Year: 2026
Type: article
Abstract: The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corput'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 progressions 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.
DOI: 10.1112/jlms.70531
Authors: Amador Martin‐Pizarro, Daniel Palacín
Institutions: University of Freiburg, Universidad Complutense de Madrid
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7147014524