[Paper Review] The abelian arithmetic regularity lemma

Sean Eberhard|arXiv (Cornell University)|Jun 29, 2016

Analytic Number Theory Research3 references3 citations

TL;DR

This paper presents a self-contained proof of the abelian arithmetic regularity lemma in the $s=1$ case, establishing that any bounded function on $[N]$ decomposes into a structured part (of controlled complexity), a uniform part (Gowers-uniform in $U^2$), and a small error. The key contribution is a simplified, elementary treatment using only the inverse theorem for the $U^2$ norm, avoiding higher-order Fourier analysis.

ABSTRACT

We give a brief exposition and proof of the arithmetic regularity lemma of Green and Tao in the abelian ($U^2$) case, over $\{1,\dots,N\}$. This may be useful to those who need just the $U^2$ case of the lemma, as the general case is significantly more involved. It may also be useful as an introduction to the general case. No originality is claimed.

Motivation & Objective

• To provide a simplified, accessible proof of the abelian arithmetic regularity lemma in the $s=1$ case, focusing on the $U^2$ norm.
• To offer a standalone exposition that avoids the full machinery of higher-order Fourier analysis, making it suitable for readers new to the topic.
• To demonstrate that the $U^2$ inverse theorem suffices for the abelian case, enabling a more elementary and transparent argument.
• To formalize the decomposition of functions into structured, uniform, and small components using $1$-complexity and $1$-measurability with growth functions.

Proposed method

• Uses the $U^2$ norm defined via averages over parallelepipeds: $\|f\|_{U^2([N])} = \left( \mathbf{E}_{a,h_1,h_2} f(a)\overline{f(a+h_1)}\overline{f(a+h_2)}f(a+h_1+h_2) \right)^{1/4}$.
• Applies the inverse theorem for $U^2$, showing that large $U^2$ norm implies correlation with a linear phase function $e(-\theta n)$.
• Introduces the concept of $1$-complexity: $f(n) = F(\theta n)$ for $F:\mathbf{T}^d \to \mathbf{R}$ with bounded Lipschitz norm and dimension $d$.
• Defines $1$-measurability with growth function $\mathcal{F}$, ensuring approximation in $L^2$ by functions of controlled $1$-complexity.
• Employs equidistribution theory via $(A,N)$-irrationality of $\theta$ to control averages of structured functions over arithmetic progressions.
• Uses truncated Fourier expansions and geometric series estimates to bound exponential sums, proving equidistribution of $\theta n$ modulo $\mathbf{T}^d$.

Experimental results

Research questions

• RQ1 How can the abelian arithmetic regularity lemma be proven using only the $U^2$ inverse theorem, without higher-order theory?
• RQ2 What is the minimal structural complexity required to approximate a function in $L^2$ up to a given error, given a growth function $\mathcal{F}$?
• RQ3 How does $U^2$-uniformity relate to equidistribution of linear phases over finite abelian groups?
• RQ4 Can the decomposition of a function into structured, uniform, and small parts be quantified using only elementary Fourier analysis?

Key findings

• If $\|f\|_{U^2} \geq \delta$, then there exists $\theta \in \mathbf{T}$ such that $\left|\mathbf{E}_{n\in[N]} f(n)e(-\theta n)\right| \gg_\delta 1$.
• For any $f:[N] \to [-1,1]$ with $\|f\|_{U^2} \geq \delta$, there exists a $1$-measurable set $E \subset [N]$ with growth $\ll_\delta 1$ such that $\left|\mathbf{E}_{n\in[N]} f(n)1_E(n)\right| \gg_\delta 1$.
• If $\theta \in \mathbf{T}^d$ is $(A,N)$-irrational and $F:\mathbf{T}^d \to \mathbf{C}$ has Lipschitz norm $\leq M$, then $\left|\mathbf{E}_{n\in P} F(\theta n) - \int F \, d\mu\right| \leq \delta$ for long progressions $P$ of density $\eta$, provided $A$ is large enough.
• For $F:[0,1] \times \mathbf{Z}/q\mathbf{Z} \times \mathbf{T}^d \to \mathbf{C}$ with Lipschitz norm $\leq M$, the average $\mathbf{E}_{n\leq N} F(n/N, n\bmod q, \theta n)$ converges to the integral $\int F \, d\mu$ as $A,N \to \infty$, uniformly in $M,q,d,\delta$.