[Paper Review] Quantitative structure of stable sets in finite abelian groups

C. Terry, Jason B. Wolf|arXiv (Cornell University)|May 17, 2018

Limits and Structures in Graph Theory11 references6 citations

TL;DR

This paper establishes a quantitative arithmetic regularity lemma for k-stable subsets of finite abelian groups, generalizing earlier results from vector spaces over finite fields. Using techniques from arithmetic combinatorics—particularly a refined almost-periodicity argument inspired by Sisask—the authors show that such sets are uniformly distributed relative to a Bohr set of bounded rank and width, yielding efficient regularity with explicit quantitative bounds.

ABSTRACT

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was recently obtained by the first author in joint work with Conant and Pillay, using model-theoretic techniques. In contrast, the approach in the present paper is highly quantitative and relies on several key ingredients from arithmetic combinatorics.

Motivation & Objective

• To provide a quantitative version of the qualitative arithmetic regularity lemma for stable sets in finite abelian groups, extending prior results from vector spaces over finite fields.
• To replace the model-theoretic methods of earlier work with a quantitative, analytic approach rooted in arithmetic combinatorics.
• To establish explicit bounds on the size and structure of Bohr sets and subgroups that regularize stable sets in finite abelian groups.
• To demonstrate that stable sets in cyclic groups of prime order are essentially trivial, justifying their difficulty in arithmetic combinatorics.
• To derive a structural corollary showing that stable sets are approximately unions of cosets of a low-codimension subgroup.

Proposed method

• Use a refined version of the Croot-Sisask almost-periodicity technique to show that stable sets are dense on some translate of a Bohr set with controlled parameters.
• Apply a Bourgain-style 'Bourgainisation' process to transfer arguments from vector spaces over finite fields to general finite abelian groups.
• Construct a model-theoretic tree of instability to witness failure of regularity, using approximate versions of the argument from [19] and [12].
• Establish that Bohr sets of bounded rank and width at least ǫ^{O_k(1)} regularize k-stable sets, with uniformity holding for all cosets.
• Bootstrap the Bohr set regularity result to obtain a subgroup regularity version by reapplying the tree argument, yielding a subgroup of index at most exp(µ^{-O_k(1)}).
• Use the inversion formula and Parseval's identity in Fourier analysis on finite abelian groups to control Fourier coefficients and support the regularity argument.

Experimental results

Research questions

• RQ1 Can the qualitative arithmetic regularity lemma for stable sets in finite abelian groups be made quantitative with explicit bounds?
• RQ2 What is the optimal quantitative relationship between stability, Bohr set parameters, and regularity in finite abelian groups?
• RQ3 To what extent do stable sets in cyclic groups of prime order resemble unions of cosets of subgroups?
• RQ4 Can the model-theoretic instability tree argument be adapted to yield quantitative bounds in the context of arithmetic regularity?
• RQ5 Is it possible to upgrade Bohr set regularity to subgroup regularity with efficient quantitative control?

Key findings

• For any k ≥ 2 and 0 < ǫ < 1, there exists N₀ = N₀(k, ǫ) such that if |G| ≥ N₀ and A ⊆ G is k-stable, then there exists a Bohr set B of width at least ǫ^{O_k(1)} and rank at most ǫ^{-O_k(1)} such that for all g ∈ G, either |(A − g) ∩ B| ≤ ǫ|B| or |B ackslash (A − g)| ≤ ǫ|B|.
• The regularizing Bohr set has rank at most ǫ^{-O_k(1)} and width at least ǫ^{O_k(1)}, with the index of the Bohr set in G bounded by exp(ǫ^{-O_k(1)}).
• A structural corollary shows that any k-stable set A ⊆ G is within symmetric difference of density ǫ of a union of cosets of a subgroup H ≤ G of index at most exp(ǫ^{-O_k(1)}).
• The paper proves that there are essentially no non-trivial stable sets in cyclic groups of prime order, providing a model-theoretic justification for their difficulty in arithmetic combinatorics.
• The main result is strengthened to a subgroup regularity version: for any µ ∈ (0,1), there exists a subgroup H ≤ G of index at most exp(µ^{-O_k(1)}) such that for all g ∈ G, either |(A − g) ∩ H| ≤ µ|H| or |H ackslash (A − g)| ≤ µ|H|.
• The proof relies on a re-implementation of the instability tree argument with quantitative control, using a refined almost-periodicity result from Sisask [17] to avoid reliance on model-theoretic localization.