An Improved Lower Bound for Arithmetic Regularity

TL;DR

This paper improves the lower bound on the size of subgroups needed in the arithmetic regularity lemma, demonstrating that the bound must grow at least exponentially with 1/epsilon, refining previous results.

Contribution

The authors provide a new example establishing a stronger lower bound, showing that the subgroup index must be at least exponential in 1/epsilon, improving upon prior tower-height bounds.

Findings

Lower bound on subgroup index is exponential in 1/epsilon

Previous bounds were tower of twos of height 1/epsilon^3

New example shows tower of height Omega(1/epsilon) is necessary

Abstract

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemer{\'e}di regularity lemma in graph theory. It shows that for any abelian group $G$ and any bounded function $f:G \to [0,1]$, there exists a subgroup $H \le G$ of bounded index such that, when restricted to most cosets of $H$, the function $f$ is pseudorandom in the sense that all its nontrivial Fourier coefficients are small. Quantitatively, if one wishes to obtain that for $1-\epsilon$ fraction of the cosets, the nontrivial Fourier coefficients are bounded by $\epsilon$, then Green shows that $|G/H|$ is bounded by a tower of twos of height $1/\epsilon^3$. He also gives an example showing that a tower of height $\Omega(\log 1/\epsilon)$ is necessary. Here, we give an improved example, showing that a tower of height $\Omega(1/\epsilon)$ is necessary.