the product ${G_S \times G'_S}$.

This concludes the Fourier-analytic proof of Theorem 1. In this paper we also give an ergodic theory proof of (a qualitative version of) Theorem 1(ii), using a correspondence principle argument adapted from this previous paper of Ziegler, and myself. Basically, the idea is to randomly generate a dynamical system on the group ${G}$, by selecting an infinite number of random shifts ${g_1, g_2, \dots \in G}$, which induces an action of the infinitely generated free abelian group ${{\bf Z}^\omega = \bigcup_{n=1}^\infty {\bf Z}^n}$ on ${G}$ by the formula

$\displaystyle T^h x := x + \sum_{i=1}^\infty h_i g_i.$

Much as the law of large numbers ensures the almost sure convergence of Monte Carlo integration, one can show that this action is almost surely ergodic (after passing to a suitable Furstenberg-type limit ${X}$ where the size of ${G}$ goes to infinity), and that the dynamical Host-Kra-Gowers seminorms of that system coincide with the combinatorial Gowers norms of the original functions. One is then well placed to apply an inverse theorem for the third Host-Kra-Gowers seminorm ${U^3(X)}$ for ${{\bf Z}^\omega}$-actions, which was accomplished in the companion paper to this one. After doing so, one almost gets the desired conclusion of Theorem 1(ii), except that after undoing the application of the Furstenberg correspondence principle, the map ${g: G \rightarrow H/\Lambda}$ is merely an almost polynomial rather than a polynomial, which roughly speaking means that instead of certain derivatives of ${g}$ vanishing, they instead are merely very small outside of a small exceptional set. To conclude we need to invoke a “stability of polynomials” result, which at this level of generality was first established by Candela and Szegedy (though we also provide an independent proof here in an appendix), which roughly speaking asserts that every approximate polynomial is close in measure to an actual polynomial. (This general strategy is also employed in the Candela-Szegedy paper, though in the absence of the ergodic inverse theorem input that we rely upon here, the conclusion is weaker in that the filtered nilmanifold ${H/\Lambda}$ is replaced with a general space known as a “CFR nilspace”.)

This transference principle approach seems to work well for the higher step cases (for instance, the stability of polynomials result is known in arbitrary degree); the main difficulty is to establish a suitable higher step inverse theorem in the ergodic theory setting, which we hope to do in future research.