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A note on lower bounds for arithmetic regularity partitions

V. Gladkova

TL;DR

This work establishes fundamental lower bounds for arithmetic regularity partitions in finite-field vector spaces. It presents a generalised lower-bound construction that yields wowzer-type codimension growth for the strong arithmetic regularity lemma, aligning arithmetic results with the graph-theoretic wowzer bounds. It further shows that, in the quadratic (higher-order) setting, the linear layer must exhibit tower-type growth, illustrating inherent complexity in both the linear and higher-order layers of arithmetic regularity partitions. The results clarify that, in general, partition sizes in arithmetic regularity lemmas inherently match the strongest growth rates observed in the graph setting, with explicit mechanisms via energy increments and structured subspace hierarchies driving the bounds.

Abstract

This paper establishes lower bounds for two kinds of arithmetic regularity partitions, building on constructions of Green [arXiv:math/0310476v2] and Hosseini, Lovett, Moshkovitz, and Shapira [arXiv:1405.4409]. The first kind occurs in the so-called strong arithmetic regularity lemma due to Bhattcharrya, Fischer, and Lovett [arXiv:1201.0330v2, Theorem 4.9], which is an arithmetic analogue of the strong regularity lemma for graphs developed by Alon, Fischer, Krivelevich, and Szegedy. Conlon and Fox [arXiv:1107.4829], as well as Kalyanasundaram and Shapira [arXiv:1107.4896v2], demonstrated that there are graphs for which any strong regularity partition must have size at least a wowzer-type function in the pseudorandomness parameter, and the primary aim of this paper is to match this bound in the setting of vector spaces over finite fields. The second kind of arithmetic regularity partition originates from higher-order arithmetic regularity lemmas. The upper bounds on the size of these partitions are known to be of tower-type growth. Previous work [arXiv:math/0310476v2, arXiv:1405.4409] demonstrated that this is unavoidable for the `linear' arithmetic regularity lemma of Green [arXiv:math/0310476v2], and the second contribution of this paper confirms that this continues to be necessary in the higher-order setting.

A note on lower bounds for arithmetic regularity partitions

TL;DR

This work establishes fundamental lower bounds for arithmetic regularity partitions in finite-field vector spaces. It presents a generalised lower-bound construction that yields wowzer-type codimension growth for the strong arithmetic regularity lemma, aligning arithmetic results with the graph-theoretic wowzer bounds. It further shows that, in the quadratic (higher-order) setting, the linear layer must exhibit tower-type growth, illustrating inherent complexity in both the linear and higher-order layers of arithmetic regularity partitions. The results clarify that, in general, partition sizes in arithmetic regularity lemmas inherently match the strongest growth rates observed in the graph setting, with explicit mechanisms via energy increments and structured subspace hierarchies driving the bounds.

Abstract

This paper establishes lower bounds for two kinds of arithmetic regularity partitions, building on constructions of Green [arXiv:math/0310476v2] and Hosseini, Lovett, Moshkovitz, and Shapira [arXiv:1405.4409]. The first kind occurs in the so-called strong arithmetic regularity lemma due to Bhattcharrya, Fischer, and Lovett [arXiv:1201.0330v2, Theorem 4.9], which is an arithmetic analogue of the strong regularity lemma for graphs developed by Alon, Fischer, Krivelevich, and Szegedy. Conlon and Fox [arXiv:1107.4829], as well as Kalyanasundaram and Shapira [arXiv:1107.4896v2], demonstrated that there are graphs for which any strong regularity partition must have size at least a wowzer-type function in the pseudorandomness parameter, and the primary aim of this paper is to match this bound in the setting of vector spaces over finite fields. The second kind of arithmetic regularity partition originates from higher-order arithmetic regularity lemmas. The upper bounds on the size of these partitions are known to be of tower-type growth. Previous work [arXiv:math/0310476v2, arXiv:1405.4409] demonstrated that this is unavoidable for the `linear' arithmetic regularity lemma of Green [arXiv:math/0310476v2], and the second contribution of this paper confirms that this continues to be necessary in the higher-order setting.
Paper Structure (6 sections, 19 theorems, 45 equations, 1 figure)

This paper contains 6 sections, 19 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A generalised lower-bound construction
  3. Proof of Theorem \ref{['thm:sarl-intro']}
  4. Size of the linear layer in higher-order regularity partitions
  5. Proof of Lemma \ref{['lemma:higher-arl-atom-version']}
  6. An upper bound for quadratic regularity partitions

Key Result

Theorem 1.3

Fix $\epsilon > 0$. There exists $C=C_{arl}(\epsilon)$ with the following property. For any function $f:\mathbb{F}_{p}^{n} \rightarrow [0,1]$ and subspace $H_0 \leqslant \mathbb{F}_{p}^{n}$, there is a subspace $H \leqslant H_0$ of codimension at most $C$ in $H_0$ such that $\mathcal{P}(H)$ is $\eps

Figures (1)

  • Figure 1: A possible choice of $A_3$ for the setting of $p=2$. In each coset of $H_2$, a codimension $1$ subspace is picked in such a way that $A_3$ is a union of cosets of $H_3$. (Here $H_3$ is depicted with a smaller codimension than defined, for greater visual clarity.)

Theorems & Definitions (48)

  • Definition 1.1: Fourier uniformity
  • Definition 1.2: Partition regularity
  • Theorem 1.3: Arithmetic regularity lemma tower-type
  • Definition 1.4: Energy
  • Theorem 1.5: Strong arithmetic regularity lemma induced-1
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • ...and 38 more