2025/03/11 by Amador Martín-Pizarro, Martin-Pizarro, Amador, Daniel Palacín +1
Computer Science · Mathematics · #03C45 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2503.08258
openalex publication_date 2025/03/11 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
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.