Polynomial extension of Van der Waerden's Theorem near zero
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916892836364288 |
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| author | Ghadimi, Ghadir Tootkaboni, Mohammad Akbari |
| author_facet | Ghadimi, Ghadir Tootkaboni, Mohammad Akbari |
| contents | 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 $p_1,\ldots,p_m \in \mathbb{Z}[x]$ are polynomials such that $p_i(0) = 0$ and there exists $δ> 0$ such that $p_i(x) > 0$ for every $x \in (0,δ)$ and for every $i=1,\ldots , m$. Then for any finite partition $\mathcal{C}$ of \( S\cap(0,1) \) and every sequence $f:\mathbb{N}\to S\cap(0,1)$ satisfying $\sum_{n=1}^\infty f(n)<\infty$, there exist a cell $C \in \mathcal{C}$, an element $a \in S$, and $F \in P_f(\mathbb{N})$ such that
\[
\{ a + p_i(\sum_{t \in F} f(t)) : i = 1,2,\ldots,m \} \subseteq C.
\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08675 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial extension of Van der Waerden's Theorem near zero Ghadimi, Ghadir Tootkaboni, Mohammad Akbari Combinatorics 05D10, 22A15, 54D35 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 $p_1,\ldots,p_m \in \mathbb{Z}[x]$ are polynomials such that $p_i(0) = 0$ and there exists $δ> 0$ such that $p_i(x) > 0$ for every $x \in (0,δ)$ and for every $i=1,\ldots , m$. Then for any finite partition $\mathcal{C}$ of \( S\cap(0,1) \) and every sequence $f:\mathbb{N}\to S\cap(0,1)$ satisfying $\sum_{n=1}^\infty f(n)<\infty$, there exist a cell $C \in \mathcal{C}$, an element $a \in S$, and $F \in P_f(\mathbb{N})$ such that \[ \{ a + p_i(\sum_{t \in F} f(t)) : i = 1,2,\ldots,m \} \subseteq C. \] |
| title | Polynomial extension of Van der Waerden's Theorem near zero |
| topic | Combinatorics 05D10, 22A15, 54D35 |
| url | https://arxiv.org/abs/2508.08675 |