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Supersimplicity and arithmetic progressions

Amador Martin-Pizarro, Daniel Palacín

TL;DR

The paper develops a bridge between van der Corput-type additive patterns and model-theoretic simplicity, proving that in a definable setup within a supersimple rank-$1$ theory, most points of an infinite definable set $X$ in an abelian group $G$ lie on infinitely many 3-term arithmetic progressions in $X$. Using this framework, the authors obtain uniform finiteness bounds for bad initial points and density-dichotomy results for constructible sets in expansions of the integers by primes (assuming Dickson's conjecture) or by square-free integers (unconditional), and extend these insights to finite and pseudo-finite fields via Lang-Weil-type estimates. They further adapt these methods to corollaries about skew-corners and Sarkozy-type results, showing that model-theoretic simplicity provides concrete, quantitative control over additive configurations. Overall, the work connects additive combinatorics with supersimple model theory to yield finitary and asymptotic results across integers and finite field settings, underpinned by constructible/definable set analysis and quantifier-elimination-type techniques.

Abstract

The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for definable groups in simple theories. In the last sections of this article, we apply our model-theoretic results to bound the number of initial points starting few arithmetic progression of length $3$ in the structure of the additive group of integers with a predicate for the prime integers, assuming Dickson's conjecture, or with a predicate for the square-free integers, as well as for asymptotic limits of finite fields. Our techniques yield similar results for the elements appearing as distances in skew-corners and for Sárközy's theorem on the distance of distinct elements being perfect squares.

Supersimplicity and arithmetic progressions

TL;DR

The paper develops a bridge between van der Corput-type additive patterns and model-theoretic simplicity, proving that in a definable setup within a supersimple rank- theory, most points of an infinite definable set in an abelian group lie on infinitely many 3-term arithmetic progressions in . Using this framework, the authors obtain uniform finiteness bounds for bad initial points and density-dichotomy results for constructible sets in expansions of the integers by primes (assuming Dickson's conjecture) or by square-free integers (unconditional), and extend these insights to finite and pseudo-finite fields via Lang-Weil-type estimates. They further adapt these methods to corollaries about skew-corners and Sarkozy-type results, showing that model-theoretic simplicity provides concrete, quantitative control over additive configurations. Overall, the work connects additive combinatorics with supersimple model theory to yield finitary and asymptotic results across integers and finite field settings, underpinned by constructible/definable set analysis and quantifier-elimination-type techniques.

Abstract

The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length ) with well-known tools and theorems for definable groups in simple theories. In the last sections of this article, we apply our model-theoretic results to bound the number of initial points starting few arithmetic progression of length in the structure of the additive group of integers with a predicate for the prime integers, assuming Dickson's conjecture, or with a predicate for the square-free integers, as well as for asymptotic limits of finite fields. Our techniques yield similar results for the elements appearing as distances in skew-corners and for Sárközy's theorem on the distance of distinct elements being perfect squares.

Paper Structure

This paper contains 3 sections, 14 theorems, 60 equations.

Table of Contents

  1. (Super-)Simplicity
  2. Some additive configurations in expansions of the integers
  3. Some additive configurations in (pseudo-)finite fields

Key Result

Lemma 1.6

Every supersimple theory of rank $1$eliminates the quantifier $\exists^\infty$, that is, for every formula $\varphi(x, y)$ with $x$ a single variable, there exists some $k_\varphi$ in $\mathbb{N}$ such that for every $b$ in $\mathbb{U}$, either $\varphi(\mathbb{U}, b)$ is infinite or it has cardinal

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.4
  • Remark 1.5
  • proof
  • Lemma 1.6
  • proof
  • Example 1.7
  • Remark 1.8
  • Definition 1.9
  • ...and 33 more