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arXiv · 2409.03447

Some negative answers to the Bergelson-Hindman's question

Qinqi Wu
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Abstract

Let $p_1,\dots,p_d$ be integral polynomials vanishing at $0$. It was asked by Bergelson and Hindman whenever $A$ is large, whether the set $\{(m,n)\in \mathbb{N}^2:m+p_1(n),m+p_2(n),\dots,m+p_d(n)\in A\}$ be large in the same sense. In this paper, we give negative answers to this question when ``large'' being the notions of ``central*'', ``IP*'', ``IP$_n$*'', ``IP$_{<ω}$*'' and ``$Δ$*''.

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Qinqi Wu. 2025-07-15. Some negative answers to the Bergelson-Hindman's question. https://arxiv.org/abs/2409.03447

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