Polynomial extension of Van der Waerden's Theorem near zero

2025/08/12 by Ghadir Ghadimi, Ghadimi, Ghadir, M. Akbari Tootkaboni +1
Mathematics · #05D10 #22A15 #54D35 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2508.08675

openalex publication_date 2025/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be a dense subring of the real numbers. In this paper we prove a polynomial version of Van der Waerden's theorem near zero. In fact, we prove that if p₁,…,p_m ∈ ℤ[x] are polynomials such that p_i(0) = 0 and there exists δ> 0 such that p_i(x) > 0 for every x ∈ (0,δ) and for every i=1,… , m. Then for any finite partition C of \( S∩(0,1) \) and every sequence f:ℕ→ S∩(0,1) satisfying ∑_n=1^∞ f(n)<∞, there exist a cell C ∈ C, an element a ∈ S, and F ∈ P_f(ℕ) such that \ a + p_i(∑_{t ∈ F} f(t)) : i = 1,2,…,m \ ⊆ C.

Citations

• A Generalized Central Sets Theorem In Partial Semigroups

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